Modeling Focused Beam Propagation in a Scattering Medium. Janaka Ranasinghesagara

Modeling Focused Beam Propagation in a Scattering Medium Janaka Ranasinghesagara

Lecture Outline Introduction Maxwell s equations and wave equation Plane wave and focused beam propagation in free space Plane wave propagation in a medium containing scatterers Focused beam propagation in a medium containing scatterers FDTD solution to Maxwell s equation Huygens Fresnel wavelets method

Introduction Incident light refracted by the lens and provides a sharp focal spot in free space Scatterers provides a secondary radiation (scattered field) Scattered field alters the focal field, limits imaging depth and resolution http://en.wikipedia.org/wiki/wavefront

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 t H J t H : lectric field : Magnetic field : Current density : Charge density : Permeability : Permittivity

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 t H J t H No flow of current No free charges H t : lectric field : Magnetic field : Current density : Charge density : Permeability : Permittivity

Origin of wave equation from Maxwell s quations In free space (no flow of current and no free charges) (Faraday s Law) (Ampere s Law) H t H t 2 2 2 where t c 1 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 ( z) exp ik zct wave number 2 k

Plane wave solution to wave equation ( z) exp ik zct t =, ( z) expikz z =, ( z) expikct ( z) expit

Plane wave solution to wave equation ( z) exp ik zct Amplitude exp ik z ct Phase Im ( z) ArcTan Re ( z) nergy flux 1 1 Re 2 2 2 H c

Polarization Polarization is described by specifying orientation of the electric field. x y expikz

Polarization Polarization is described by specifying orientation of the electric field. x y expikz cos expikz sin expikz u v

Focused beam propagation in free space nergy incident on the lens is equal to the energy that leaves (, ) i n cos 1 n m n cos 1 (, ) i cosn sinn n m Parallel component Perpendicular component max 2 1 (,, z) ikf exp( ikf) (, )exp ikzcos ksincos( ) sin d d 2 Phase at the origin w.r.t. lens lectric field at lens surface Phase at,, w.r.t. origin Richards and Wolf. Proc. Royal Soc. Lond. A 253(1274) 1959

Airy Disk Radius and Numerical Aperture(NA) Airy disk radius r.61 / NA In confocal microscopy r.4 / NA lateral r 1.41 n / NA axial m 2 http://zeiss-campus.magnet.fsu.edu/tutorials/basics/airydiskformation/index.html Numerical Aperture (NA) NA n m sin( ) max NA Resolution Probing depth http://zeiss-campus.magnet.fsu.edu/tutorials/basics/numericalaperturelightcones/index.html

Wave propagation in a medium containing scatterers Consider a linearly polarized M wave with unit amplitude propagate along the z-axis impinging on a scatterer Scattered electric field in the far field is given by exp( ) (, ) ikr S (, ) S (, ) ikr s 2 i 3 i exp( ) (, ) ikr S (, ) S (, ) ikr s 4 i 1 i where cos i i sin i i

Wave propagation in a medium containing scatterers Consider a linearly polarized M wave with unit amplitude propagate along the z-axis impinging on a scatterer Scattered electric field in the far field is given by s (, ) exp( ikr) i S(, ) s (, ) ikr i where S(, ) S S (, ) S (, ) (, ) S (, ) 2 3 4 1 Amplitude scattering matrix

Wave propagation in a medium containing spherical scatterers Consider a linearly polarized M wave with unit amplitude propagate along the z-axis impinging on a spherical scatterer Scattered electric field in the far field is given by exp( ikr) s (, ) S2 ( ) ikr exp( ikr) s (, ) S1( ) ikr i i s (, ) exp( ikr) S2 ( ) i s (, ) ikr S1( ) i

Mie solution to Maxwell s equations Mie solution is an analytic solution to Maxwell s equation in a spherical geometry for a plane wave incident It describes the scattering of M radiation by a sphere The solution is an infinite series, but converges It provides both internal and external scattering fields At the boundary, Internal field = Incident field xternal field

Mie solution to Maxwell s equation Amplitude scattering matrix components 1 2n1 Pn cos d 1 S1 an bn Pn cos n1 nn1 sin d 1 2n1 Pn cos d 1 S2 bn an Pn cos n1 nn1 sin d where a b n n mka ka m mka ka mka ka m mka ka n n n n n n n n n n n n m mka ka mka ka m mka ka mka ka n n n n P 1 cos : 1 st derivative of Legendre polynomials n Van de Hulst, H. C., Light scattering by small particles, Dover publications (1981)

Rayleigh scattering and polarization Parallel Perpendicular

Mie solution to Maxwell s equations Inputs for far field solution Size parameter ka n n Relative ref. index, m n Scattering angle 2 m p m a Inputs for near field solution Size parameter ka n n Relative ref. index, m n Scattering angle 2 m a Distance parameter kr n p m 2 m r Size parameter 2 1.331 1.37 Relative ref. index 1.33

FDTD solution to model light propagation Finite Difference Time Domain (FDTD) solution for Maxwell s equation In FDTD, Maxwell s equations are implemented in Cartesian space grids FDTD uses time stepping leapfrog approach It simulates continuous electromagnetic waves in a finite spatial region It is good for rigorously modeling optical wave interactions with inhomogeneous tissue structures in small volumes No linear algebra is involved FDTD domain can be parallelized easily

FDTD solution to model light propagation In free space H t H t 2 1 c 2 t 2 2 In a non magnetic medium : r r relative permittivity H t H r t 2 1 v 2 t 2 2 Speed of light in medium, v 1 c r r c n m

1D FDTD Solution for Maxwell s equations H t t, H r Let s consider 1D model, i j k x x y z, y z ( Hx Hy Hz ) i j k t i j k x H H H x y z ( x y z ) i j k t Solutions H y x x z H t r t z y Relationship between and H x y r t z H y x t z Relationship between and

1D FDTD Solution for Maxwell s equations H x z r t y, y x H t z Wartak M.S., Computational Photonics, Cambridge press, (213)

3D FDTD Solution for Maxwell s equations Perfectly matched layerabsorbing boundary condition PML-ABC 3D voxelized grid Single voxel scat Hscat r r 1 t t inc dges: -field Surfaces: H-field http://en.wikipedia.org/wiki/finite-difference_time-domain_method

FDTD solution vs. Mie Solution Grid spacing = /1 /2 /3 M. Starosta, Dissertation, UT Austin 21

Plane wave propagation in a scattering medium Dunn et al. J Biomed.Optics 23(2), 1997

Focused Beam propagation in a scattering medium x(m) Non scattering medium x(m) Medium with scatterers z(m) Simulation time: 5 processor hours Starosta and Dunn, Opt. xpress 17(15), 29

Focused Beam propagation in a scattering medium x(m) z(m) Simulation time: 53 processor hours Starosta and Dunn, Opt. xpress 17(15), 29

Limitations of FDTD Approach FDTD requires enormous computational resources The size of the voxel and the time step has to be small to satisfy leapfrog integration and to provide stable and accurate results Digitization errors on non rectangular objects http://www.remcom.com/xf7-fdtd-method/

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 (HF wavelets) Plane wave Focused beam HF Ray representation: ach HF wavelet is represented by an infinite number of rays radiating from its center

HF Ray based lectric Field Superposition (HF-RFS) Implementation in a non scattering medium Generate uniformly distributed points (HF radiating source locations) in the spherical cap Project rays from each radiating source to a detector point Phase advances with traveling distance

HF Ray based lectric Field Superposition (HF-RFS) Verifying results in a non scattering medium with the analytical solution (A) Analytical Solution (B) HF-RFS (A) (B) Simulation parameters : 8nm, nm:1.33, f:5 m, NA:.667

HF wavelets to model focus beam propagation Focused beam can be represented as a linear combination of plane waves

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 rays 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

HF wavelets to model focus beam propagation Sim : 8

Focal spot displacement & amplitude change Non scattering Single scatterer

Pros and cons of HF-RFS Pros: 2-4 orders of magnitude faster than FDTD solution High performance computer systems are not necessary Does not require to simulate complete volume to obtain results Provides a quick snapshot of electric field distortion Cons: Require complete amplitude scattering matrix data

Summary Maxwell s equation and solutions Plane and focused beam propagation in free space FDTD solution to model focused beam propagation Huygens Fresnel wavelets to model focused beam propagation