Modal expansion methods for modeling of photonic...

Modal expansion methods for modeling of photonicdevices

Jirı PetracekBrno University of Technology

Technicka 2, 616 69 Brno, Czech Republicpetracek@fme.vutbr.cz

May 30, 2007

Outline1 Introduction

2 Eigenmode expansion3 Analysis of 1D structure – ε(y)

TE and TM modesTransfer matrix methodOrthogonality relationsE- and H-type modes

4 Propagation in 2D structure – ε(y, z)Field expansionMode matchingS-matrix methodPerfectly matched layer (PML)Bloch modes

5 Modes of 3D waveguide – ε(x, y)Full vector expansion in uniform sectionMode matching and numerical technique

6 Conclusions

Outline1 Introduction2 Eigenmode expansion

3 Analysis of 1D structure – ε(y)TE and TM modesTransfer matrix methodOrthogonality relationsE- and H-type modes

6 Conclusions

Outline1 Introduction2 Eigenmode expansion3 Analysis of 1D structure – ε(y)

6 Conclusions

How does light propagate in photonic device?

Answer: Maxwell equations.Strategy:

we suppose time-harmonic field ∼ exp (iωt) with given ω or λ, i.e.we solve in frequency domain

the device is described with refractive index n(x, y, z) or relativepermittivity ε(x, y, z) – these functions are complex and depend onω

then we solve Maxwell equations to find ~E, c ~B

Maxwell equations

Convention: To keep formulation as simple as possible we use c ~B insteadof ~H and dimensionless coordinates

(x, y, z) =2πλ(X, Y, Z) =

c(X, Y, Z)

β = neff , kvacuum = 1

~∇× c ~B = iε ~E (1)~∇× ~E = −ic ~B (2)

Rigorous methods

Finite-difference

Finite-element

Modal expansion methods (“Mode-matching”)

Method of lines

Spectral index

Tasks to be solved

1 Searching for waveguide modes and propagation constants - “Modesolvers”– stationary state, eigenvalue problem

2 Modeling of light propagation (i.e. evolution of the em field) inphotonic devices (e.g. “BPM” = Beam propagation method)– evolving state

Mode solvers

��

ε(x, y, z) = ε(x, y) (2D task) (3)

ε(x, y, z) = ε(y) (1D task) (4){~E(x, y, z)

c ~B(x, y, z)

{~Eν(x, y)

c ~Bν(x, y)

}exp (−iβνz) (5)

mode profile? propagation constant?

Example: Mode profiles of step-index optical fiber

n1 = 1, 5, n2 = 1, 495, a/λ = 10

Light propagation

– device can change along z– we know incident field i.e. ~E, c ~B at z = 0– we search for ~E, c ~B for any zTwo kinds of methods:(i) one-way methods use paraxial approximation, reflections are neglected(ii) bi-directional methods can deal with reflected field, can calculatereflected and transmitted power and loss.

BPM example

6 Conclusions

Eigenmode expansion

See e.g. [1].Suppose that the waveguide structure is uniform for z ∈ (z1, z2). Generalsolution of the Maxwell equations in (z1, z2) is a sum of forward andbackward-traveling modes: {

~E(x, y, z)c ~B(x, y, z)

=∑ν∈N

{~Eν(x, y)

c ~Bν(x, y)

}exp (−iβνz) + bν

{~E−ν(x, y)

c ~B−ν(x, y)

}exp (iβνz)

6 Conclusions

One-dimensional structure

ε = ε(y) (7)

If we suppose∂

}= 0 (8)

we obtain TE and TM solutions (modes) of the Maxwell equations.

6 Conclusions

TE modes

ϕhk(y)000

βhkϕhk(y)−iϕ′hk(y)

exp (−iβhkz) (9)

ϕ′′hk(y) + ε(y)ϕhk(y) = β2hkϕhk(y) (10)

TM modes

0−βekϕek(y)/ε(y)

iϕ′ek(y)/ε(y)ϕek(y)00

exp (−iβekz) (11)

ε(y)ϕ′ek(y)

]′+ ϕek(y) = β2ek

1ε(y)

ϕek(y) (12)

Eigenvalue problem

Eqs. (10) and (12) can be written in unique form

Lp(y)ϕpk(y) = β2pkηp(y)ϕpk(y) (13)

Lp(y)ϕpk(y) ≡[ηp(y)ϕ

′pk(y)

] ′ + ηp(y)ε(y)ϕpk(y) (14)

ηp(y) ≡{1 p = h1/ε(y) p = e

Note:– β2pk and ϕpk(y) are eigenvalues and eigenfunctions of Lp(y)– for each β2pk there are two modes ±βpk propagating in ±z– to solve the problem in 〈ymin, ymax〉 we need to know boundaryconditions at ymin and ymax

Boundary conditions

Name Condition at ymin or ymax

Open B.C. ϕpk, ϕ′pk are finite

Closed B.C. – Electric wall ~Et = 0 i.e. ϕhk = 0, ϕ′ek = 0

Closed B.C. – Magnetic wall c ~Bt = 0 i.e. ϕ′hk = 0, ϕek = 0

6 Conclusions

Transfer matrix method

is technique for solving Eq.(13) in multilayer waveguide [4, 5, 6, 7]. Thewaveguide consists of N layers. dn and εn are thickness and relativepermittivity of layer n, n = 0..N − 1.

y0ε0, d0

y1ε1, d1

y2ε2, d2

yNεN−1, dN−1

yN−1

Solution in layer n

Choose yn = 0

ϕn(y) = fn exp (−iαny) + bn exp (iαny) (16)

= fn exp [iαn (dn − y)] + bn exp [−iαn (dn − y)] (17)

= An cos (αny) +Bn

αnsin (αny) (18)

= An cos [αn (dn − y)]− Bn

αnsin [αn (dn − y)] (19)

= · · · (20)

αn =√

εn − β2. (21)

Transfer matrix and “matching” conditions

Realizing that An and Bn are values of function ϕ(y) and its derivativeat y = yn and using continuity conditions for normal and tangentialcomponents of electric and magnetic field we obtain(

Bn+1ηn+1

(cos(αndn) sin(αndn)/ (αnηn)

−αnηn sin(αndn) cos(αndn)

) (22)

for any n = 0..N − 1.

Dispersion equation

As a consequence if we know β and An, Bn for some n we can calculatethe all An, Bn. This can be used to find radiation modes and reflectionand transmission coefficients. On the other hand if we know A0, B0 andAN , BN we can obtain implicit dispersion equation for unknown β2. Wechoose layer n and compare A+n , B

+n with A−

n , B−n . A+n , B

+n are

calculated from known A0, B0 and A−n , B

−n are calculated from known

AN , BN . Values of A0, B0, AN and BN depend on type of boundaryconditions used.

∆n(β2) = A+n B−

n −A−n B+n = (B

−n , −A−

(A+nB+n

)= 0 (23)

Root searching

Roots on real axis β2 (bound modes of a waveguide with real indexprofile) are searched using standard numerical routines [8]. Complexroots (leaky modes, active or passive waveguide) are searched using“root-tracking” technique [9] or rigorous technique that uses analyticityof the dispersion function [10, 11, 12, 13], see also [14, 15, 16, 17].

6 Conclusions

Orthogonality relations

For simplicity we suppose closed boundary conditions.ymax∫

ηp(y)ϕpk(y)ϕpl(y)dy = δkl (24)

1β2el

ymax∫ymin

ηe(y)ϕhk(y)ϕ′el(y)dy +

1β2hk

ymax∫ymin

ηe(y)ϕ′hk(y)ϕel(y)dy = 0 (25)

The both relations can be derived from (13) [3, 2]. Note that theserelations are valid for ε complex. (For ε real we can also use relationswith complex conjugate fields.)

6 Conclusions

E- and H-type modes

See e.g. [2]We do not suppose (8). As ε does not depend on x and z the solution ofthe Maxwell equations can be expressed using

~kpk = (kxpk, 0, kzpk) (26)

Instead of (5) we have{~E(x, y, z)

c ~B(x, y, z)

{~Epk(y)

c ~Bpk(y)

}exp (−ikxpkx− ikzpkz) (27)

Choose system (u, y, v) in such a way that ~kpk = (0, 0, βpk)

β2pk = k2xpk + k2zpk (28)

Rotation of TE mode

��

��*

AAAAAAAAAU

AAAAAAAU~k

?kz~ez

��

��*~E

H-modes

Transformation of (9)

kzhkϕhk(y)0

−kxhkϕhk(y)−ikxhkϕ

′hk(y)

β2hkϕhk(y)−ikzhkϕ

′hk(y)

exp (−ikxhkx− ikzhkz) (29)

E-modes

Transformation of (11)

ikxekηe(y)ϕ′ek(y)−β2ekηe(y)ϕek(y)ikzekηe(y)ϕ′ek(y)

kzekϕek(y)0

−kxekϕek(y)

exp (−ikxekx− ikzekz) (30)

6 Conclusions

Propagation in 2D structure – ε(y, z)

The technique is often called “Bi-directional eigenmode expansion andpropagation method” – BEP, [18, 19]

The 2D waveguide is divided into a sequence of M uniform sectionswhich are separated by vertical lines. The refractive index in eachsection is function of y coordinate only. The total field in eachsection is expanded into set of TE or TM modes of that section(= local eigenmodes). The mode spectrum is discretised by closingcomputational domain in y direction by electric or magneticconductors.

(“Mode Matching”) At the interfaces between all neighbouringsections the conditions for continuity of electric and magnetic fieldare used.

Model of 2D structure

- z⊗

ε(m−1)(y)

d(m−1)

ε(m)(y)

ε(m+1)(y)

d(m+1)

z(m) z(m+1)

· · · · · ·

Electric ormagnetic wallsPPPPPi

��

Approximations

1 The continuous part of local eigenmodes spectra has to bediscretised and this is achieved by closing computational domain intransverse direction by suitable artificial boundaries.

2 Spatial resolution of the method depends on the number of localeigenmodes used [20].

3 Staircase approximation is used to model curvature interfaces.

Note that 1 does not apply for devices which are closed or periodic intransverse direction as the local eigenmodes are naturally discrete.However, for open devices this approximation causes serious problemwhich can be solved using various techniques, for example the PerfectlyMatched Layers (PML). Approximations 2 and 3 do not causecomplications in most cases as fast convergence of results with increasingnumber of local eigenmodes or stairs is usually observed.

6 Conclusions

Field expansion in section m

TE solution

Ex(y, z) =∑

uk(z)ϕk(y) (31)

cBy(y, z) = i∑

u′k(z)ϕk(y) (32)

cBz(y, z) = −i∑

uk(z)ϕ′k(y) (33)

TM solution

cBx(y, z) =∑

uk(z)ϕk(y) (34)

Ey(y, z) = −iη(y)∑

u′k(z)ϕk(y) (35)

Ez(y, z) = iη(y)∑

uk(z)ϕ′k(y) (36)

choose z(m) = 0

uk(z) = fk exp(−iβkz) + bk exp(iβkz) (37)

= fk exp [iβk(d− z)] + bk exp [−iβk(d− z)] (38)

= Ak cos(βkz) +Bk

βksin(βkz) (39)

= Ak cos [βk(d− z)]− Bk

βksin [βk(d− z)] (40)

u′k(z) = · · · (41)

6 Conclusions

Mode matching

Compare field at z(m+1):Continuity of Ex or cBx, see (31) or (34)∑

u(m)k (z(m+1))ϕ(m)k (y) =

u(m+1)k (z(m+1))ϕ(m+1)k (y)∑

A(m)k ϕ

(m)k =

A(m+1)k ϕ

(m+1)k (42)

multiply η(m)ϕ(m)l and integrate

ymax∫ymin

A(m)k ϕ

(m)k η(m)ϕ

(m)l dy =

ymax∫ymin

A(m+1)k ϕ

(m+1)k η(m)ϕ

(m)l dy

use (24)

A(m)l =

Q(m,m+1)lk A

(m+1)k (43)

Q(m,n)lk ≡

ymax∫ymin

η(m)ϕ(m)l ϕ

(n)k dy (44)

Alternatively Eq. (42) can be solved with respect to A(m+1)k

A(m+1)l =

Q(m+1,m)lk A

(m)k (45)

Continuity of cBy or Ey

B(m)l =

O(m,m+1)lk B

(m+1)k , (46)

B(m+1)l =

O(m+1,m)lk B

(m)k , (47)

O(m,n)lk ≡

ymax∫ymin

η(n)ϕ(m)l ϕ

(n)k dy. (48)

Relations among overlap integrals

It follows from (43) – (46)[O(n,m)lk

]−1= O

(m,n)lk ≡ Q

(n,m)kl , (49)[

Q(n,m)lk

]−1= Q

(m,n)lk ≡ O

(n,m)kl . (50)

For TE only:

Q(m,n)lk = O

(m,n)lk , (51)

O(m,n)lk = Q

(n,m)kl . (52)

6 Conclusions

S-matrix method

Motivation: solution using transfer matrix is unstable because we addexp (iβkz) and exp (−iβkz) and these can be big and small numbers.The source of instability:

(Big + Small)− Big = 0 not Small!

Solution: S- or R-matrix technique [22]. S-matrix is used in e.g. [19, 21].

What is S-matrix?

It provides relation between amplitudes of output and input modes(f (2)

(t rr t

)(f (1)

)≡ S

(f (1)

-f (1)

�b(1)

-f (2)

�b(2)

-zz(1) z(2)

Composition law

← b(1)

→ f (1)

← b(2)

→ f (2)

← b(3)

→ f (3)

- zS(1) S(2)

← b(1)

→ f (1)

← b(3)

→ f (3)

- zS = S(1) ⊗ S(2)

(f (2)

)= S(1)

(f (1)

(f (3)

)= S(2)

(f (2)

(f (1)

S ≡ S(1) ⊗ S(2) =(

t(2)J−1t(1) t(2)r(1)K−1t(2) + r(2)

t(1)r(2)J−1t(1) + r(1) t(1)K−1t(2)

)J ≡ 1− r(1)r(2), K ≡ 1− r(2)r(1)

Definitions

u(m)(z) = {u(m)k (z)}

f (m) = {f (m)k } b(m) = {b(m)k }

K(m) = {K(m)kk } = {iβ(m)k },

P (m)(d) = exp(−K(m)d

Expression for the field

u(m)(z) = P (m)(z − z(m)

)f (m) + P (m)

(z(m+1) − z

)b(m)[

u(m)(z)]′= −K(m)

[P (m)

(z − z(m)

)f (m) − P (m)

(z(m+1) − z

S-matrix of uniform section

(f (m)

(P (m)

0 P (m)(d(m)

) )( f (m)

)≡ S(m)

(f (m)

S-matrix at boundary

It follows from (45) and (47)

f (m+1) + b(m+1) = Q(m+1,m)(f (m) + b(m)

K(m+1)(f (m+1) − b(m+1)

)= O(m+1,m)K(m)

(f (m) − b(m)

This can be rewritten into form(f (m+1)

)= S(m,m+1)

(f (m)

b(m+1)

S(m,m+1) ≡

(2(K(m)D(+)−1

)T −(D(−)D(+)−1

)TD(+)−1D(−) 2D(+)−1K(m+1)

)D(±) ≡ O(m+1,m)K(m) ±K(m+1)Q(m+1,m)

S-matrix of the whole structure

S = S(0) ⊗ S(0,1) ⊗ S(1) ⊗ S(1,2) ⊗ . . .⊗ S(M−2,M−1) ⊗ S(M−1)

(f (M)

(f (0)

Example: Waveguide with Bragg grating

Calculate modal power reflectance R, transmittance T and lossL = 1−R− T . For details see: www.ure.cas.cz/dpt130/cost268 and[23].

Convergence with increasing number of local modes

Poynting vector

6 Conclusions

How to represent radiation modes?

Continuous spectrum of radiation modes can be discretised using

closed boundary conditions – electric or magnetic walls

leaky modes [24, 25]

direct sampling [26]

adaptive sampling [27, 28]

transparent boundary condition [29] (TBC), originally used in BPM[30]

Perfectly matched layer (PML)

Problem: We should avoid parasitic reflections.PML is an artificial material that can absorb radiation without anyparasitic refection at its interface, regardless of wavelength, incidenceangle or polarisation [31, 32, 33, 34].According to [29], the most efficient way is to use closed boundaryconditions + PML

Complex coordinate stretching

is one form of PML [33, 35] particularly suitable for our method [19, 29].Thickness of PML (d0 or dN−1) is complex. (β2 gets complex, we needto use root-tracking technique in which we change Im(d)).If ε0 = ε1 there are no reflections at y1. The wave inside of PML

exp(−iαy)

is absorbed because y is complex, e.g.

Im(y) =Re(y)Re(d)

Movement of roots during searching of PML modes

Example: calculation with and without PML

6 Conclusions

Bloch modes

Solution of the Maxwell equations in periodic media [36]. If we knowS-matrix of the period with length a then{

{f (0)

}≡ exp (−ikFBa)

{f (0)

γf (0)

γ−1b(M)

(f (0)

t rr t

)(f (0)

)This can be rewritten into (linear) form of the generalized eigenvalueproblem (

t r0 1

)(f (0)

(1 0r t

)(f (0)

). (54)

Bloch modes can be used to improve BEP performance see e.g. [37].

Example: Modeling of 2D photonic crystals

– One of the most exciting developments in physics is discovery ofphotonic crystals [36].– Plane wave method, PWM– Hexagonal lattice, Γ-K, empty pillars in InP with ε=10.5, crystal lengthis 10a, area of holes/area of crystal is 0.4

Dependence R, T on a/λ for TE wave

Electric field profiles

In band-gap a/λ = 0.280.

high T : a/λ = 0.351

Band structure

Example: Line defect

6 Conclusions

Modes of 3D waveguide – ε(x, y)

The technique is often called “Mode matching method” – MMM,[2, 38, 7, 1, 20].The waveguide cross-section is divided into a sequence of M uniformsections which are separated by vertical lines. Each section can be viewedas a part of a 2D waveguide (the refractive index in each section isfunction of y coordinate only) and TE and TM modes (= local modes)of such a 2D waveguide can be found and normalized. The modespectrum is discretised by closing computational domain in y direction byelectric or magnetic walls. The computational domain is not limited in xdirection, except possible walls resulting from waveguide symmetry.MMM is based on the expansion of the unknown modal field into localmodes in each section. Consequently tangential components of themodal field are matched at the section interfaces and this results in anonlinear eigenvalue problem. The spatial resolution of the methoddepends on the number of local mode pairs used [20].

Model of the waveguide cross-section

- x⊙

ε(m−1)(y)

d(m−1)

ε(m)(y)

ε(m+1)(y)

d(m+1)

x(m) x(m+1)

· · · · · ·

Electric ormagneticwalls

PPPPPi

��

6 Conclusions

Full vector expansion in section m

As (8) is not valid we have to use E- and H-modes (29) and (30). Thesemodes can be viewed as “propagating” in ±x directions so that (6) takesthe form{

~E(x, y, z)c ~B(x, y, z)

}=∑pk

{~Epk(y, z)

c ~Bpk(y, z)

}exp (−ikxpkx) +

{~E−pk(y, z)

c ~B−pk(y, z)

}exp (ikxpkx)

Ex(x, y) = kz

uhk(x)ϕhk(y)−1

u′ek(x)ϕ′ek(y) (55)

Ey(x, y) = − 1ε(y)

uek(x)β2ekϕek(y) (56)

Ez(x, y) = −i∑

u′hk(x)ϕhk(y) +ikz

uek(x)ϕ′ek(y) (57)

cBx(x, y) =∑

u′hk(x)ϕ′hk(y) + kz

uek(x)ϕek(y) (58)

cBy(x, y) =∑

uhk(x)β2hkϕhk(y) (59)

cBz(x, y) = −ikz

uhk(x)ϕ′hk(y)− i

u′ek(x)ϕek(y) (60)

kxpk = (β2pk − k2z)

1/2 (61)

upk(x) =Apk

cos(kxpkx) +Bpk

kxpksin(kxpkx) (62)

cos [kxpk(d− x)]−Bpk

kxpksin [kxpk(d− x)] (63)

u′pk(x) = · · · (64)

cos(kxpkd) +Bpk

kxpksin(kxpkd) (65)

Bpk = −kxpkApk

sin(kxpkd) +Bpk cos(kxpkd) (66)

Matrix formulation

diagonal matrices T(m)p and S

(m)p in section m

T(m)pkk ≡

k(m)xpk

β(m)2pk tan

[k(m)xpkd

] (67)

S(m)pkk ≡

k(m)xpk

β(m)2pk sin

[k(m)xpkd

] (68)

rewrite (65) a (66) into stable form used in R-matrix technique [22, 7]

B(m)p = −T (m)p A(m)p + S(m)p A(m)p (69)

B(m)p = −S(m)p A(m)p + T (m)p A(m)p (70)

Boundary conditions at z(0) or z(M)

If A(0)p = 0 or A(M−1)p = 0, Eqs. (69), (70) turn into form

B(M−1)p = −T (M−1)

p A(M−1)p , (71)

B(0)p = T (0)p A(0)p . (72)

If B(0)p = 0 or B(M−1)p = 0, we use again (71), (72), with

T(b)pkk ≡ −

k(b)xpk

β(b)2pk

tan[k(b)xpkd

], b = 0,M − 1.

For open boundary

u(b)′pk = ±ik

(b)xpku

(b)pk , b = 0,M − 1

we use again (71), (72), with

T(b)pkk ≡ ±

k(b)xpk

β(b)2pk

6 Conclusions

Mode matching

A(m)h = O

(m,m+1)hh A

(m+1)h (73)

B(m)h = O

(m,m+1)hh B

(m+1)h − kzO

(m,m+1)he A(m+1)e (74)

A(m)e = O(m,m+1)ee A(m+1)e (75)

B(m)e = O(m+1,m)Tee B(m+1)e + kzO(m+1,m)The A

(m+1)h (76)

Overlap integrals

O(m,n)pplk ≡

ymax∫ymin

η(n)p ϕ(m)pl ϕ

(n)pk dy (77)

Q(m,n)pplk ≡

ymax∫ymin

η(m)p ϕ(m)pl ϕ

(n)pk dy (78)

X(m,n)helk ≡ 1

β(n)2ek

ymax∫ymin

η(n)e ϕ(m)hl ϕ

(n)′ek dy (79)

Y(m,n)helk ≡ 1

β(n)2hk

ymax∫ymin

η(m)e ϕ(m)el ϕ

(n)′hk dy (80)

O(m,n)helk ≡ Y

(n,m)The +X

(m,n)he (81)

Dispersion equation

Using (69), (70) (0 < m < M − 1), (71), (72) and (73), (75) we removeB, B and A from Eqs. (74) and (76). The result is set of nonlinearequations for eigenvector A

(m)p and eigenvalue kz [7]

M(kz)U = 0 (82)

This problem can be solved by searching for roots of determinant ofMwith the inverse iteration technique. The “root tracking” technique wasused to find eigenvalues in the complex plane.

A(M−1)h

A(M−1)e

P(1)h Z

(1)h R

Z(1)e P

(1)e 0 R

W(2)h 0 P

(2)h Z

(2)h R

W(2)e Z

(2)e P

(2)e 0 R

......

W(M−1)h 0 P

(M−1)h Z

(M−1)h

W(M−1)e Z

(M−1)e P

(M−1)e

Dimension ofM is 2(M − 1)L

P (m)p ≡ O(m−1,m)Tpp T (m−1)p O(m−1,m)pp + T (m)p ,

W (m)p ≡ −O(m−1,m)Tpp S(m−1)p ,

R(m)p ≡ −O(m,m+1)pp S(m)p ,

Z(m)h ≡ kzO

(m−1,m)Thh O

(m−1,m)he ,

Z(m)e ≡ −kzO(m−1,m)Tee O

(m,m−1)The

Example: Dominant modes of rib waveguide

Example: Surface plasmon mode

6 Conclusions

Conclusions

Advantages

straightforward, no hidden parameters

no need to generate mesh

accurate, no approximations

can be used for quick estimation

relatively fast

Disadvantages

complicated formulation

nonlinear eigenvalue problem

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