Boundary conditions - uni-jena.de · Boundary conditions Problems appear if the fields at the boundary have to be evaluated H x E y E z (Finite) Computational domain For keeping the

Boundary conditions

Problems appear if the fields at the boundary have to be evaluated

Hx Ey Ez

(Finite) Computational domain

For keeping the discretized mesh treatable on a computer, we have to limit its size

For a proper determination of the field components that are positioned directly at the boundary of the computational domain, we need actually information about field components outside

But

Choosing proper boundary conditions

36

Boundary conditionsThe easiest boundary conditions: perfectly conducting material (E or H)

The field cannot penetrate the structure

Setting the field values outside the structure equal to zero

Hx Ey Ez

(physical grid)

37

Boundary conditionsFloquet-Bloch periodic boundaries have to be applied for periodic objects (gratings, photonic crystals)

Hx Ey Ez

(physical grid)

ΛIncident plane wave (arbitrary propagation direction)

Floquet-Bloch boundaries in the frequency domain

Boundary conditionsFloquet-Bloch periodic boundaries applied for calculating the band structure of a PC

Geometry of a PC Unit cell with cylindrical inclusion

Periodic boundary with a particular andEInc ∝ eıkxxeıkyyeıkzze−ıωt (200)Ψ̃(x + mΛx, y + mΛy,ω) = Ψ̃(x, y, ω)eıkxmΛxeıkynΛy (201)

ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)

E(x + Λx, y, t) = E(x, y, t)eıkΛx (203)m = n = 1 (204)

L−f = ∂xf − c−1∂tf = 0 (205)L−f = ∂2xtf − c−1∂2ttf + 0.5c(∂2yyf + ∂2zzf) = 0 (206)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)

k3y(fn0,j−1,k − 2fn0,j,k + fn0,j+1,k + fn1,j−1,k − 2fn1,j,k + fn1,j+1,k) + (208)k3z(fn0,j,k−1 − 2fn0,j,k + fn0,j,k+1 + fn1,j,k−1 − 2fn1,j,k + fn1,j,k+1) (209)

k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥

=η2 cos θi − η1 cos θtη2 cos θi + η1 cos θt

(227)

9

EInc ∝ eıkxxeıkyyeıkzze−ıωt (200)Ψ̃(x + mΛx, y + mΛy,ω) = Ψ̃(x, y, ω)eıkxmΛxeıkynΛy (201)

ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

39

Boundary conditionsLunching an arbitrary field distribution and recording the evolving pattern on some discrete points in the space

Initial field distribution Time evolution of the field

m = 3 (305)d = 16∆x (306)

(! = 13, R = 0.3a) (307)(308)

13

40

Boundary conditionsLunching an arbitrary field distribution and recording the evolving pattern on some discrete points in the space

Time evolution of the field

m = 3 (305)d = 16∆x (306)

(! = 13, R = 0.3a) (307)(308)

13


41

Boundary conditions


m = 3 (305)d = 16∆x (306)

(! = 13, R = 0.3a) (307)(308)

13

All the frequencies which do not satisfy the periodic boundaries are annihilated and only the modes that are allowed to propagate persist

Spectra obtained as a FFT

42

Boundary conditions

Band structure computationm = 3 (305)d = 16∆x (306)

(! = 13, R = 0.3a) (307)(308)

13

Spectra obtained as a FFT

Scanning the k-space and tracing the frequencies that persist as modes delivers the band structure via FDTD

43

Boundary conditionsBy neglecting the vectorial aspect, each field component obeys the scalar wave equation

with

11

Boundary conditionsFor a propagation in the +/- x-direction the operators are written as

Wave propagating in the –x-direction

Wave propagating in the +x-direction

Engquist-Madja exact ABC(Mathematics of computation, Vol. 31, 629, 1977)

Direct implementation of the operator is not possible, but the square-root can be expanded as a Taylor-series 12

Boundary conditions

First order approximation Second order approximation

13

Boundary conditions

First order approximation Second order approximation

Nearly plane wave propagating in the x-direction

13

Boundary conditions

First order approximation

Nearly plane wave propagating in the x-direction

Second order approximation

Boundary conditionsWriting the differential operators as finite differences(G. Mur, IEEE Trans. Electromagnetic Compatibility, Vol. 32, 377, 1981)

Discretizing the operator a half spatial step in front of the boundary (example of the boundary at x=0)

Averaging the second time derivatives at x=0 and x=Δx

Same holds for the second time derivatives in y and z direction 15

Boundary conditions


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9

Inserting all those difference scheme leads to


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

9

Example for the boundary x=0, similar equations are used for the others boundaries

Fields have to be stored for 2 different time steps 16

Boundary conditionsSimplification by using only the first order Taylor approximation(skipping the derivatives along the y and z directions)


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

9

Only the fields components that are evaluated at this (most outer) boundary have to be updated with this equation (the tangential components of the E-field e.g. at the boundary x=0)

Reflection coefficients are in the order of 10 -2

Easy to implement 17

Boundary conditionsMost efficient boundary conditions are the PML

Published by Berenger in 1994 with reflections about 3000 times less than with 2nd order Mur boundary

Basic idea: constructing a media which absorbs light and at whose interfaces with the region of interest no reflections taking place

Impedance matching of the medium with the surrounding

Impedance of a medium having both electric and magnetic conductivity


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

(Electrical conductivity)

(Magnetic conductivity)

18

Boundary conditionsLet the loss-less region being region 1, characterized by and


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γ = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

Impedance:


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

If the condition holds, the impedances are equal


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

19

Plane waves propagating in each of the regions are characterized by their propagation constants, given as (by choosing and )


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

k =ω

c

√"µ k =

ω

c

√"µ

√(1− iσe

ω"

) (1− iσm

ωµ

)

And the velocity of propagation is given by

The same as in free space but light is additionally absorbed

k =ω

c

√"µ + iησe

Boundary conditions

The following derivation is done as an example for 2D FDTD in TE polarization

R|| =Er||Ei||

=η2 cos θt − η1 cos θiη2 cos θt + η1 cos θi

(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t

+ σeEy = −∂Hz∂x

(233)

Hz = Hzx + Hzy (234)

µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t

+ σeyEx =∂(Hzx + Hzy)

∂y(237)

&∂Ey∂t

+ σexEx = −∂(Hzx + Hzy)

∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

20absorb light only in one direction, fields shall propagate parallel to the surface

(with )

The reflections coefficients of the interface are given generally by


ψ(x + Λx, y + Λy, t) = ψ(

x, y,ΛxvPhx

+ΛyvPhy

+ t)

(202)



fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) + (207)


k1 =c∆t−∆xc∆t + ∆x

(210)

k2 =2∆x

c∆t + ∆x(211)

k3y =(c∆t)2∆x

2∆y2(x∆t + ∆x)(212)

fn+10,j,k = −fn−10,j,k + k1(f

n+11,j,k + f

n−10,j,k) + k2(f

n1,j,k + f

n0,j,k) (213)

η =√

µ̃

%̃(214)

η =√

µ′ − ıµ′′%′ − ı%′′ (215)

%′′ =σeω

(216)

µ′′ =σmω

(217)

η =√

µ

%(218)

γ = ıω√

µ% (219)% = %′ (220)

µ = µ′ (221)γ = ıω

√µ% (222)

γl = ıω√

µ%

√(1− ıσe

ω%

) (1− ıσm

ωµ

)(223)

σe%

=σmµ

(224)

ηl = η (225)γl = ıω

√µ% + ησe (226)

R⊥ =Er⊥Ei⊥


(227)

9

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

Reflection is 0, only if the angle of incidence = the angle of transmittance

n1 sin θi = n2 sin θt

Boundary conditionsBerenger’s idea: splitting the H field into a x and y component(x-derivative of the E field drives the component and vice versa)

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

Introduction of an anisotropy of all properties

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

R|| =Er||Ei||


(228)

γ1 sin θi = γ2 sin θt (229)η1 = η2 (230)

µ∂Hz∂t

+ σmHz = −(

∂Ey∂x

− ∂Ex∂y

)(231)

&∂Ex∂t

+ σeEx =∂Hz∂y

(232)

&∂Ey∂t


(233)


µ∂Hzx

∂t+ σmxHzx = −

∂Ey∂x

(235)

µ∂Hzy

∂t+ σmyHzx =

∂Ex∂y

(236)

&∂Ex∂t


∂y(237)

&∂Ey∂t


∂x(238)

(239)

10

It can be shown, that the impedance of the Berenger medium equals the impedance of the free space, regardless of the angle of propagation

21

Boundary conditions

Choosing an appropriate absorption profile for and

wx =1− ıσexω#1− ıσmxωµ

(254)

wy =1− ıσeyω#1− ıσmyωµ

(255)

Hzx0 = E0wx cos2 φ

ηG(256)

Hzy0 = E0wy sin2 φ

ηG(257)

η =√

µ

#(258)

Hz0 = Hzx0 + Hzy0 = E0G

η(259)

ηPML =E0Hz0

=η

G(260)

σex#

=σmxµ

(261)

σey#

=σmyµ

(262)

wx = wy = 1 (263)

G =√

wx cos2 φ + wy sin2 φ = 1 (264)

ηPML = η (265)

ψ = ψ0eıωte−ıω1

νG (1−ı σexω$ ) cos φx (266)

e−ıω1

νG (1−ıσeyω$ ) sin φy (267)

ψ = ψ0eıω(t−x cos φ+y sin φ

νG )e−ηG σex cos φx (268)

e−ηG σey sin φy (269)


ν )e−η(σex cos φx+σey sin φy) (270)

e−ıω√

µ# sin φy = e−ıω√

µ#(1−ı σeyω$ ) sin φy (271)σey = 0 (272)σmy = 0 (273)

G = 1 (274)(275)

e−ıω√


µ#(1−ı σeyω$ ) sin φy (276)x = 0 (277)

σey = 0 (278)σmy = 0 (279)

σex =(x

d

)mσemax (280)

(281)

11


(254)


(255)

Hzx0 = E0wx cos2 φ

ηG(256)

Hzy0 = E0wy sin2 φ

ηG(257)

η =√

µ

#(258)


η(259)

ηPML =E0Hz0

=η

G(260)

σex#

=σmxµ

(261)

σey#

=σmyµ

(262)

wx = wy = 1 (263)

G =√

wx cos2 φ + wy sin2 φ = 1 (264)

ηPML = η (265)



e−ıω1







e−ıω√



G = 1 (274)(275)

e−ıω√


µ#(1−ı σeyω$ ) sin φy (276)x = 0 (277)

σey = 0 (278)σmy = 0 (279)

σex =(x

d

)mσemax (280)

(281)

11


(254)


(255)

Hzx0 = E0wx cos2 φ

ηG(256)

Hzy0 = E0wy sin2 φ

ηG(257)

η =√

µ

#(258)


η(259)

ηPML =E0Hz0

=η

G(260)

σex#

=σmxµ

(261)

σey#

=σmyµ

(262)

wx = wy = 1 (263)

G =√

wx cos2 φ + wy sin2 φ = 1 (264)

ηPML = η (265)



e−ıω1







e−ıω√



G = 1 (274)(275)

e−ıω√


µ#(1−ı σeyω$ ) sin φy (276)x = 0 (277)

σey = 0 (278)σmy = 0 (279)

σex =(x

d

)mσemax (280)

(281)

11

Polynomial scaling


(254)


(255)

Hzx0 = E0wx cos2 φ

ηG(256)

Hzy0 = E0wy sin2 φ

ηG(257)

η =√

µ

#(258)


η(259)

ηPML =E0Hz0

=η

G(260)

σex#

=σmxµ

(261)

σey#

=σmyµ

(262)

wx = wy = 1 (263)

G =√

wx cos2 φ + wy sin2 φ = 1 (264)

ηPML = η (265)



e−ıω1







e−ıω√



G = 1 (274)(275)

e−ıω√


µ#(1−ı σeyω$ ) sin φy (276)x = 0 (277)

σey = 0 (278)σmy = 0 (279)

σex =(x

d

)mσemax (280)

2

Boundary conditions

Important: waves propagating along the y-axis are not absorbed in the x-boundary

x

y

z

Region of interest

d=

10∆

x(2

82)

R=

1E−

16(2

83)

PM

L(σ

ex,σ

mx,σ

ey,σ

my)

(284

)P

ML

(σex,σ

mx,0

,0)

(285

)P

ML

(0,0

,σey,σ

my)

(286

)(2

87)

12

d=

10∆x

(282)R

=1E−

16(283)

PM

L(σ

ex ,σ

mx ,σ

ey ,σ

my )

(284)P

ML

(σex ,σ

mx ,0,0)

(285)P

ML

(0,0,σey ,σ

my )

(286)(287)

12

d = 10∆x (282)R = 1E − 16 (283)

PML(σex,σmx,σey,σmy) (284)PML(σex,σmx, 0, 0) (285)PML(0, 0,σey,σmy) (286)

(287)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

12

PEC or periodic

23

Boundary conditions

Splitting the field in the boundaries

Adding the electrical and magnetic conductivity to the equations as material parameters

Only the components normal to the boundary are absorbed, adjusting the proper absorption profile for each component

Electric and magnetic fields are evaluated a half discretization step apart, hence the absorption profile is evaluated likewise at different spatial coordinates

(J.P. Berenger, Journal of Computational Physics, Vol. 114, 185 (2D))

24

Boundary conditions

Extensions for TM polarization and 3D is straight forward

d = 10∆x (282)R = 1E − 16 (283)


(287)

("

∂

∂t+ σey

)Exy =

∂

∂y(Hzx + Hzy) (288)

("

∂

∂t+ σez

)Exz = −

∂

∂z(Hyx + Hyz) (289)

("

∂

∂t+ σex

)Eyx = −

∂

∂x(Hzx + Hzy) (290)

("

∂

∂t+ σez

)Eyz =

∂

∂z(Hxy + Hxz) (291)

("

∂

∂t+ σex

)Ezx =

∂

∂x(Hyx + Hyz) (292)

("

∂

∂t+ σey

)Ezy = −

∂

∂y(Hxy + Hxz) (293)

(294)

("

∂

∂t+ σmy

)Hxy = −

∂

∂y(Ezx + Ezy) (295)

("

∂

∂t+ σmz

)Hxz =

∂

∂z(Eyx + Eyz) (296)

("

∂

∂t+ σmx

)Hyx =

∂

∂x(Ezx + Ezy) (297)

("

∂

∂t+ σmz

)Hyz = −

∂

∂z(Exy + Exz) (298)

("

∂

∂t+ σmx

)Hzx = −

∂

∂x(Eyx + Eyz) (299)

("

∂

∂t+ σmy

)Hzy =

∂

∂y(Exy + Exz) (300)

(301)

12

d = 10∆x (282)R = 1E − 16 (283)


(287)

("

∂

∂t+ σey

)Exy =

∂

∂y(Hzx + Hzy) (288)

("

∂

∂t+ σez

)Exz = −

∂

∂z(Hyx + Hyz) (289)

("

∂

∂t+ σex

)Eyx = −

∂

∂x(Hzx + Hzy) (290)

("

∂

∂t+ σez

)Eyz =

∂

∂z(Hxy + Hxz) (291)

("

∂

∂t+ σex

)Ezx =

∂

∂x(Hyx + Hyz) (292)

("

∂

∂t+ σey

)Ezy = −

∂

∂y(Hxy + Hxz) (293)

(294)

("

∂

∂t+ σmy

)Hxy = −

∂

∂y(Ezx + Ezy) (295)

("

∂

∂t+ σmz

)Hxz =

∂

∂z(Eyx + Eyz) (296)

("

∂

∂t+ σmx

)Hyx =

∂

∂x(Ezx + Ezy) (297)

("

∂

∂t+ σmz

)Hyz = −

∂

∂z(Exy + Exz) (298)

("

∂

∂t+ σmx

)Hzx = −

∂

∂x(Eyx + Eyz) (299)

("

∂

∂t+ σmy

)Hzy =

∂

∂y(Exy + Exz) (300)

(301)

12

25

Boundary conditions

(J.P. Berenger, Journal of Computational Physics, Vol. 127, 363 (3D))26

Inclusion of other materials

FDTD is not directly applicable for materials with (metals)

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)(320)

15

Material properties depend strongly on the wavelength (dispersion)

Inclusion of nonlinear (instantaneous or non-instantaneous) response of the material

Their exist a great diversity of approaches, but they require usually all the simulation of an additional quantity

So far we have taken into account and

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

(325)

15

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

(325)

15

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

(325)

15

Current density

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

(325)

15

Polarization

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

(325)

15

Displacement

27

FDTD for metals

28

Maxwell:

ωP Plasma frequency

1 γ Relaxation time

B. Circular Cylinder near an InterfaceProximity between particles modifies their respective sur-face plasmons.68 This effect is of great interest insurface-enhanced Raman scattering because higher fieldenhancements can be achieved. An irregular particlepresents a rich spectrum of plasmon resonances by itself,and the spectrum resulting from the interaction with an-other object is even more complex. Hence it may be dif-ficult to interpret this spectrum (especially if one takesinto account that plasmons have complex resonance fre-quencies and therefore the modes broaden and overlap).For this reason, we restrict our study to the interaction ofa simple structure (cylinder with circular cross section)with a substrate.

The simulated system is an infinitely long silver cylin-der of radius r ! 25 nm, near the interface between twodifferent media. This interface is the XZ plane, and thecylinder’s axis is parallel to the Z axis (Fig. 8 insets).The medium above the interface is vacuum, whereasbelow the interface, ! ! 2.25. For simplicity, a Drudemodel was used to represent the dielectric constant ofsilver:

!Ag"#$ ! 1 "i%&p

2

2'#"1 # i2'%#$, (2)

with % ! 1.45 $ 10#14 s and &p ! 1.32 $ 1016 s#1. This!Ag(#) function suffices for our purposes, but for more re-alistic results the experimental values of the dielectricconstant should be used. The excitation is a plane wavewith k ! #!k!ey , and the electric field vector is con-tained in the XY plane. To study the influence of the cou-pling strength on the spectrum, we performed the compu-tations for various distances between the interface andthe cylinder. The distance between the interface and thecylinder’s closest point to the interface is h (positive if thecylinder lies in vacuum). In Fig. 8, the scattering crosssection as a function of the exciting field frequency is plot-ted for several values of h.

For h ! %(, a single maximum is found, which shiftsaccording to the background medium. The amplitude ofthe electric field is enhanced by a factor 246 in vacuum(h ! "() and by a factor 167 in a dielectric (h ! #().For h ! "5 nm, the spectrum presents a bump (#0

"5

! 1.26 PHz) in addition to two distinct maxima (#1"5

! 1.3785 PHz, #2"5 ! 1.441 PHz). The field patterns for

#1"5 and #2

"5 are different: The field lines of the firstmode have a four-fold pattern, whereas the second mode

Fig. 7. Comparison of the electric field amplitude computed byusing two different techniques. The plots show the amplitude(normalized to the incident amplitude) along the dotted line (seethe insets). The circles represent Green’s tensor technique withfinite elements, and the curves represent the MMP with theAMS. (a) ) ! 331 nm and (b) ) ! 456 nm.

Fig. 8. Scattering cross section as a function of the frequency:(a) cylinder above the interface and (b) cylinder below the inter-face. The parameter h denotes the distance between the cylin-der and the interface.

Moreno et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. A 107

+ eqn. relating current and electric field (Drude model, mean drift velocity of electrons in a field)

FDTD for metals



!Ag"#$ ! 1 "i%&p

2

2'#"1 # i2'%#$, (2)



"5



#1"5 and #2







!Ag"#$ ! 1 "i%&p

2

2'#"1 # i2'%#$, (2)



"5



#1"5 and #2





28

Maxwell:





!Ag"#$ ! 1 "i%&p

2

2'#"1 # i2'%#$, (2)



"5



#1"5 and #2






Scattering cross section of a circular silver cylinder

(r=25nm, TM)

FDTD for metals

29





!Ag"#$ ! 1 "i%&p

2

2'#"1 # i2'%#$, (2)



"5



#1"5 and #2





Maxwell:


FDTD for Lorentz-materialsAssuming a 2D geometry (y-z-plane) with TM polarization

and three-dimensional PBG studies, which are currentlyin progress. The FDTD approach allows one to obtainthe frequency response of finite PBG structures over awide set of frequencies in a single simulation as well as acomplete visualization of the time evolution of all the as-sociated field and material quantities. Since we are alsocurrently studying nanostructure waveguides formedfrom defects in finite-sized PBG’s, the FDTD approachpermits us to investigate the temporal evolution of thepropagation of the associated electromagnetic guidedwaves.

2. FINITE-DIFFERENCE TIME-DOMAINSIMULATORWe assume that the PBG structure varies only along the zaxis and is uniform on any x–y plane. Thus all the elec-tromagnetic waves are planar, with the electric and mag-netic fields being constant in any x–y plane and the di-rection of propagation being along the z axis. We takethese plane waves to be x polarized (electric field alongthe x axis and magnetic field along the y axis) throughout.

A. Lorentz ModelTo include dispersion in the materials, we introduce theLorentz model for the polarization field P. Since theelectric field has the form E ! Exx̂, the polarization fieldhas the form P ! Pxx̂ and satisfies the equation

!2Px! t2

" "!Px! t

" #02Px ! $0#p

2%LEx , (1)

where " is the damping coefficient, #0 is the resonancefrequency, #p is the plasma frequency, and %L is relatedto the dc value of the electric susceptibility %dc as %dc! (#p

2%L)/#02. The associated frequency domain expres-

sion

Px,# !$0#p

2%L

#02 # #2 " j#"

Ex,# (2)

is obtained by Fourier transform of Eq. (1). Figure 1 in-dicates the relation between the frequency and the refrac-tive index n(#) ! &1 " %(#)'1/2, where the frequency-domain susceptibility %(#) ! Px,# /$0Ex,# when #0 ! #p! #C , " ! 0.01#C and %L ! 1 and where #C acts as a

reference frequency and will be called the center fre-quency. These values will be assumed throughout all ofthe numerical calculations unless otherwise indicated.Referring to Fig. 1, one finds essentially four regions of in-terest: (1) The real part of the index is increasing onlywith the frequency; (2) the real and imaginary parts ofthe index are very large (near resonance); (3) the real partof the index is nearly zero and the imaginary part is large;and (4) the real part of the index is increasing but is lessthan one and the imaginary part is nearly zero.

If #0 $ #, then the polarization field takes a simpleform: Px,# ! ($0#p

2%L /#02)Ex,# . In this dipole approxi-

mation, Px,# ! $0#dcEx,# and the electrical susceptibilityis constant, so the refractive index is ndc ! (1" %dc)1/2. On the other hand, as the frequency ap-proaches infinity, clearly the electrical susceptibility goesto zero. Our interest is mainly in the frequency region# % #C , where these approximations are invalid and thefull model must be used.

B. Finite-Difference Time-Domain FormulationIn the FDTD approach the simulation space (region of in-terest) is discretized into cells of length ( z, and time isdiscretized into intervals of length (t. Using the stan-dard leapfrog in time and the staggered-grid approach,the electric field is taken at the edge of a cell and at inte-ger time steps so that Ex(z, t) ! Ex(k( z, n( t) is repre-sented by Ex

n(k), and the magnetic field is taken at thecenter of the cell at half-integer time steps so thatHy(z, t) ! Hy&(k " 1/2)( z, (n " 1/2)( t' is representedby Hy

n"1/2(k " 1/2). The FDTD formulation for thepropagation of 1D electromagnetic waves in the PBGstructure is then obtained directly from Maxwell’s equa-tions and can be expressed as follows:

Hyn"1/2! k " 12 " ! Hyn#1/2! k " 12 "

#(t)0

Exn*k " 1 + # Ex

n*k +( z

, (3)

Exn"1*k + ! Ex

n*k + #(t$0

& #Hyn"1/2*k " 1/2+ # Hyn"1/2*k # 1/2+( z" Jx

n"1/2*k +$ . (4)The Lorentz model [Eq. (1)] is incorporated

self-consistently11 into Maxwell’s equations by introduc-ing the equivalent first-order system through the defini-tion of the polarization current; i.e.,

Jx !!Px! t

, (5)

! Jx! t

" "Jx ! $0#p2%LEx # #0

2Px . (6)

The electric current and the polarization field are takenat the same spatial location as the electric field, but theelectric current is taken at the same time value as themagnetic field while the polarization field is taken at thesame time value as the electric field; i.e., Jx(z, t) andPx(z, t) are represented, respectively, by Jx

n"1/2(k) andFig. 1. Refractive index versus normalized driving frequency inthe case in which #0 ! #p ! #C , " ! 0.01#C and %L ! 1.

R. W. Ziolkowski and M. Tanaka Vol. 16, No. 4 /April 1999 /J. Opt. Soc. Am. A 931




!2Px! t2

" "!Px! t

" #02Px ! $0#p

2%LEx , (1)



sion

Px,# !$0#p

2%L

#02 # #2 " j#"

Ex,# (2)









Hyn"1/2! k " 12 " ! Hyn#1/2! k " 12 "

#(t)0

Exn*k " 1 + # Ex

n*k +( z

, (3)

Exn"1*k + ! Ex

n*k + #(t$0

& #Hyn"1/2*k " 1/2+ # Hyn"1/2*k # 1/2+( z" Jx



Jx !!Px! t

, (5)

! Jx! t

" "Jx ! $0#p2%LEx # #0

2Px . (6)







!2Px! t2

" "!Px! t

" #02Px ! $0#p

2%LEx , (1)



sion

Px,# !$0#p

2%L

#02 # #2 " j#"

Ex,# (2)









Hyn"1/2! k " 12 " ! Hyn#1/2! k " 12 "

#(t)0

Exn*k " 1 + # Ex

n*k +( z

, (3)

Exn"1*k + ! Ex

n*k + #(t$0

& #Hyn"1/2*k " 1/2+ # Hyn"1/2*k # 1/2+( z" Jx



Jx !!Px! t

, (5)

! Jx! t

" "Jx ! $0#p2%LEx # #0

2Px . (6)




Lorentz dispersion (frequency domain)

(the same as Drude-model but resonance frequency is not at 0)




!2Px! t2

" "!Px! t

" #02Px ! $0#p

2%LEx , (1)



sion

Px,# !$0#p

2%L

#02 # #2 " j#"

Ex,# (2)









Hyn"1/2! k " 12 " ! Hyn#1/2! k " 12 "

#(t)0

Exn*k " 1 + # Ex

n*k +( z

, (3)

Exn"1*k + ! Ex

n*k + #(t$0

& #Hyn"1/2*k " 1/2+ # Hyn"1/2*k # 1/2+( z" Jx



Jx !!Px! t

, (5)

! Jx! t

" "Jx ! $0#p2%LEx # #0

2Px . (6)




Lorentz dispersion (time domain)

Fourier-transformation

(R.W. Ziolkowski et al., JOSA A, Vol. 16, No. 4, 980) 30

FDTD for nonlinear materials

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

h2ω2

c2

(309)

Γ (310)M (311)K (312)0 (313)

= (314)2π

a√

3(315)

2πa3

(316)

ωa/2πc (317)(318)

# < 1 (319)$J (320)$P (321)$D (322)$E (323)$H (324)

$P = #0(ξ(1) $E + ξ(2) $E2 + ξ(3) $E3 + · · ·) (325)

∇× $E + $̇B = 0 (326)

∇× $H = + $̇D +$j (327)∇ $D = $ρ (328)∇ $B = 0 (329)

$D = #0 $E + $P (330)(331)

15

−ê1ê′2ê1ê

′1

[a]h′1h′2h1

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