Optical Imaging Chapter 5 – Light Scattering

Gabriel Popescu

University of Illinois at Urbana-Champaign Beckman Institute

Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu

Principles of Optical Imaging Electrical and Computer Engineering, UIUC ECE 460 – Optical Imaging 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

Chapter 5: Light Scattering 2 ECE 460 – Optical Imaging 5.1 Scattering on Simple Particles

N=1 s i n Ej [ r ]

Ej [ r ]

Direct Problem: given n [ r ] , what is E [ s ] ? Inverse Problem: given E [ s ] , what is n [ r ] ? This problem is more difficult but it provides a diagnostics tool i⋅ k ⋅ n 2π ErEe[ ]= ⋅ = plane move incident; k = (5.1) j 0 λ

Chapter 5: Light Scattering 3 ECE 460 – Optical Imaging 5.1 Scattering on Simple Particles

Particles can be characterized by a complex dielectric constant ε ε ε [r ]= Re[ [ ri ]] + Im[ [ r ]] (5.2) ε []r= n2 [] r (5.3) Recall: Re[n] Refraction Im[n] Absorption d 2 The scattered “far” field (R > ) λ ei⋅ k ⋅ R Er[]= E ⋅ ⋅ fsi [,] (5.4) S 0 R f[ s , i ]= scattering amplitude = defines amplitude, phase, and po larization of the scattered field. Chapter 5: Light Scattering 4 ECE 460 – Optical Imaging 5.1 Scattering on Simple Particles

Differential Cross section 2 S s 2 σ d [,]si= lim[ R ] = fsi [,] R → ∞ (5.5) S i Only defined in far-field

Si and S s are the Pointing vectors (incident and scattered.) 1 2 S= E is,2η is , µ (5.6) η η ==impedance ( =Ω 377 ) ε v a cu u m From (eq. 5.5) differential cross-section:

(5.7) σ σb= d [ − i , i ] = back scattering section

Chapter 5: Light Scattering 5 ECE 460 – Optical Imaging 5.1 Scattering on Simple Particles

Scattering Cross section σσ =[s , i ] d Ω (5.8) s∫ d 4π 2 [σ s ]=m = area = meaning of probability of scatt ering

Phase Function 2 σ [s , i ] f[ s , i ] p[,]4 s i = π π d = 4 (5.9) σ σ ∫ ps[ , i ] d Ω = 4π = normailzation

Chapter 5: Light Scattering 6 ECE 460 – Optical Imaging 5.1 Scattering on Simple Particles

In the presence of absorption (5.10) σ σ σ=a + s = Total Cross Section Area 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.

Chapter 5: Light Scattering 7 ECE 460 – Optical Imaging 5.2 Rayleigh Scattering

If a << λ , the dipole approximation works well 2 Scattering Amplitude ε = n

2 k 3(ε r − 1)  fsi[,]=  ⋅⋅−×× Vssi [()] (5.11) 4π ε r + 2 

It describes the “doughnut” shape p p p = induced dipole moment

isotropic in the (think Lorentz) plane⊥ p no radiation along p

Chapter 5: Light Scattering 8 ECE 460 – Optical Imaging 5.2 Rayleigh Scattering

Question: is the sky polarized? 2 8(ka ) 4 (ε − 1) 2 r (5.12) σ πs =a ⋅ 3ε r + 2 Note: 2π λ σ ∼ k 4 = ⇒ small scatter more ⇒ blue λ 4

6 2 σ s ∼ a= V

Chapter 5: Light Scattering 9 ECE 460 – Optical Imaging 5.3 The Born Approximation

Recall the Helmholtz equation (Chapter 2, eq. 2.50)

∇2Ur ω[,] ω ω + knr 2 2 [,][,]0 Ur = 2 π (5.13) k=; ε = n 2 λ Scalar eq. is good enough for our purpose The Scattering Potential 1 Fr ω[,]ω = knr2 ⋅ ([,]1) 2 − (5.14) 4π => Equation 5.13 can be rewritten: ∇2Ur ω[,] ω π ω ω + kUr 2 [,] =− 4 Fr [,][,] ⋅ Ur (5.15)

Chapter 5: Light Scattering 10 ECE 460 – Optical Imaging 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 ) . (5.12) Let’s find the Green’s function, i.e. impulse response in space domain. Space -Pulse = (3) (x,y,z ) δ [r ] z y The elementary equation becomes: x δ δ δ [xx−′ ]. [ yy − ′ ]. [ zz − ′ ] 2 2 (3) ∇Hr[,] ω δω + kHr [,] =− [] r (5.16)

Take the Fourier Transform w.r.t. r ⇒−⋅qHq2[,] ωω +⋅ kHq 2 [,] =− 1 (5.17)

Chapter 5: Light Scattering 11 ECE 460 – Optical Imaging 5.3 The Born Approximation

(Eq. 5.17 from last slide) ⇒−⋅qHq2[,] ωω +⋅ kHq 2 [,] =− 1 (5.17) 1 ⇒H[ q ,ω ] = (5.18) q2− k 2 H = transfer function => impulse response. hr[,] ωω = FHq {[,]} (5.19) eik⋅ r ⇒h[ r ,ω ] = A ⋅ r

This is the Spherical Wavelet we used in diffraction (section 3.11)

Chapter 5: Light Scattering 12 ECE 460 – Optical Imaging 5.3 The Born Approximation

Scattering and diffraction are 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 Fr [,] ω ω ⋅ Ur [,] : eik⋅ r − r ' Ur()S [,] ω ω ω =∫ Fr [',] ⋅ Ur [',] ⋅ dr3 ′ (5.20) (V ) r− r ' = superposition of wavelets Assuming the plane wave incident U ( i ) = e ik ⋅ s 0 ⋅ , r the total field in the for zone: U (s ) eik⋅ r − r ' ωUr ω [,]ω =+ eik⋅ s0 ⋅ r ∫ Fr [',][',] ⋅ Ur ⋅ dr3 ′ (5.21) (V ) r− r ' Chapter 5: Light Scattering 13 ECE 460 – Optical Imaging 5.3 The Born Approximation

r− r ' P

r ' θ r s O z

Useful Approximation (for zone) 2 2 2 r′ r ′  rr−=+−' rr′ 2 rr ′θ cosθ =+ r 1 2 cos +   r r  Since r ′ << r and 1− 2x 1 − x 0 rr−' rr − ′ cos θ (5.22a)

Equivalently r− r' rsr − ⋅ ' (5.22b)

Chapter 5: Light Scattering 14 ECE 460 – Optical Imaging 5.3 The Born Approximation

Equation 5.22b implies that ik⋅ r − r ' ik⋅ r e e ik⋅ s ⋅ r e (5.23) r− r' r The scattered field U ( S ) in equation 5.21 becomes eik⋅ r Urs()S ω[,] ω ω = Fr [',] ⋅ Ur [',] ⋅ e−ik ⋅ s ⋅ r ' dr 3 ′ (5.24) r ∫ Equation 5.24 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 ik⋅ s ⋅ r '

Chapter 5: Light Scattering 15 ECE 460 – Optical Imaging 5.3 The Born Approximation

With U [ r ', ω ] = e ik ⋅ s ⋅ r ', equation 5.24 becomes the Born Approximation:

()S −⋅ik( s − s ) ⋅ r ' 3 U[ ss , ]= AFre [ '] ⋅ 0 dr ′ (5.25) 0 ∫

Scattering vector: ks q q= ks( − s 0 ) momentum transfer ks Note: p= p 0 elastic scattering 0 Equation 5.25 becomes: U()S[ g ]= AFre∫ [ '] ⋅ − iqr ⋅ ⋅ '3 dr ′ (5.26)

Chapter 5: Light Scattering 16 ECE 460 – Optical Imaging 5.3 The Born Approximation

Fourier Relationship: U(S ) [ g ]= ℑ [ Fr [ ']] 1 Fr[ ']= k2 ⋅ ( nr 2 [ '] − 1) 4π F.T. is nice because it is easy to compute and bi -directional

kS k q 0 n[r′ ]

θ 2 k θ 0 g= 2 k sin 0 2 θ = scattering angle Chapter 5: Light Scattering 17 ECE 460 – Optical Imaging 5.3 The Born Approximation

So measuring I[ θ ] we can get info about n[r] because Fr[ ']= ℑ [ Uq [ ]] (5.27)

Historically the Born approximation was first used in neutron scattering and then in x -rays Recall dispersion relationship n[ω]

1 ω ω0 limn [ω ]= 1 for x rays, most materials are within Born Approx ω→∞ Spectroscopy Lab uses light scattering to detect early cancer!!

Chapter 5: Light Scattering 18 ECE 460 – Optical Imaging 5.4 The Spatial Correlation Function

U()s( q )= AFre∫ ( ')− iq ⋅ r '3 dr ' provides the scattered field U (s )

2 The measured quantity is U()s= I () s 2 I()()s= U s =ℑ[ Fr ( )] ℑ [ Fr ( ')] *

Recall correlation theorem: GG* = ℑ[ g ⊗ g ] Product of Fourier Transformation equals ℑ of correlation. Fr( ')= ℑ [ Ug ( )] becomes: 3 I(s ) = ℑ[ Fr ( ) ⊗ Fr ( )] where Cr()=∫ frfr (')(' − rdr ) ' I(s )() q= ∫ Cre () − iqr ⋅ dr 3 (4.28)

Chapter 5: Light Scattering 19 ECE 460 – Optical Imaging 5.4 The Spatial Correlation Function

I(s )() q= ∫ Cre () − ig ⋅ r dr 3 is another form of the Born approximation It connects measurable quantity I ( s ) ( g ) with the spatial correlation function C ( r ) . I(s )() q= ∫ Cre () − igr ⋅ dr 3 applies to both continuous and discrete media.

n(x) C(x) I(θ )

θ x x θ

Refractive Index Correlation Scattering

Chapter 5: Light Scattering 20 ECE 460 – Optical Imaging 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 Cr()=∫ frfr (')(' − rdr ) ' (5.29) Use some geometry to show: The overlap vol is C(x) 2x of thespherical a “cap” r x 3 2r 2r 3r 1  r  σ d()q= I d () q C(r)=A[1-+   ] (5.30) 4R 16  R 

The differential cross section of such particle is σ d()q= I d () q

Chapter 5: Light Scattering 21 ECE 460 – Optical Imaging 5.5 Single Particle Under Born Approximation Using: I(s )() q= ∫ cre () − iqr ⋅ dr 3

1 2 σ d (q )= A [sin( qa ) − ga cos( qa )] (5.31) q4 v 6 1 σ (q )= A [sin( qa ) − ga cos( qa )] 2 d q4 v 6 is the solution for Rayleigh - Gauss particles (Born approx. for spheres).

Applies for kd( n − 1) 1 Greater applicability than Rayleigh(Rayleigh is a particular case of Rayleigh-Gaus, for a 0)

Chapter 5: Light Scattering 22 ECE 460 – Optical Imaging 5.6 Ensemble of particles

Example: n( x )

x1 x2 xk x

nx()= fx () v [∑ δ ( xx − i )] Convolution (5.32) i

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: (5.33) Fr( )= Fr0 ( ) v ∑ drr ( − i )

Chapter 5: Light Scattering 23 ECE 460 – Optical Imaging 5.6 Ensemble of particles

Particles have refractive index given by f(x), but are distributed in space at positions Consider volume distribution of identical particles: (5.33) Fr( )= Fr0 ( ) v ∑ drr ( − i )

F0 ( r ) = scattering potential a single particle. v convolution The scattered field is (Eq. 5.26)

(s ) (5.34) Ug()= ℑ [ Fr0 () v ∑ drr ( − i )] i =ℑ[Fr0 ()][ ℑ∑ drr ( − i )] i F( g ) S( g )

Chapter 5: Light Scattering 24 ECE 460 – Optical Imaging 5.6 Ensemble of particles

The scattered field is (Eq. 5.26)

(s ) (5.34) Ug()= ℑ [ Fr0 () v ∑ drr ( − i )] i =ℑ[Fr0 ()][ ℑ∑ drr ( − i )] i F( g ) S( g ) F( g ) = Form factor = Describe the scattering from one particle

S( g ) = Structure factor – interference = Defines the arrangement of scattering

Chapter 5: Light Scattering 25 ECE 460 – Optical Imaging 5.6 Ensemble of particles

Similarly a) Yang experiment

;

b) X-ray scattering on crystals - each scattering is Rayleigh are isotropic in angle (atom) - Structure factor provides info about lattice

Chapter 5: Light Scattering 26 ECE 460 – Optical Imaging 5.7 Mie Scattering

Provides exact solution σ s for spherical particles of arbitrary refractive index and size. ∞ 2 2 2 σ π =a2 (2 nab + 1)( + ) (5.35) s2 ∑ n n α n=1 with , = Mie coefficients = complicated functions of α = ka an bn β ω= k a The above equation is an infinite series n Converges slower for larger particles ω = n Computes routines easily available 0

Chapter 5: Light Scattering 27 ECE 460 – Optical Imaging 5.7 Mie Scattering

Scattering Efficiency σ Q = s (5.36) π a2 Scattering from large particles:

y y cos(θ )= 0 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.

Chapter 5: Light Scattering 28 ECE 460 – Optical Imaging 5.8 Multiple Light Scattering

Attenuation through scattering medias is ...... quantified by the scattering mean free path : ...... P . . . P 1 i . . . f (5.37) . . . ls = Nσ s L N = Concentration of particles [m -3] 1 Scattering Coefficient: σµ = = N s ls The power left unscattered upon passing through the

medium: −µs L Pf= Pe i (5.38) = Lambert-Beer

Chapter 5: Light Scattering 29 ECE 460 – Optical Imaging 5.8 Multiple Light Scattering

Transport mean free path :

ls (5.39) lt = g = cos(θ ) 1− g = average of scatter angle

is renormalizing ; takes into account the directional lt ls scattering of large particles. Note: g ∈(0.75;0.95) for tissue! l t ∈(10;20)-important factor ls

Chapter 5: Light Scattering 30 ECE 460 – Optical Imaging 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. z ϕ(,rΩ ,) t = 1 dϕ dV +Ω∇+ ϕ µ ϕ µϕ (,,)rt Ω− [(' Ω→Ω )(,',)] rtdSrt Ω Ω= ' (,,) Ω N(r,t) v dt ∫ s = angular 4π Leakage r Loss due Source flux out of dV to scott V Scattered into (5.40) y direction Ω x Even this simple equation is typically solved numerically. One more approximation analytic solutions.

Chapter 5: Light Scattering 31 ECE 460 – Optical Imaging 5.10 The Diffusion Approximation

Integrate both sides of the equation S (, r Ω ,) t with respect to solid angle Ω 1 dφ +∇Jrt(,) +µφ µ φ (,) rt = (,) rt + Srt (,) (5.41) v dt s ϕ φ = photon flux = ∫ (r , R , t ) d Ω J= current = ∫ Ωϕ ( rRtd , , ) Ω 4π 4π

Chapter 5: Light Scattering 32 ECE 460 – Optical Imaging 5.10 The Diffusion Approximation

Diffusion Approximation : Assume the angular flux is only linearly anisotropic. 1 3 φ ϕ (,,)rtΩ (,) rt + Jrt (,) ⋅Ω (5.42) π4π 4 cos(θ ) 1 ∂J << vµ If J ∂ t t = slow variation of J, an equation that relates J and φ can be derived: 1 Jrt(,)=− φ ∇φ (,) rt =− Dr ()(,) ∇ rt (5.43) 3µt l 1 s = average cosine. µt=; lt = ; g =Ω⋅Ω ' lt 1− g D = Photon diffusion Coefficient Eq 5.43 is Flick’s Law

Chapter 5: Light Scattering 33 ECE 460 – Optical Imaging 5.10 The Diffusion Approximation

The diffusion equation can be derived: 1 dφ −∇∇[()Dr µ φφ (,)] rt + ()(,) r rt = Srt (,) v dt a (5.44) Equation in only one variable: φ =vr( r , t ) = photon flux v = photon velocity n = photon density [m−3 ] µa = absorption coefficient For homogeneous medium and no absorption: dφ( r , t ) −vD ∇2φ(,) r t = vS (,) r t (5.45) dt Time resolved equation

Chapter 5: Light Scattering 34 ECE 460 – Optical Imaging 5.10 The Diffusion Approximation

Diffusion equation often used for light-tissue introduction

Applicability: L>> l s ⇔ many scattering events

µµ a<< s → absorption not too high φ = measurable quantity Diffusion model gives insight into the medium/tissue

Measurable parameters: µ µs , a , g , etc

Chapter 5: Light Scattering 35 ECE 460 – Optical Imaging 5.11 Solutions of the Diffusion Equation

Slab of thickness d: Diffusion medium r PR r ' PT

Power reflected : 5 r 2 − − 2 −µa s 4Ds PrsdR (,,)= Ase ⋅ e fdzR (,) e d 5 r 2 (5.46) − − 2 −µa s 4Ds PrsdT (,,)= Ase ⋅ e fdzT (,) e

With: s = path-length = vt

ze = extrapolated length boundary condition fR, f T reflection and transmission functions Laser pulse investigation has been used for tissue Stationary investigation also used spatially resolved

Chapter 5: Light Scattering 36 ECE 460 – Optical Imaging 5.12 Diffusion of Light in Tissue

Tissue is a highly scattering medium; direction & polarization are randomized.

Typical values: ls =100 µm,g 0.9 Measurable quantities:

µa = absorption factor g = anysotropy factor µ s = scattering coefficient Reflection and geometry is suitable for in vivo diagnostics a) Time resolved:

P( s )= path-legth distribution

TISSUE

Chapter 5: Light Scattering 37 ECE 460 – Optical Imaging 5.12 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.

Chapter 5: Light Scattering 38