Modeling Focused Beam Propagation in scattering media. Janaka Ranasinghesagara, Ph.D.

Transcription

1 Modeling Focused Beam Propagation in scattering media Janaka Ranasinghesagara, Ph.D.

2 Teaching Objectives The need for computational models of focused beam propagation in scattering media Introduction to the principles and mathematical models underlying focus beam propagation. Learn how fundamental concepts are applied to develop an efficient focused beam propagation model.

3 Spatial Scales of Imaging Ntziachristos, Nat. Methods, 7 (21) Leigh, Chen and Liu, Biomed Opt. Exp., 5 (6) (214)

4 Focused Beam Propagation in Free Space Amplitude Analytical solution* Phase * Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) 1959

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

6 Focused Beam Propagation in Scattering Media Scattering distorts the excitation volume Scattering is the main limiting factor for penetration depth in microscopy

7 Existing Focused Beam Propagation Models Monte Carlo Simulation Propagates photons towards the focal point Ignores the wave nature of light Considers far-field phase function Provides mean behavior Requires large number of photons Song et al, Appl. Opt.. 38(13) (1999) Blanca et. Al Appl. Opt. 37(34) (1998) Dunn et al, Appl. Opt.. 39(7) (2) Hayakawa et al, Biomed. Opt. Exp. 2(2) (211) Cai et al, Prog. in Electromag. Res. 142 (213) Cai et al, JBO, 19(1) (214)

8 Existing Focused Beam Propagation 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 Considers far-field phase function Provides mean behavior Requires large number of photons Solve Maxwell s equations rigorously in a voxelized space Size of the voxel has to be small Need enormous computational resources (15m 15m 5m 12GB, 5hours*) Stair case errors Song et al, Appl. Opt.. 38(13) (1999) Blanca et. Al Appl. Opt. 37(34) (1998) Dunn et al, Appl. Opt.. 39(7) (2) Hayakawa et al, Biomed. Opt. Exp. 2(2) (211) Cai et al, Prog. in Electromag. Res. 142 (213) Cai et al, JBO, 19(1) (214) *Starosta and Dunn, Opt. Express 17(15), (29) Capoglu et al. Opt. Express 21(1), (213) Elmaklizi et al. JBO 19(7) (214)

9 Key Concepts, Equations and Properties Electromagnetic wave Maxwell s equations Plane wave solution to Maxwell s equations Properties of plane wave

10 Light is an Electromagnetic Wave

11 Maxwell s Equations Provide exact model for EM 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 E t E H J t E H E: Electric field H: Magnetic field J: Current density ρ: Charge density μ : Permeability ε : Permittivity

12 Origin of Wave Equation from Maxwell s Equations Provide exact model for EM wave propagation Provide theoretical foundation of optics Model wave interference, diffraction and polarization (Faraday s Law) (Gauss Law) (Gauss Law for Magnetism) H E t No flow of current (Ampere s Law) H E H J t E H No free charges E t E 2 2 E 2 where t Wave equation in free space 2 E c 1 c 2 t E E

13 Plane Wave Wave equation 2 E 1 c 2 t E 2 2 Plane wave solution E ( z) E cos k z ct OR E ( z) E sin k z ct

14 Plane Wave Wave equation Amplitude 2 E 1 c 2 t E 2 2 Phase Plane wave solution E ( z) E cos k z ct OR E ( z) E sin k z ct E ( z) E cos kz t Amplitude Phase 1 1 Intensity ReE H ce k: wave number = 2/ ω: angular frequency ε : Permittivity c: Speed of light in vacuum Plane wavefront

15 Complex Representation of Waves Complex representation of wave enable us to combine the amplitude and the phase into a single function. Complex number X iy E ( z) E cos kz t ie sin kz t Aexp( i ) {Euler s Formula} real {Original function} imaginary Amplitude = Phase () = E E Im E( z) Re E( z) 2 2 Re ( z) Im ( z) E ArcTan kz t E( z) E expi

16 Polarization Polarization is described by specifying orientation of the electric field. E y y x = z E x E expikz Considering x-z plane E E x E E E y expikz Considering y-z plane E E y E E x E expikz

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

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

19 Plane Wave Incident on a Spherical Scatterer Scattered field Incident field Mie solution to Maxwell s equation (Mie theory) describes the scattering of plane wave by a homogeneous sphere

20 Mie Solution to Maxwell s Equations (Mie Theory) Scattering efficiency (Q sca ) 2 Q (2n 1) a b 2 2 sca 2 n n x n1 Scattering cross section (σ s ) s Q sca a 2 Far-field amplitude scattering matrix components S2 ( ) S( ) S1( ) 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 nn 1 sin d Phase function p ave ( ) S Q sca S x 2 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)

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

22 Diameter of Sphere and Wavelength Diameter < λ Diameter > λ Uniform electric field throughout sphere Non-uniform electric field throughout sphere

23 Rayleigh Limit of Mie Scattering Diameter < λ Parallel (p) to incident polarization Polarization Perpendicular (s) to incident polarization y-z plane x-z plane incident polarization Hertzian Dipole 18 Mie Simulator GUI 18 Mie Simulator GUI

24 Rayleigh Limit of Mie Scattering Diameter < λ Parallel (p) to incident polarization Polarization Perpendicular (s) to incident polarization y-z plane x-z plane incident polarization Hertzian Dipole 18 Mie Simulator GUI 18 Mie Simulator GUI

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

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

27 Plane Wave Incident on a Spherical Scatterer Incident field on scattered plane E i cos sin Einc sin cos E i Scattered electric field at A is given by 1 E s r,, exp( ikr) S2 r, E i ikr 1 Es r,, exp( ikr) S1 r, Ei ikr A 1/r Phase {Sph. Wave} Scattering amplitude E s(, ) 1 S2 ( r, ) E exp( ) i ikr E (, ) ikr S ( r, ) E 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)

28 An Efficient Focused Beam Propagation Model Huygens-Fresnel (HF) Principle Focused beam as a summation of plane waves Constructive and destructive interference of HF waves HF-WEFS model algorithm

29 Focused Beam as a Summation of Plane Waves Huygens-Fresnel (HF) principle: Each point of an advancing wavefront act as a source of outgoing secondary spherical waves HF Wavelet: A small section of a secondary spherical wave Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) (1959) Capoglu et al. Opt. Express 16(23), (28) Elmaklizi et al. JBO 19(7) (214)

30 Interference and Airy pattern formation Constructive interference Destructive interference Airy disk radius (r) (the distance between the central maximum and the first minimum) r.61 / NA

31 HF Wavelets to Model Focused Beam Propagation 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 E (, ) cos sin Einc E (, ) E (, )exp( ikd j ) (, ) sin cos E A E(, ) E(, )exp( ikd j ) A E (, ) E i E j E k x y z E (, ) E i E j E k A x y z and polarization calculation HF wavelet propagation Wave summation (interference)

32 Verifying Results in a Non Scattering Medium HF Wavelet based Electric Field Superposition (HF-WEFS) model and analytical solution (A) Analytical Solution (B) HF-WEFS* (A) (B) Simulation parameters : 8nm, n m :1.33, f:5m, NA:.667 *Ranasinghesagara et al, JOSA A 31(7) 214

33 Focused Beam Simulator GUI: Analytical solution

34 Focused Beam Simulator GUI: Huygens-Fresnel Approach

35 Calculation of Scattering Field What information is necessary to find the total electric field at point A? - Incident electric field A - Scattering field (from Mie Theory) - Distance to point A A

36 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 Incident Field E (, ) cos sin Einc (, ) sin cos E P E (, ) E (, )exp( ikd i ) P E(, ) E(, )exp( ikd i ) Scattered Field P E s (, ) 1 S2( r, ) E exp( ) i ikr P s (, ) ikr S1( r, ) E Ei Total Field

37 HF-WEFS in a medium with spherical scatterers

38 Comparing HF-WEFS results with FDTD A B C D Simulation parameters: : 8nm, n m :1.33, f:5m, NA:.667 HF-WEFS FDTD 1µm Ranasinghesagara et al, JOSA A 31(7) (214)