Modeling Focused Beam Propagation in scattering media. Janaka Ranasinghesagara

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1 Modeling Focused Beam Propagation in scattering media Janaka Ranasinghesagara

2 Teaching Objectives Understand the need of focused beam computational models. Understand the concepts, equations and principles applied in focus beam propagation. Learn plane wave propagation in scattering media. Learn Huygens-Fresnel principle. Learn how fundamental concepts are applied to develop a focused beam propagation model.

3 Need of Focused Beam Propagation Models Non scattering medium (free space) Scattering medium? Analytical solution Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) 1959

$(layer thickness, scatterer size, refractive index and number$ density) is given, can we predict reflectance and transmission?

4 Need of Focused Beam Propagation Models If all the information (layer thickness, scatterer size, refractive index and number density) is given, can we predict reflectance and transmission? Dermis pidermis??? Pencil beam Collimated beam (Planer beam) Focused beam

5 Need of Focused Beam Propagation Models If all the information (layer thickness, scatterer size, refractive index and number density) is given, can we predict reflectance and transmission? Dermis pidermis? Pencil beam Collimated beam (Planer beam) Focused beam

6 Need of Focused Beam Propagation Models Images by Mihaela Balu Develop/design methods to mitigate scattering Design better illumination/detection schemes

7 xisting Focused Beam Models Monte Carlo Simulation Propagates photons towards the focal point Ignores the wave nature of light Provides mean behavior Requires large number of photons Song et al, Appl. Optics. 38(13) (1999) Blanca and Saloma Appl. Optics. 37(34) (1998) Dunn et al, Appl. Optics. 39(7) (2000) Hayakawa et al, Biomed. Opt. xpress. 2(2) (2011) Cai et al, Prog. in lectromagnetics Res. 142 (2013) Cai et al, JBO, 19(1) (2014)

xisting Focused Beam Models Monte Carlo Simulation Finite Difference Time Domain (FDTD) Solution for Maxwell s equations Propagates

Optics. 38(13) (1999) Blanca and Saloma Appl. Optics. 37(34) (1998) Dunn et al, Appl. Optics. 39(7) (2000) Hayakawa et al, Biomed. Opt. xpress.

142 (2013) Cai et al, JBO, 19(1) (2014) Solve Maxwell s equation rigorously in a voxelized space Size of the voxel has to be small Need

8 xisting Focused Beam Models Monte Carlo Simulation Finite Difference Time Domain (FDTD) Solution for Maxwell s equations Propagates photons towards the focal point Ignores the wave nature of light Provides mean behavior Requires large number of photons Song et al, Appl. Optics. 38(13) (1999) Blanca and Saloma Appl. Optics. 37(34) (1998) Dunn et al, Appl. Optics. 39(7) (2000) Hayakawa et al, Biomed. Opt. xpress. 2(2) (2011) Cai et al, Prog. in lectromagnetics Res. 142 (2013) Cai et al, JBO, 19(1) (2014) Solve Maxwell s equation rigorously in a voxelized space Size of the voxel has to be small Need enormous computational resources (15m 15m 50m 12GB, 500hours*) Stair case errors *Starosta and Dunn, Opt. xpress 17(15), (2009) Capoglu et al. Opt. xpress 21(1), (2013) lmaklizi et al. JBO 19(7) (2014)

9 Modeling Problem: Distortion by a Single Scatterer Nominal focal point Z X-Y plane Spherical Scatterer X X-Z plane Focus beam Plane wave (x-polarized) Lens (Focal length f )

10 Modeling Single Scatterer Problem 1 Plane wave incident on a spherical scatterer 2 Focused beam propagation in free space 3 Modeling Problem: Distortion by a Single Scatterer

11 Key Concepts, quations and Properties lectromagnetic wave Maxwell s equations Plane wave solution to Maxwell s equations Properties of plane wave

12 Light is an lectromagnetic Wave

13 Maxwell s quations Provide exact model for M wave propagation Provide theoretical foundation of optics Model wave interference, diffraction and polarization (Faraday s Law) (Ampere s Law) (Gauss Law) (Gauss Law for Magnetism) H 0 t H J 0 t H 0 0 : lectric field : Magnetic field : Current density : Charge density : Permeability : Permittivity

14 Maxwell s quations Provide exact model for M wave propagation Provide theoretical foundation of optics Model wave interference, diffraction and polarization (Faraday s Law) (Gauss Law) (Gauss Law for Magnetism) H 0 t H J 0 t (Ampere s Law) 0 H 0 0 No flow of current No free charges H t 0 : lectric field : Magnetic field : Current density : Charge density : Permeability : Permittivity

15 Origin of Wave quation from Maxwell s quations In free space (no flow of current and no free charges) (Faraday s Law) (Ampere s Law) H 0 t H 0 t where t c Wave equation in free space 2 1 c 2 t 2 2 speed of light in vacuum

16 Origin of Wave quation from Maxwell s quations In free space (no flow of current and no free charges) (Faraday s Law) (Ampere s Law) H 0 t H 0 t where t c Wave equation in free space 2 1 c 2 t 2 2 speed of light in vacuum Solutions to wave equation in free space General form r () f rct Plane wave solution 0 ( z) cosk zct OR 0 ( z) sink zct 0 ( z) cos kzt k: wave number : angular frequency Plane wavefront

17 Plane wave Amplitude 0 ( z) cos kzt Amplitude Phase Phase Intensity 1 1 Re H c 0 0 k: wave number = 2/ : angular frequency : Permittivity c: Speed of light in vacuum

18 Complex Representation of Waves Complex representation of wave enable us to combine the amplitude and the phase into a single function. 0 0 ( z) cos kz t i sin kz t real {Original function} imaginary Amplitude = Phase () = Im ( z) Re ( z) 2 2 Re ( z) Im ( z) ArcTan kz t 0 ( z) expi 0 {uler s Formula}

19 Polarization of Plane wave Polarization is described by specifying orientation of the electric field. x 0 y 0 expikz Considering x-z plane x 0 0 y expikz Considering y-z plane 0 y x 0 expikz

20 Polarization of Plane wave Polarization is described by specifying orientation of the electric field. x u cos sin sin cos v y Rotation matrix Considering u-w plane: cos u sin v x x y y sin cos

21 Plane Wave Incident on a Spherical Scatterer Mie Solution to Maxwell s quations (commonly known as Mie Theory) Mie Simulator GUI

Mie Solution to Maxwell s quations Mie solution is an analytic solution to Maxwell s equation for an incident plane wave Describes M wave scattering by a sphere The solution is a convergent infinite

22 Mie Solution to Maxwell s quations Mie solution is an analytic solution to Maxwell s equation for an incident plane wave Describes M wave scattering by a sphere The solution is a convergent infinite series Provides both internal and external scattering fields xternal field Internal field + xternal field = Incident field Incident field Van de Hulst, H. C., Light scattering by small particles, Dover publications (1981) Bohren and Huffman, Absorption and Scattering of Light by Small Particles (1983)

23 Mie Solution to Maxwell s quations Scattering efficiency (Q sca ) 2 Q (2n1) a b 2 2 sca 2 n n x n1 Scattering cross section ( ) s Q sca a 2 where a b n n k mx x m mx x mx x m mx x n n n n n n n n m n mx n x n mx n x m mx x mx x 2 x ka n n n n (size parameter) Van de Hulst, H. C., Light scattering by small particles, Dover publications (1981) Bohren and Huffman, Absorption and Scattering of Light by Small Particles (1983)

Mie Solution to Maxwell s quations Far-field amplitude scattering matrix components 1 2n1 Pn cos d 1 S1 an bn Pn cos n1 nn 1 sin d 1 2n1 Pn cos d 1 S2 bn an Pn cos n1 nn1 sin d S( ) S2 ( ) 0 0 S1( )

24 Mie Solution to Maxwell s quations Far-field amplitude scattering matrix components 1 2n1 Pn cos d 1 S1 an bn Pn cos n1 nn 1 sin d 1 2n1 Pn cos d 1 S2 bn an Pn cos n1 nn1 sin d S( ) S2 ( ) 0 0 S1( ) Phase function p ave ( ) S Q sca S x 2 where P n1 cos : 1 st derivative of Legendre polynomials Van de Hulst, H. C., Light scattering by small particles, Dover publications (1981) Bohren and Huffman, Absorption and Scattering of Light by Small Particles (1983)

Mie Simulator GUI s=nss s = s(1-g) Ns g http://virtualphotonics.codeplex.

25 Mie Simulator GUI s=nss s = s(1-g) Ns g Poly disperse: Gelebart et al. Pure Appl. Opt., 5 (1996)

26 Rayleigh Limit of Mie Scattering (Diameter <<λ) Q sca a 4 4 Parallel (p) Polarization Perpendicular (s) Dipole radiation 0 0 Hertzian Dipole

27 Plane Wave Incident on a Spherical Scatterer A Detector A x-polarized incident plane wave

28 Plane Wave Incident on a Spherical Scatterer Incident field on scattered plane i cos sininc sin cos 0 i A HC Van de Hulst, Light Scattering by small particles Dover, (1981)

29 Plane Wave Incident on a Spherical Scatterer Incident field on scattered plane i cos sininc sin cos 0 i Scattered electric field at A is given by 1 s r,, exp( ikr) S2 r, i ikr 1 s r,, exp( ikr) S1 r, i ikr A 1/r Phase {Sph. Wave} Scattering amplitude HC Van de Hulst, Light Scattering by small particles Dover, (1981)

30 Plane Wave Incident on a Spherical Scatterer Incident field on scattered plane i cos sininc sin cos 0 i Scattered electric field at A is given by 1 s r,, exp( ikr) S2 r, i ikr 1 s r,, exp( ikr) S1 r, i ikr A 1/r Phase {Sph. Wave} Scattering amplitude s (, ) 1 S2 ( r, ) 0 exp( ) i ikr (, ) ikr 0 S ( r, ) s 1 i For non-spherical scatterers S2(, r, ) S3(, r, ) S4(, r, ) S1(, r, ) HC Van de Hulst, Light Scattering by small particles Dover, (1981)

31 Plane Wave Incident on a Spherical Scatterer A inc scat A : lectric field at A

32 Plane Wave Incident on a Spherical Scatterer A inc scat A : lectric field at A Vector decomposition inc inc i 0 j 0 k i j k i j k s xs ys zs s xs ys zs

33 Plane Wave Incident on a Spherical Scatterer A inc scat A : lectric field at A Vector decomposition inc inc i 0 j 0 k i j k i j k s xs ys zs s xs ys zs x y z

34 Two Scatterer Problem: Plane wave incident Detector A According to superposition theorem*, N s scat A inc j j1 : lectric field at A N s : Number of scatterers *Chapter 8: The Mie Theory: Basics and Applications, Springer 2012

35 Two Scatterer Problem: Plane wave incident Primary scattering A 1 2 Secondary scattering According to superposition theorem*, N s scat A inc j j1 : lectric field at A N s : Number of scatterers Tertiary scattering scat scat *Chapter 8: Multiple Scattering of Light by Particles, Cambridge 2006

36 Multi-Scatterer Problem: Plane wave incident Detector

37 Modeling Problem: Distortion by a Single Scatterer Plane wave incident on a spherical scatterer Focused beam propagation in free space Modeling Problem: Distortion by a Single Scatterer 1 2 3

38 Focused Beam Propagation Focused beam propagation in free space HF Wavelets to Model Focus Beam Propagation in free space in a scattering medium

39 Focused Beam Propagation in Free Space max Geometrical representation Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) 1959

Focused Beam Propagation in Free Space max Analytical solution: 1 (,, ) kf z exp( ikf) (, )exp cos sincos( ) sin 2 i kz k d d i max 2 0 0 Geometrical

40 Focused Beam Propagation in Free Space max Analytical solution: 1 (,, ) kf z exp( ikf) (, )exp cos sincos( ) sin 2 i kz k d d i max Geometrical representation Phase at the focal point w.r.t. lens lectric field at lens surface Phase at,, w.r.t. focal point Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) 1959

41 Focused Beam Propagation in Free Space Focused Beam Simulator GUI: Analytical solution

42 Numerical Aperture (NA) NA n m sin( ) max Low NA High NA Low resolution High resolution

43 HF Wavelets to Model Focus Beam Propagation Huygens-Fresnel (HF) principle: ach point of an advancing wavefront act as a source of outgoing secondary spherical waves Plane wave HF Wavelet: A small section of a secondary spherical wave Focused beam

44 HF Wavelets to Model Focus Beam Propagation Huygens-Fresnel (HF) principle: ach point of an advancing wavefront act as a source of outgoing secondary spherical waves Plane wave Focused beam HF Wavelet: A small section of a secondary spherical wave

45 HF Wavelets and Airy Disk Radius Airy disk radius r 0.61 / NA m -1 m -1 m 0 m 0

HF Wavelets in a non scattering medium Implementation in a non scattering medium Generate uniformly distributed points (HF radiating source locations) on the spherical cap* Project wavelets from each

46 HF Wavelets in a non scattering medium Implementation in a non scattering medium Generate uniformly distributed points (HF radiating source locations) on the spherical cap* Project wavelets from each radiating source to a detector point Phase advances with traveling distance A A (, ) cos sininc (, ) (, )exp( ikd j ) (, ) sin cos 0 A (, ) (, )exp( ikd j ) A (, ) xi y j zk (, ) i j k A x y z *Koay, C. J. Comput. Sci. 2, (2011).

47 HF Wavelet based lectric Field Superposition (HF-WFS) Verifying results in a non scattering medium with the analytical solution (A) Analytical Solution (B) HF-WFS* (A) (B) Simulation parameters : 800nm, nm:1.33, f:500 m, NA:0.667 *Ranasinghesagara et al, JOSA A 31(7) 2014

48 HF Wavelet based lectric Field Superposition (HF-WFS) Verifying results in a non scattering medium with the analytical solution

49 Detection of Scattering Fields Primary and secondary scattering detection

HF Wavelets to Model Focus Beam Propagation Implementation in a medium with spherical scatterers Generate uniformly distributed points (HF radiating source locations) in the spherical cap Project

50 HF Wavelets to Model Focus Beam Propagation Implementation in a medium with spherical scatterers Generate uniformly distributed points (HF radiating source locations) in the spherical cap Project wavelets from each radiating source to a scatterer Phase advances with traveling distance Find scattering angle and distance from scatterer to the detector point Calculate scattered field contribution at the detector from Mie solution P s (, ) 1 S2 (, r ) 0 exp( ) i ikr P s (, ) ikr 0 S1( r, ) i (, ) cos sin (, ) sin cos 0 (, ) (, )exp( ikd ) P inc 1 (, ) (, )exp( ikd ) P 1

51 Modeling Problem: Distortion by a Single Scatterer Plane wave incident on a spherical scatterer Focused beam propagation in free space Modeling Problem: Distortion by a Single Scatterer 1 2 3

52 HF Wavelets to Model Focus Beam Propagation Simulation parameters : 800nm, n m :1.33, f:500m, NA:0.667 Ranasinghesagara et al, JOSA A 31(7) 2014

53 Focal Spot Displacement & Amplitude Change Non scattering Single scatterer

54 Pros and Cons of HF-WFS Pros: 2-4 orders of magnitude faster than FDTD High performance computer systems are not necessary Propagation throughout the entire volume is not necessary to obtain results Provides a quick snapshot of electric field distortion Cons: May not see a huge speed gain for thick samples Require complete amplitude scattering matrix data

55 HF Wavelet based lectric Field Superposition (HF-WFS)

Basic optics. Geometrical optics and images Interference Diffraction Diffraction integral. we use simple models that say a lot! more rigorous approach

$Basic optics. Geometrical optics and images Interference Diffraction Diffraction integral. we use simple models that say a lot! more rigorous approach$ Basic optics Geometrical optics and images Interference Diffraction Diffraction integral we use simple models that say a lot! more rigorous approach Basic optics Geometrical optics and images Interference