Finite-Difference Time-Domain Simulation of Electromagnetic waves - - PowerPoint PPT Presentation

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Finite-Difference Time-Domain Simulation of Electromagnetic waves

Yoon Tiem Leong School of Physics, USM, Penang May 17, 2007 Presented at “Mathematica Expository Workshop” Physics Department, UPM

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Table of contents

1 1D scalar wave 2 (Sourceless) Maxwell Equation

2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

3 2D EM wave propagation from point source 4 Absorbing Boundary Condition in 1D

Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

5 2D EM plane waves 6 Total field/scattered field method 7 Putting everything together 8 Further applications

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

One-dimension scalar wave equation:

∂2u ∂t2 = c2 ∂2u ∂x2 .

u = u(x, t) one-dimension scalar wave. Use finite difference method to discretize the scala wave equation

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

Taylor expand u(x, tn) about xi, keeping time tn fixed, u(xi + ∆x)|tn = u|xi,tn + ∆x · ∂u ∂x |xi,tn + (∆x)2 2 · ∂2u ∂x2 |xi,tn + (∆x)3 6 · ∂3u ∂x3 |xi,tn + (∆x)4 24 · ∂4u ∂x4 |ξ1,tn (1) u(xi − ∆x)|tn = u|xi,tn − ∆x · ∂u ∂x |xi,tn + (∆x)2 2 · ∂2u ∂x2 |xi,tn + (∆x)3 6 · ∂3u ∂x3 |xi,tn − (∆x)4 24 · ∂4u ∂x4 |ξ2,tn (2)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

Combining both Eqs, u(xi + ∆x)|tn + u(xi − ∆x)|tn = 2u|xi,tn + (∆x)2 · ∂2u ∂x2 |xi,tn + (∆x)4 12 · ∂4u ∂x4 |ξ3,tn where xi − ∆x ≤ ξ3 ≤ xi + ∆x.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

Short-hand notation: un

i ≡ u(xi, tn) ≡ u(i∆x, n∆t)

un±1

i±1 ≡ u(xi ± ∆x, tn ± ∆t) ≡ u[(i ± 1)∆x, (n ± 1)∆t]

Second-order accurate, central difference approximation to ∂2u

∂x2 |xi,tn:

∂2u ∂2x |xi,tn = un

i+1 − 2un i + un i−1

(∆x)2 + O

• (∆x)2

(3) Likewise, ∂2u ∂2t |xi,tn = un+1

i

− 2un

i + un−1 i

(∆t)2 + O

• (∆t)2

(4)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

Plug the difference approximation of ∂2u

∂2x |xi,tn and ∂2u ∂2t |xi,tn into the

• ne-dimension scalar wave equation:

∂2u ∂t2 = c2 ∂2u ∂x2 , we then obtain the iterative difference equation for the scalar wave un+1

i

= (c∆t)2[un

i+1 − 2un i + un i−1

(∆x)2 ] + 2un

i − un−1 i

+ O[(∆x)2 + (∆t)2] (5)

• Eq. (5) allows us to implement numerical iteration to simulate

propagation of scalar wave.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

1D scalar wave

∆x = xi − xi−1 = xi+1 − xi is the spatial interval (usually expressed in unit of wavelength, λ, e.g. ∆x = λ

10.

∆t time step 1D grid points :

x0

• x1
• x2
• · · ·

xi−∆x

• xi•

xi+∆x

• · · ·

xilast

• Given un

i is known, un+1 i

can be calculated by iteration Say un

i=i0 = sin(nω∆t)

hyperlink to 1Dscalar.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

(Sourceless) Maxwell Equation

Gauss’s Law for electric field

• ∇ ·

D = 0. Gauss’s Law for magnetic field

• ∇ ·

B = 0.

• D = ǫ

E

• D :electric flux density,

E: electrin field σ: electric conductivity, σ∗: equivalent magnetic loss ǫ: electrical permitivity, µ: magnetic permeability

• B = µ

H

• B: magnetic flux density,

H: magnetic field ∂ H ∂t = − 1 µ

• ∇ ×

E − 1 µσ∗ H (6) ∂ E ∂t = 1 ǫ

• ∇ ×

H − 1 ǫ σ H (7)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

(Sourceless) Maxwell Equation

The system of six coupled partial differential equations of the curl operator in Eq. (6) and Eq. (7) forms the basis of the FDTD numerical algorithm for electromagnetic wave interactions with general 3-D objects. ∂Hx ∂t = 1 µ[∂Ey ∂z − ∂Ez ∂y − σ∗Hx] ∂Hy ∂t = 1 µ[∂Ez ∂x − ∂Ex ∂z − σ∗Hy] ∂Hz ∂t = 1 µ[∂Ex ∂y − ∂Ey ∂x − σ∗Hz] (8)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

(Sourceless) Maxwell Equation

∂Ex ∂t = 1 ǫ [∂Hz ∂y − ∂Ey ∂z − σEx] ∂Ey ∂t = 1 ǫ [∂Hx ∂z − ∂Hz ∂z − σEy] ∂Ez ∂t = 1 ǫ [∂Hy ∂x − ∂Hx ∂y − σEz] (9) A general medium is characterized by ǫ(i, j, k), µ(i, j, k), σ(i, j, k), σ∗(i, j, k) A total of 6 fields present in the most general 3D case: Ex, Ey, Ez, Hx, Hy, Hz

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

(Sourceless) Maxwell Equation

The main purpose of FDTD: Given a set of fields Ex, Ey, Ez, Hx, Hy, Hz are known for the entire domain of known spatial structure in terms of ǫ(i, j, k), µ(i, j, k), σ(i, j, k) at time tn times march the fields Ex, Ey, Ez, Hx, Hy, Hz to the next time step, tn+1

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

2D TMz and TEz modes

Assuming: ∂u

∂z =0, where u = {Ex, Ey, Ez, Hx, Hy, Hz}, the 6

partial equations, Eq.(8,9), reduces to ∂Hx ∂t = 1 µ − ∂Ez ∂y − σ∗Hx] ∂Hy ∂t = 1 µ[∂Ez ∂x − σ∗Hy] ∂Ez ∂t = 1 ǫ [∂Hy ∂x − ∂Hx ∂y − σEz] (10) ∂Ex ∂t = 1 ǫ [∂Hz ∂y − σEx] ∂Ey ∂t = 1 ǫ [∂Hx ∂z − σEy] ∂Hz 1 ∂Ex ∂Ey

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

2D TMz and TEz modes

TMz mode: ∂Hx ∂t = 1 µ − ∂Ez ∂y − σ∗Hx] ∂Hy ∂t = 1 µ[∂Ez ∂x − σ∗Hy] ∂Ez ∂t = 1 ǫ [∂Hy ∂x − ∂Hx ∂y − σEz] (12) {Hx, Hy, Ez} propagating on the x-y plane (∂u

∂x , ∂u ∂y = 0)

TEz mode: ∂Ex ∂t = 1 ǫ [∂Hz ∂y − σEx] ∂Ey ∂t = 1 ǫ [∂Hz ∂x − σEy] ∂Hz ∂t = 1 µ[∂Ex ∂y − ∂Ey ∂x − σ∗Hz] (13) {Ex, Ey, Hz} propagating on the x-y plane (∂u

∂x , ∂u ∂y = 0)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

1D sourceless Maxwell Equations

∂u ∂y = 0, ∂Hx ∂t |t=0 = 0,

⇒ x−directed, z−polarised TEM mode: ∂Hy ∂t = 1 µ[∂Ez ∂x − σ∗Hy] ∂Ez ∂t = 1 ǫ [∂Hy ∂x − σEz] {Hy, Ez} propagating along the x-direction ( ∂u

∂x = 0) ∂u ∂y = 0, ∂Ex ∂t |t=0 = 0

⇒ x−directed, y-polarized TEM mode: ∂Ey ∂t = 1 ǫ [∂Hz ∂x − σEy] ∂Hz ∂t = 1 µ[−∂Ey ∂x − σ∗Hz] {Ey, Hz} propagating along the x-direction ( ∂u

∂x = 0)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

1D TEM mode

How to simulate the time marching of the x−directed {Ez, Hy} fields in z-polarised TEM Mode? Yee’s algorithm Discretize the equation ∂Hy ∂t = 1 µ[∂Ez ∂x − σ∗Hy] ∂Ez ∂t = 1 ǫ [∂Hy ∂x − σEz] (14) using central difference method

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

1D Discretisation

Central difference method ∂u ∂t |i,n = u[i∆x, (n + 1/2)∆t] − u[i∆x, (n − 1/2)∆t] ∆t + O[(∆t)2] ∂u ∂x |i,n = u[(i − 1/2)∆x, n∆t] − u[(i + 1/2)∆x, n∆t] ∆x + O[(∆x)2] Semi-implicit approximation u|n

i = u|n+1/2 i

− u|n−1/2

i

2

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

1D Discretisation

After some algrabra, iterative expressions of the fields Ez and Hy is

• btained from Eq. (14):

Ez|n+1/2

i−1/2

= Ca|i−1/2Ez|n−1/2

i−1/2 + Cb|i−1/2(Hy|n i − Hy|n i−1)

Hy|n+1

i

= Da|iHy|n

i + Db|i(Ez|n+1/2 i+1/2 − Ez|n+1/2 i−1

)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

Ca, Cb, Da, Db characterises the medium

Ca|i−1/2 = (1 −

σi−1/2∆t 2ǫi−1/2 )

(1 +

σi−1/2∆t 2ǫi−1/2 )

Cb|i−1/2 = (

∆t ǫi−1/2∆x )

(1 +

σi−1/2∆t 2ǫi−1/2 )

Da|i = (1 − σ∗

i ∆t

2µi )

(1 + σ∗

i ∆t

2µi )

Db|i = ( ∆t

µi∆x )

(1 + σ∗

i ∆t

2µi )

(15) Ca, Cb, Da, Dd encode the physical characteristics of the medium through which the EM wave propages Dispersive ones σ = σ(ω), Dissipative ones, σ = 0 Non-vacuum medium, ǫ > ǫ0 Magnetic medium, µ > µ0. Usually, σ∗ = 0

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

Leap-frog

Note that the fields Hy|n+1

i

, Ez|n+1/2

i−1/2 are evaluted at different grid

points and timestep (e.g. i vs. i ± 1/2, n vs. n ± 1/2) Begin with some given intial profile, e.g. Ez|1/2

i−1/2, i = {iini, iini+1, · · · , iilast} →

Do Loop Hy|1

i for i = {iini, iini+1, · · · , iilast} →

Do Loop Ez|3/2

i−1/2 for i = {iini, iini+1, · · · , iilast} → · · ·

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

Flow chart of Yee FDTD scheme

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

Fields leapfrog in time and space

Figure: Space-time chart of the Yee algorithm for a 1D wave propagation showing the use of central differences for the space derivatives and leapfrog for the time derivatives

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization

Demo of 1D TEM z-polarized propagation

hyperlink to 1DTEMz.nb to see sinusoidal waveform generated by the hard source: Ez[iini − 1/2, n] = sin[2πfn∆t]

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Iterative difference equations for 2D TMz mode

It is straigthforward to generalise the 1D case to 2D As an illustratiohn, take the TMz mode of Eq.(12) The iterative difference for the 2D TMz mode are:

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Iterative difference equations for 2D TMz mode

Ez|n+1/2

i−1/2,j+1/2

= Ca(i − 1/2, j + 1/2)Ez|n−1/2

i−1/2,j+1/2 +

Cb(i − 1/2, j + 1/2)[Hy|n

i,j+1/2 − Hy|n i−1,j+1/2 +

Hx|n

i−1/2,j − Hx|n i−1/2,j+1/2]

Hx|n+1

i−1/2,j+1

= Da(i − 1/2, j + 1)Hx|n

i−1/2,j+1 +

Db(i − 1/2, j + 1)[Ez|n+1/2

i−1/2,j+1/2 − Ez|n+1/2 i−1/2,j+3/2

Hy|n+1

i,j+1/2

= Da(i, j + 1/2)Hy|n

i,j+1/2 +

Db(i, j + 1/2)[Ez|n+1/2

i+1/2,j+1/2 − Ez|n+1/2 i−1/2,j+1/2

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Iterative difference equations for 2D TMz mode

The structure matrices, Ca, Cb, Da, Db are similar to that of Eq. (15) except that now their values depend on i, j.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Courant factor, S

For the numerical model to converge, the time step ∆t has to

• bey the Courant-Freidrichs-Lewy (CFL) stability criterion:

∆t ≤ 1 c

• 1

(∆x)2 + 1 (∆y)2 + 1 (∆z)2

This translates into the choice of ∆t that is tied to ∆x via ∆t = ∆x cS In general, S ≥ √ N, where N is the dimensionality of the problem For the 2D case, S ≥ √ 2

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Procedures

Prepare the memory a 2D domain {iini, · · · i · · · ilast}, {jini, · · · j · · · jlast}, Empty all initial values of the fields, u(i, j) Important: Generate the medium Ca(i, j), Cb(i, j), Da(i, j), Db(i, j) and store them in the memory Generate a point hardsource at the point i0, j0 the point hardsource can be sinusoildal, a Gaussian pulse or anything, e.g. Ez|n

i0,j0 = sin(2πfn∆t) in the n-loop

hyperlink to 2DTMzdemo.nb hyperlink to 2DTMzcontour.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 1D

An EM wave crossing from Medium 1 to Medium 2 will suffer minimal reflection if σM2 ǫM1 = σ∗

M2

µM1 (16) is “perfectly matched” at the boundary. The relationship forces the wave impedence to match with that of the free space - causing reflectionless transmission We artificially place a PML around the outer boundary to absorb all wave fall upon the edges of the boundary

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Reflectionless transmission of a plane wave at a PML/free-space interface

Figure: No reflected EM wave if the perfectly matched Eq. (16)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 1D

One possible piecewise graded PML is as followed:

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 1D

As an example, apply this to the x−directed z−polarised 1D TEM mode, {Ez, Hy} hyperlink: 1DTEMz.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

(Graded) PML can also be implemented in a 2D domain albeit the generalisation is not straight forwardly trivial Oblique incident angles to the PML may not be fully attenuated The conductivity of the PML must have a certain anisotropy characteristic to ensure reflectionless transmission

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Figure: Structure of a 2D PML FDTD grid.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Take the x−directed TEz mode {Ex, Ey, Hz} as illustration To absorb obliquely incident waves, do the following TEz mode: ǫ∂Ex ∂t + σEx = ∂Hz ∂y ǫ∂Ey ∂t + σEy = − ∂Hz ∂x µ∂Hz ∂t + σ∗Hz = ∂Ex ∂y − ∂Ey ∂x

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Split Hz = Hzx + Hzy and rewrite the TEz Maxwell equations ǫ∂Ex ∂t + σyEx = ∂(Hzx + Hzy) ∂y ǫ∂Ey ∂t + σxEy = −∂(Hzx + Hzy) ∂x µ∂Hzx ∂t + σ∗

xHzx

= −∂Ex ∂y µ∂Hzy ∂t + σ∗

yHzy

= −∂Ex ∂y Note that we got 4 difference equations instead of 3

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Descretisation yield 1 E n+1

x

|i+1/2,j = Cay(i, j)E n

x |i+1/2,j + Cby(i, j)(Hn+1/2 zx

|i+1/2,j+1/2 + Hn+1/2

zy

|i+1/2,j+1/2 − Hn+1/2

zx

|i+1/2,j−1/2 − Hn+1/2

zy

|i+1/2,j−1/2) E n+1

y

|i,j+1/2 = Cax(i, j)E n

y |i,j+1/2 + Cbx(i, j)(Hn+1/2 zx

|i−1/2,j+1/2 + Hn+1/2

zy

|i−1/2,j+1/2 − Hn+1/2

zx

|i+1/2,j+1/2 − Hn+1/2

zy

|i+1/2,j+1/2) (17)

1J.P. Berenger, “A perfectly matched layer for the absorption of

electromagnetic waves,” J. Comp. Phys., 114 185 (1994). The following formula are adopted from Sadiku (2001) pg. 185

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Hn+1/2

zx

|i+1/2,j+1/2 = Dax(i, j)Hn−1/2

zx

|i+1/2,j+1/2 +Dbx(i, j)(E n

y |i,j+1/2 − E n y |i+1,j+1/2)

(18) Hn+1/2

zy

|i+1/2,j+1/2 = Day(i, j)Hn−1/2

zy

|i+1/2,j+1/2 +Dby(i, j)(E n

x |i+1/2,j+1 − E n x |i+1/2,j)

(19)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Perfectly Matching Layers (PML) for 2D

Cay(i, j) = e−σy(j)∆t/ǫ(i,j) Cby(i, j) = 1 − eσy(j)∆t/ǫ(i,j) σy(j)∆x Cax(i, j) = e−σx(i)∆t/ǫ(i,j) Cbx(i, j) = 1 − eσx(i)∆t/ǫ(i,j) σx(i)∆x Day(i, j) = e−σ∗

y (j+1/2)∆t/µ(i,j)

Dby(i, j) = 1 − eσ∗

y (j+1/2)∆t/µ(i,j)

σ∗

y(j + 1/2)∆x

Dax(i, j) = e−σ∗

x (i+1/2)∆t/µ(i,j)

Dbx(i, j) = 1 − eσ∗

x (i+1/2)∆t/µ(i,j)

σ∗

x(i + 1/2)∆x

(20)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

2D Perfectly Matching Layers (PML)

Note that in order to implement the 4 iterative difference equations, we first need to generate 8 matrices for the medium Cax(i, j), Cbx(i, j), Dax(i, j), Dbx(i, j), Cay(i, j), Cby(i, j), Day(i, j), Dby(i, j) For C’s,D’s in the active domain, σx,y(i, j), σ∗

x,y(i, j), ǫ(i, j), µ(i, j) assume vacuum values

For C’s,D’s in the surrounding graded PML, σx,y(i, j), σ∗

x,y(i, j), ǫ(i, j), µ(i, j) fulfil the perfectly matched

condition of Eq.(16) hyperlink to 2DPML.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

2D Perfectly Matching Layers (PML)

With the active domain surrounded by the graded PML, the EM waves propagating outwards will be almost perfectly attenuated (thus never reflected back into the active domain) when they hit the edges This simulates an “artifitial infinity” as in most realisitic scenario

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

2D TEz mode from point (hard) source in PML-surrounded domain

Model the propagation of TEz mode EM waves generated by a sinusoidal point (hard) source based on the 4 iterative equations in a PML-surrounded medium Note that now there is no visible reflection from the boundary, thanks to the layered PML hyperlink to 2DPoint3DPML.nb hyperlink to 2DPointCountourPML.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML)

Ilustration: 2D TEz mode from point source interacting with a square block of PEC

Place a square of Perfect Electric Conductor (PEC) at the center of the active domain to interact with the cylindrical wave hyperlink to 2DPointCountour PECbloc PML.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Look-up method to generate 2D EM plane waves

Look-up table method: Our purpose: to generate a 2D plane EM waves (TMz mode for illustration) that move in an arbitrary direction m at an angle φ with respect to x. Look-up table method save us from evaluating the EM fields at each grid points using the iterative difference equations. First, generate a 1D EM waves along a reference straight line at an adjustable angle φ across the domain. The grid points along the 1D reference straight line is indexed by integer m, starting from m0 − 2 to some mmax. The origin m0 coincide with the origin of the 2D grid.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Look-up method to generate 2D EM plane waves

The point source is located at m = m0 − 2, Einc|n

m0−2 = E0 sin(2πfn∆t).

From m = m0 − 2, a 1D Einc|n

m field will propagate along the

• m direction at every m, wherease a 1D Hinc|n

m+1/2 field will

propagate in the m direction at every m + 1/2. The 1D fields along m is equivalent to a 1D TEM m−directed EM waves, with component Hinc|n

m+1/2 and Einc|n m

Don’t forget to apply PML at the mmax end to avoid the 1D reference wave from bouncing back

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Look-up method to generate 2D EM plane waves

At a general grid coordinates {i′, j′}, d can be calculated for that point via d = (i′ − iini) cos φ + (j′ − jini) sin φ. The TEM electric field at that point is denoted Einc|n

d(≡ Einc|n i′,j′)

Einc|n

d can be calculated by taking weigted average of Einc|n m′

and Einc|n

m′+1, where m′ ≤ d ≤ m′ + 1.

Likewise for Hinc|n

d(≡ Hinc|n i′,j′)

From Einc|n

d, Hinc|n d, we then can workout the three incident

fiels in TMz mode via Ez,inc|n

d = Einc|n d,

Hx,inc|n

d = −Hinc|n d sin φ,

Hx,inc|n

d = Hinc|n d cos φ

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Look-up method to generate 2D EM plane waves

hyperlink to incidentplane.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Ez,inc|n, Hx,inc|n, Hy,inc|n are background

The 2D plane waves are called incident waves. Their evolution is based on the m-directed 1D reference wave They are running in the background and are independent from the iterative difference equations of Eqs. (17), (18), (19). In other words, the incident fields can’t be used to interact with the medium Then what good are they for?

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Ideas

The physical (measurable) total electric and magnetic fields

• Etotal =

Einc + Escat

• Htotal =

Hinc + Hscat

• Einc,

Hinc are the incident plane waves that are generated using look-up table as discussed earlier They are assumed to be present at every grid point (actually

• nly the

Einc, Hinc field near the TF/SF boundary will be required ) in the computer momory when simulation of the EM wave propagation through the FDFT grid

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Ideas

Figure: Zoning of total-field /scattered field of the FDTD grid

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Ideas

Divide the FDTD grid into (1) region of total field, (2) region

• f scattered field

Take E n+1

x

|i+1/2,j = Cay(i, j)E n

x |i+1/2,j + Cby(i, j)(Hn+1/2 zx

|i+1/2,j+1/2 + Hn+1/2

zy

|i+1/2,j+1/2 − Hn+1/2

zx

|i+1/2,j−1/2 − Hn+1/2

zy

|i+1/2,j−1/2) as example

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Illustration

For[i = iiniprime, i <= ilastprime, i++, For[j = jiniprime, j <= jlastprime, j++, Ex[i + 1/2, j] = Ex[i + 1/2, j]*Cay[i + 1/2, j] + (Hzx[i + 1/2, j + 1/2]+Hzy[i + 1/2, j+1/2]- Hzx[i + 1/2, j - 1/2]-Hzy[i+1/2,j-1/2])* Cby[i + 1/2, j]; ];];(*end for i, j*)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Illustration

Ex[i + 1/2, j] is understood as scattered field for {i + 1/2, j} in the scattered field region but as total field for {i + 1/2, j} in the total field region (the TF/SF boundary belongs to the TF region) Note that to update the LHS Ex[i + 1/2, j], we need Hz[i + 1/2, j − 1/2] and Hz[i + 1/2, j + 1/2] on the RHS

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Boundary condition for consistency

For iini+1/2 ≤ i’+1/2 ≤ ilast-1/2, no problem when j run from jiniprime till jini-1 since all the fields in the RHS are scattered field However, right at j = jini, a consistency problem arises Right at j = jini, Ex[i′ + 1/2, jini] becomes a total field but Hz[i′ + 1/2, jini − 1/2] is a scattered field

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Figure: Detail of field component locations in a 2D FDTD grid

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Boundary condition for consistency

Re-express Hz,s[i′ + 1/2, jini − 1/2] = Hz,t[i′ + 1/2, jini − 1/2] − Hz,inc[i′ + 1/2, jini − 1/2] Practically, this means that in the FDTD code we add a boundary condition such that whenever j = jini, we have to replace Hz[i′ + 1/2, j − 1/2] by Hz[i′ + 1/2, j − 1/2] − Hz,inc[i′ + 1/2, j − 1/2] after exiting the loop

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Boundary conditions for consistency

For[i = iiniprime, i <= ilastprime, i++, For[j = jiniprime, j <= jlastprime, j++, Ex[i + 1/2, j] = Ex[i + 1/2, j]*Cay[i + 1/2, j] + (Hzx[i + 1/2, j + 1/2]+Hzy[i + 1/2, j+1/2]- Hzx[i + 1/2, j - 1/2]-Hzy[i+1/2,j-1/2])* Cby[i + 1/2, j]; ];];(*end for i, j*) Ex[i + 1/2, jini] = Ex[i + 1/2, jini] - Hzinc[i + 1/2, jini - 1/2]* Cby[i + 1/2, jini]

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Boundary conditions for consistency

There are a total of 8 such boundary conditions for consistency at the TF/SF boundary, i.e. i = iini,ilast, j=jini,jlast Now you see why we need Hinc and Einc near or on the TF/SF boundary These incident plane waves will excite total fields that will interact with the structure within the total field region

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Fields evolution in the absence of structure

In the absence of any structure in the total field region, i.e. the total field region is a vacuum, we should see only a 2D plane waves Ex, Ey, Hz that are exactly as that of incident plance waves Einc,x, Einc,y, Hinc,z The scattered fields in the SF region should void of any perturbations (approximately). However, due to numerical artifact effect, we still see some leakage especially at the sharp corners of the TF/SF In the program that i will show, the leakage is too serious, and this may be caused by some programming mistake when I implemented the 8 boundary conditions (appology) hyperlink to TFSFvacuum.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Revision

1 Prepare PML to surround the FDTD grid 2 Generate a 2D background incident fields Einc, Hinc 3 Use TF/SF method to excite 2D total fields that will interact

with the medium via the interative difference equations of Eqs.(17, 18, Sadiku4). In particular, implementation of the boundary conditions at the TF/SF surface is essential (but the programming could be cubersome) Once you have programmed the Mathematica for procedures as mentioned above, the task left to do is to construct any medium of your interest (by programming the C’s and D’s matrices of Eq. (20)) to investigate how a 2D EM waves interact with the medium constructed.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

EM waves interacting with a square PEC block

A simple example of a structure located within the total field region that will interact with the incident EM plane waves is a square PEC block that will reflect the incident EM waves that fall upon its surface hyperlink to TFSFPECblock.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Further simulation results

hyperlink to TFSFPECwall.nb hyperlink to TFSFsingleslit.nb hyperlink to tangyong.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Varios Medium

hyperlink to medium.nb

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Generalisation to 3D

Generalisation to 3D is possible but probably would be too texting for Mathematica illustrate with an example of a triple loop storing values to a matrix M[i,j,k] for i=[1,200],i=[j,200],k=[1,200] compare.nb c.f. compare.f95 My opinion (may be wrong): Mathematica may be good for graphical illustration but not optimised for numerical intensive programe such as FDTD (esp. for large grid or for 3D cases) - too slow and get ‘saturated’ easily

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Sub-wavelength diffractive optical element with binary profile

Application: Use FDTD to simulate 3D EM waves when interacting with a sub-wavelength diffractive optical element (SWDOE) with binary profile (2D or 3D)

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w

1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications

Sub-wavelength diffractive optical element with binary profile

Subwavelength diffraction optical elements (SWDOEs) with binary features are artificially structured on a suitable optical substrate. Controlling the dimensions of a SWDOE determine whether it will form a polarizer, waveplate or polarisation dependent filter.

Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w