Nonlinear Light Scattering Nonlinear Light Scattering using the - - PowerPoint PPT Presentation

Nonlinear Light Scattering Nonlinear Light Scattering using the FDTD Method g

Seán Meenehan

Second Harmonic Generation

Incident fields at frequency ω, if of sufficient intensity,

Incident fields at frequency ω, if of sufficient intensity, can generate scattered fields at 2ω.

A common example is green laser pointers Source term for radiation is the second-order

polarization, with the dominant contribution being of h f the form:

k j ) ( ijk ) ( i

E E χ P

2 2

=

k j ijk i

E E χ P

Nonlinear Maxwell’s Equations

r r B P D

f

r r r ∇ + = ⋅ ∇

) 2 (

ρ B E B r r ∂ ∇ = ⋅ ∇ P D t E r r ∂ ∂ ∂ − = × ∇

) 2 (

J t P t D H r r + ∂ ∂ + ∂ ∂ = × ∇

) 2 (

FDTD Method

Basic prescription:

as c p esc pt o :

Discretize space and time Express temporal and spatial derivatives in the Maxwell curl

i fi i diff equations as finite differences

Split fields into scattered and incident Rearrange equations to get scattered fields at time t = (n+1)Δt

ea a ge equat o s to get scatte ed e ds at t e t ( + ) t in terms of earlier scattered fields and incident fields.

Solve for fields in solution space for each time step, alternating

between evaluating E and H and store results between evaluating E and H, and store results.

Handle boundary fields seperately at each time step

Advantages of FDTD

Easy to simulate arbitrary scattering geometries Easy to simulate arbitrary scattering geometries

(w/ staircasing error) and inhomogeneous media.

Easy to simulate any time dependent incident

field with an analytical expression (e.g. laser field with an analytical expression (e.g. laser pulse which is Gaussian in time and/or spatial intensity) y)

For multiple frequency fields, we can solve for

the whole field at once, instead of superposing , p p g individual frequency solutions

Assumptions

Medium is linearly isotropic (or diagonally Medium is linearly isotropic (or diagonally

anisotropic)

Medium is nonmagnetic (i e H = B) Medium is nonmagnetic (i.e H = B) No dispersion (dielectric response is equal for all

frequencies) frequencies)

Example (linear case)

∂ (

) ( )

) (

scatt inc scatt inc scatt inc

E E E E t H H + + + ∂ ∂ = + × ∇ σ ε r r r r r r r r ) ( ) (

, , , , , , scatt x inc x xx scatt x inc x xx scatt y scatt z

E E t E t E y H y H + + ∂ ∂ + ∂ ∂ − = ∂ ∂ − ∂ ∂ σ ε ε ε ) 1 , , ( ) , , ( ) , 1 , ( ) , , (

, 2 1 , 2 1 , 2 1 , 2 1 s y n s y n s z n s z n

k j i H k j i H k j i H k j i H − − − −

+ + + +

) ( ) ( ) , , ( ) , , ( ) , , ( ) , , (

, , 1 , , , , , s x n inc x n inc x n s x n s x n y y

E E E E E z j j y j j + + ∂ − + − = Δ − Δ

−

σ ε ε ε ) ( ) (

, , s x inc x xx

• xx

xx

E E t t + + ∂ + Δ σ ε ε ε

Stability Issues

Rule of thumb for spatial discretization size is ~λ/10

Rule of thumb for spatial discreti ation si e is λ/10 for highest frequency of interest (and no bigger than λ/4)

Need multiple samples per wavelength, also smaller grid

spacing minimizes “grid dispersion” F i i C di i i 3D

For time spacing, use Courant condition in 3D:

2 2 2

1 z y x c t Δ + Δ + Δ = Δ

Boundary Issues

In real-life, the scattered fields propagate out to In real life, the scattered fields propagate out to

infinity, but we necessarily truncate our solution space p

Need to simulate boundaries that absorb the

• utgoing waves to minimize reflection errors
• utgoing waves to minimize reflection errors

Popular scheme is the Mur boundary condition,

which estimates the fields at the boundary by which estimates the fields at the boundary by interpolating past boundary fields and interior fields

Far Zone Scattering

Approximate surface integral of tangential fields Approximate surface integral of tangential fields

• ver the boundary at each time step

Relate these to transverse radiation fields far Relate these to transverse radiation fields far

away from the scatterer

Nonlinear Problem

The presence of the nonlinear polarization’s The presence of the nonlinear polarization s

time derivative gives us two unknowns at time step n+1 with only one equation p y q

For a single incident frequency, the time

derivative just becomes i2ω times the nonlinear derivative just becomes i2ω times the nonlinear polarization at time step n