ME equations. Cylindrical symmetry. Bessel functions 1 kind Bessel functions 2 kind Modifies Bessel functions 1 kind Modifies Bessel functions 2 kind

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Myles Gardner
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1 Δϕ=0 ME equations ( 2 ) Δ + k E = 0 Quasi static approximation Dynamic approximation Cylindrical symmetry Metallic nano wires Nano holes in metals Bessel functions 1 kind Bessel functions 2 kind Modifies Bessel functions 1 kind Modifies Bessel functions 2 kind Dielectric Fibers Metallic nano wires Metallic nano spferes Spherical symmetry Spherical Bessel functions 1 kind Spherical Bessel functions 2 kind Dielectric Micro sphere Complex shapes

2 Δϕ=0 ME equations ( 2 ) Δ + k E = 0 Quasi static approximation Dynamic approximation Cylindrical symmetry Metallic nano wires Nano holes in metals Bessel functions 1 kind Bessel functions 2 kind Modifies Bessel functions 1 kind Modifies Bessel functions 2 kind Dielectric Fibers Metallic nano wires Metallic nano spferes Spherical symmetry Spherical Bessel functions 1 kind Spherical Bessel functions 2 kind Dielectric Micro sphere Complex shapes

3 Computational Methods Maxwell s equations

4 Computational Methods Maxwell s equations Periodic objects Non-periodic objects

5 Computational Methods Maxwell s equations Periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Non-periodic objects Mode expansion + amplitude matching (MMP)

6 Computational Methods Maxwell s equations Periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Non-periodic objects Mode expansion + amplitude matching (MMP) Single multipole (Mie-theory)

7 Computational Methods Maxwell s equations Periodic objects Non-periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Disretization and time marching (FDTD) Mode expansion + amplitude matching (MMP) Single multipole (Mie-theory)

8 Computational Methods Maxwell s equations Periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Disretization and time marching (FDTD) Volume integral (MoM, Greens method) E = E i + E s Non-periodic objects Mode expansion + amplitude matching (MMP) Single multipole (Mie-theory)

9 Computational Methods Maxwell s equations Periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Disretization and time marching (FDTD) Volume integral (MoM, Greens method) E = E i + E s Non-periodic objects Mode expansion + amplitude matching (MMP) Green s II identity Surface Integral (BEM) Single multipole (Mie-theory)

10 This lecture Maxwell s equations Periodic objects Mode expansion + amplitude matching (FMM, RCWA, C- Method) Disretization and time marching (FDTD) Volume integral (MoM, Greens method) E = E i + E s Non-periodic objects Mode expansion + amplitude matching (MMP) Green s II identity Surface Integral (BEM) Single multipole (Mie-theory)

11 Mie theory for cylinders Gustav Mie proposed already 1908 a quasi-analytical solution of the diffraction problem of a wave at a sphere or cylinders (and applied it to the problem of metallic nano particles) So far we have expanded fields mainly in plane waves or Bloch-waves Basic idea: Spherical objects Expansion of all fields in spherical waves k n 2 R n 1 y x r θ

12 Mie theory for cylinders Fields are solution of the scalar wave equation (treating E-field only in the case of TE polarization, E-field parallel to the cylinder axis, and H-Field only in the case of TM polarization, H-field parallel to the cylinder axis) r u i = e ıkr Bessel-Fkt. Writing the incident field as a superposition of cylindrical waves Only problem Sum has to be truncated

13 Mie theory for cylinders Bessel function of the first kind No singularity in the origin Does NOT fulfill Sommerfeld s radiation condition Expansion used for illuminating beam and field inside the cylinder

14 Mie theory for cylinders H 0 m(x) = J 0 m(x) + ıj 1 m(x) (apologize for this inconsistency, first order has index 0, second order has index 1) Hankel function of the first kind Singularity in the origin Does fulfill Sommerfeld s radiation condition Expansion used for the scattered field outside of the cylinder

15 Target: Mie theory for cylinders Plane wave m=0..5 m=0 m=0..2 m=0..10!!! Sufficient number of orders!!!

16 Mie theory for cylinders Strategy for Mie theory Writing the field outside the cylinder as a superposition of the illuminating field and the scattered field The field inside the cylinder is given as the transmitted field In each domain the fields are expanded and are a solution to the wave-equation (analytically known) Matching the amplitudes at the boundary such that the continuity are fulfilled (done numerically)

17 Mie theory for cylinders Solution to the problem of the scattering of an illuminating wave field on cylinder All fields are expanded: Incident field, scattered field and field inside (choosing the expansion such that singularities are avoided) Plane wave r>rad r<rad Hankel-Fkt. Bessel-Fkt.

18 Mie theory for cylinders The entire field is written as a superposition of cylindrical waves = +. But how to determine the amplitudes of the modes?

19 Mie theory for cylinders Boundary conditions TE TM

20 Mie theory for cylinders Two equations with two unknowns Analytically solvable ) a m = J ( m 0 2π n λ 2r ) ( J 0 2π n m λ 1r ) ( p 1 Jm 0 2π n λ 1r ) ( ( ) ( ) ( ) J 0 ( 2π n m λ 2r ) ) p 2 ( Hm 0 2π n λ 1r ) ( J 0 2π n m λ 2r ) ( p 2 Hm 0 2π n λ 1r ) ( Jm 0 2π n λ 2r ) ( ) ( ) ( ) ( ) p1 ) ( ) ( ) ( ) ( ) ( 2π b m = H0 m n λ 1r ) ( J 0 2π n m λ 1r ) ( p 2 Hm 0 2π n λ 1r ) ( Jm 0 2π n λ 1r ) p2 ( Hm 0 2π n λ 1r ) ( J 0 2π n m λ 2r ) ( p 2 Hm 0 2π n λ 1r ) ( Jm 0 2π n ) λ 2r ) p1 ) r = Rad

21 Mie theory for cylinders All fields can be calculated subsequently via r>rad r<rad

22 Mie theory for cylinders A plane wave illuminates a cylinder made of silver as a function of the frequency At plasmon resonance the field amplitude is enhanced (dipole, quadrupole, hexapole) and the scatering cross section (SCS) is large (r=25nm, TM)

23 Mie theory for cylinders Useful for the simulation of a larger number of diffraction events on the same structure Time consuming finding appropriate position of the multipoles Investigation of the force on nano particles Particle in a Gaussian beam (r=0.3λ, n=1.2, ω=λ, TE)

$Mie theory for cylinders Useful for the simulation of a larger number of diffraction events on the same structure Time consuming finding$

24 Mie theory for cylinders Useful for the simulation of a larger number of diffraction events on the same structure Time consuming finding appropriate position of the multipoles Investigation of the force on nano particles Particle in a Gaussian beam (r=0.05λ, n=1.2, ω=λ, TE)

25 Mie theory for coupled cylinders For coupled particles Writing the incident field as a superposition of the illuminating field and the scattered field from each sphere

26 Mie theory for coupled cylinders λ=303 nm λ=186 nm λ=217 nm λ=243 nm λ=199 nm λ=209 nm Light distribution inside a PC with a line defect Two coupled silver particles (r=25nm d=5nm Drude)

appropriate λ that illuminates a quasi periodic photonic crystal Propagation

27 Mie theory for coupled cylinders Used for calculating the response of a larger number of cylinders Amplitude in db of a plane wave (TE) with appropriate λ that illuminates a quasi periodic photonic crystal Propagation of light through a meta material that consists of randomly distributed metallic nano particles

28 Mie theory for spheres All fields are expanded in terms of vector spherical harmonics based on electric and magnetic potential Magnetic potential Electric potential Angular momentum operator

29 Mie theory for spheres ψ is a solution to a wave equation Different Bessel functions BC: Continuity of the normal displacements and the parallel electric and magnetic field

30 Mie theory for spheres Analyzing metallic nano particles in 3D with or without covering layers E x E z E y in the x-z-plane at λ=633nm, r Core = 15 nm, n Shell = 1.5, r Shell = 30 nm

Multiple Multipole Method For arbitrary shaped particles Basically exactly the same; but more multipoles are necessary for fulfilling the boundary conditions (multipoles = points around which the

31 Multiple Multipole Method For arbitrary shaped particles Basically exactly the same; but more multipoles are necessary for fulfilling the boundary conditions (multipoles = points around which the fields are expanded in spherical waves) Fields in homogenous domains are written as superposition of multipoles Fulfillment of the boundary conditions gives the amplitudes of each mode Multipoles outside describe the field inside Multipoles inside describe the field outside Problems in finding appropriate position for multipoles and the number of expansion orders

32 Multiple Multipole Method Useful for the simulation of a larger number of diffraction events on the same structure Time consuming finding appropriate position of the multipoles Investigation of the torque on nano particles Particle in a Gaussian beam (r Ref =0.01λ, n=1.5, ω=λ, TE)

33 Boundary Integral Method General problem: y x Homogeneous scatterer described by its contour is illuminated by an incident field u inc TE: u=e y TM: u=h y z = + Fields in each region are a solution of the wave equation

34 Green s second identity Having a field that is a solution to the wave equation ( 2 + k 2 )U = 0 And any other field that is a solution to the same equation V ( + ) = 0 ) ( 2 + k 2 )U = 0 ) (U 2 U U 2 U)dV = (U U S n U U ) ds n ( ) Green s second identity (V is any volume bound by the surface S) Volume integral Reduction by a single dimension Surface integral

35 Boundary Integral Method After application of Green s second identity Normal derivatives Incident wave field Free space Green s function

36 Boundary Integral Method Additional restrictions Boundary conditions TE TM Rayleigh-Sommerfeld radiation condition in the absence of a scatterer

37 Boundary Integral Method Final boundary integral equation Solving by the boundary element method Expanding the field in terms of interpolation functions

38 Boundary Integral Method Linear interpolation functions Approximated contour Introduction of these interpolations into BIM leads to a set of linear equations

39 Boundary Integral Method Matrix elements e.g. After solving the system of linear equations, the scattered field is calculated by

40 Boundary Integral Method 30 nm 30 nm 42 nm silver triangle

Nanophotonics: principle and application. Khai Q. Le Lecture 4 Light scattering by small particles

Nanophotonics: principle and application Khai Q. Le Lecture 4 Light scattering by small particles Previous lecture Drude model, Drude-Sommerfeld model and Drude-Lorentz model for conducting media (metal):