2 modelling of ridge waveguide bendstexpower.sourceforge.net/gallery/methodoflines.pdf · bessel...

Modelling of Ridge Waveguide Bendsfor Sensor Applications

Wilfrid PascherFernUniversitat, Hagen, Germany

R

n 3

n 1 n 2

Modelling of Ridge Waveguide Bendsfor Sensor Applications

Wilfrid PascherFernUniversitat, Hagen, Germany

R

n 3

n 1 n 2• Radiation losses

• Evanescent field

of a rib waveguide

• are precisely modelled

by the Method of Lines

Why employ the Method of Lines?

The MoL is a semianalytic approach



• Analytic solution in one coordinate direction(perpendicular to the layers)




• Discretization in the other direction(s)(with Finite Differences)=⇒ 3D problem −→ 2D discretization




• Discretization in the other direction(s)(with Finite Differences)=⇒ 3D problem −→ 2D discretization

For reasons of technology, waveguide structures are

• multilayered (e.g., planar waveguides)

• cascaded (e.g., waveguide circuits)

Advantages and Disadvantages

+ precise modeling

+ low memory and computing time

Advantages and Disadvantages

+ precise modeling

+ low memory and computing time

– reduced flexibility

=⇒ different geometries require new algorithms

=⇒ extension to hybrid methods

Transition to Cylindrical Coordinates

1n

n 2

R

n2

1n


1n

n 2

z�

Propagation

�

ϕ�P or� p� i

�a� g� a� t on

R

n2

1n


1n

n 2

z�

Propagation

�

ϕ�P or� p� i

�a� g� a� t on

R

n2

1n

Propagation z → ϕ

exp(−jβz)→ exp(−jνϕ)

Discretization of Straight / Bent Waveguides


x�

Ana

lytic

Sol

utio

n

ψh

ψe

Discretization

y�

Dis

cret

izat

ion

z

ψψ

h

e

Analytic Solution r


x�

Ana

lytic

Sol

utio

n

ψh

ψe

Discretization

y�

Dis

cret

izat

ion

z

ψψ

h

e

Analytic Solution r

Discretization x→ z

Px → Pz


x�

Ana

lytic

Sol

utio

n

ψh

ψe

Discretization

y�

Dis

cret

izat

ion

z

ψψ

h

e

Analytic Solution r

Discretization x→ z

Px → Pz

Analytic solution y → r

sin(kyy)→ Jν(λr)

The Method of Lines (MoL)

for circular bends in waveguides

1. Transition cartesian −→ cylindrical (x, y, z) −→ (z, r, ϕ)




2. Discretization of the wave equation∂2

∂z2 → −Pz

(3. Transformation to diagonal form) → diag (λ2k)





∂z2 → −Pz


4. Solution of the wave equation Jν

(λkr)

+ . . .





∂z2 → −Pz



(λkr)

+ . . .

5. Field computation

(6. Inverse transformation)





∂z2 → −Pz



(λkr)

+ . . .

5. Field computation

(6. Inverse transformation)

7. Characteristic equation in C

−→ radiation loss L ∝ Im(neff )

Wave equations in coordinate free form

Vector MoL with two potentials Πe, Πh

+++ +++ +++ accurate fulfillment of the continuity conditions

for all field components





−→ coordinate free approach:

a) Helmholtz equation for Πh{∆ + εr(z)k2

0

}Πh = 0





−→ coordinate free approach:

a) Helmholtz equation for Πh{∆ + εr(z)k2

0

}Πh = 0

b) Sturm-Liouville differential equation for Πe{∆ + εr(z)k2

0 −1

εr(z)grad εr(z) · div

}Πe = 0

Wave equations in cylindrical coordinates

Potentials with one component in z direction only

Πe,h = k−20 exp(−jνϕ) ψe,h az

• order proportional to effective model index ν = neffR

using normalized coordinates: e.g. R = k0R

Wave equations in cylindrical coordinates

Potentials with one component in z direction only

Πe,h = k−20 exp(−jνϕ) ψe,h az

• order proportional to effective model index ν = neffR

using normalized coordinates: e.g. R = k0R

Consideration of the radiation losses

⇒ neff , ν complex

+++ +++ +++ no artificial increase in the guiding

Discretization of the wave equation

in the cartesian z direction

Partial differential equations in cylindrical coordinates{1r

∂

∂r

(r∂

∂r

)− ν2

r2 + εr(z) +∂2

∂z2

}ψh = 0

{1r

∂

∂r

(r∂

∂r

)− ν2

r2 + εr(z) + εr(z)∂

∂z

(1

εr(z)∂

∂z

)}ψe = 0



Partial differential equations in cylindrical coordinates{1r

∂

∂r

(r∂

∂r

)− ν2

r2 + εr(z) +∂2

∂z2

}ψh = 0

{1r

∂

∂r

(r∂

∂r

)− ν2

r2 + εr(z) + εr(z)∂

∂z

(1

εr(z)∂

∂z

)}ψe = 0

Potentials and dielectric constants

continuous −→ discretized

ψe , ψh −→ Ψe ,Ψh (column vector)

εr(z) −→ εe , εh (diagonal matrix)



Differential operators −→ difference operators

∂2

∂z2 ψh −→ −Pzh Ψh (tridiagonal)

εr(z)∂

∂z

(1

εr(z)∂

∂z

)ψe −→ −Pε

ze Ψe (tridiagonal)



Differential operators −→ difference operators

∂2

∂z2 ψh −→ −Pzh Ψh (tridiagonal)

εr(z)∂

∂z

(1

εr(z)∂

∂z

)ψe −→ −Pε

ze Ψe (tridiagonal)

−→ Coupled ordinary differential equations1r

d

dr

(rd

dr

)I − ν2

r2 I + εe,h −

Pεze

Pzh

︸︷︷︸(tridiagonal)

Ψe,h = 0

Transformation to diagonal form

with Tεe−1

(tridiagonal)︷︸︸︷(εe −Pε

ze) Tεe = λ2

e = diag (λ2e,k)


with Tεe−1


ze) Tεe = λ2

e = diag (λ2e,k)

−→ Decoupled ordinary differential equations{1r

d

dr

(rd

dr

)I − ν2

r2 I + λ2e

}Ψe = 0

with the transformed potential

Ψe = Tεe−1Ψe


with Tεe−1


ze) Tεe = λ2

e = diag (λ2e,k)

−→ Decoupled ordinary differential equations{1r

d

dr

(rd

dr

)I − ν2

r2 I + λ2e

}Ψe = 0

with the transformed potential

Ψe = Tεe−1Ψe

completely analogous for Ψh = Th−1Ψh

Solution of the Bessel differential equation

1r

d

dr

(rd

dr

)+

(λ2e,k −

ν2

r2

) Ψe,k = 0 with ν = neffRb

for one component Ψe,k of the transformed potential


�

� �


�

� �

��

Solution in three regions


�

� �

��


1 finite for r → 0

Ψe,k = Ak Jν(λe,kr)


�

� �

��




2 Ψe,k = Bk Jν(λe,kr)

+ Ck Yν(λe,kr)


�

� �

��




2 Ψe,k = Bk Jν(λe,kr)

+ Ck Yν(λe,kr)

3 radiation condition

Ψe,k = Dk H(2)ν (λe,kr)

Radial derivation matrices Γ

Transmission line equation

∂

∂r

A

B

= Γ

A

B

with Γ =jλ

pν

rν WA

−WB qν

Radial derivation matrices Γ

Transmission line equation

∂

∂r

A

B

= Γ

A

B

with Γ =jλ

pν

rν WA

−WB qν

The radial derivation matrix Γ is computed from the crossproducts and the Wronskian of the Bessel functions

pν = Jν(rA)Yν(rB)− Jν(rB)Yν(rA)

qν = Jν(rA)Y ′ν(rB)− J ′ν(rB)Yν(rA)

rν = J ′ν(rA)Yν(rB)− Jν(rB)Y ′ν(rA)

WA,B =2

πrA,Bwith rA,B = jλrA,B

Programming of the cylinder functions

The problem

1. very high complex order: ν ≈ 105 − 5j• with small imaginary part


The problem


2. order nearly equal to argument:ν ≈ λ1r for the first eigenvalue


The problem



The solution

1, 2 ⇒ Uniform asymptotic series (high argument and order)


The problem



The solution

1, 2 ⇒ Uniform asymptotic series (high argument and order)

Alternative for the ring regions⇒ Multiplication formulas for cross products

Uniform asymptotic expansions

Bessel functions, e.g. Abramowitz-Stegun (9.3.35)

Jν(νy) ∼(

4ζ1− y2

)1/4Ai(ν2/3ζ)

ν1/3

∞∑i=0

ai(ζ)ν2i

+Ai′(ν2/3ζ)ν5/3

∞∑i=0

bi(ζ)ν2i

ν →∞

with 23ζ

3/2 = log1 +

√1− y2

y−√

1− y2



Jν(νy) ∼(

4ζ1− y2

)1/4Ai(ν2/3ζ)

ν1/3

∞∑i=0

ai(ζ)ν2i

+Ai′(ν2/3ζ)ν5/3

∞∑i=0

bi(ζ)ν2i

ν →∞

with 23ζ

3/2 = log1 +

√1− y2

y−√

1− y2

Coefficients ai(ζ), bi(ζ) descending and independent of ν



Jν(νy) ∼(

4ζ1− y2

)1/4Ai(ν2/3ζ)

ν1/3

∞∑i=0

ai(ζ)ν2i

+Ai′(ν2/3ζ)ν5/3

∞∑i=0

bi(ζ)ν2i

ν →∞

with 23ζ

3/2 = log1 +

√1− y2

y−√

1− y2


⇒ Series terms decrease by1ν2≈ 10−10

+++ +++ +++ Series can be truncated already after i = 2

for double precision arithmetic



Jν(νy) ∼(

4ζ1− y2

)1/4Ai(ν2/3ζ)

ν1/3

∞∑i=0

ai(ζ)ν2i

+Ai′(ν2/3ζ)ν5/3

∞∑i=0

bi(ζ)ν2i

ν →∞

with 23ζ

3/2 = log1 +

√1− y2

y−√

1− y2


⇒ Series terms decrease by1ν2≈ 10−10

+++ +++ +++ Series can be truncated already after i = 2

for double precision arithmetic

Yν(νy), J ′ν(νy), Y ′ν(νy) computed analogously

Multiplication theorem

for direct calculation of cross products

Bessel function of outer radius rB = µrA

Cν(rB) = Cν(µrA) = µ−ν∞∑i=0

(δrA)i

i!Cν−i(rA)

with δ = (µ2 − 1)/2 Abramowitz-Stegun (9.1.74)





(δrA)i

i!Cν−i(rA)


yields series for cross products pν , qν pν

qν

= µ−ν∞∑i=0

(δrA)i

i!

Pν,i(rA)

Qν,i(rA)





(δrA)i

i!Cν−i(rA)



qν

= µ−ν∞∑i=0

(δrA)i

i!

Pν,i(rA)

Qν,i(rA)

P , Q are computed from a three term recurrence relationOther cross products rν , sν are determined from pν , qν





(δrA)i

i!Cν−i(rA)



qν

= µ−ν∞∑i=0

(δrA)i

i!

Pν,i(rA)

Qν,i(rA)

P , Q are computed from a three term recurrence relationOther cross products rν , sν are determined from pν , qν

=⇒ Two independent procedures for checking

Field computation

• in every subregion separately• Matching on the cylinders r = rA and r = rB

Field computation


• Transfer of the fields from cylinder r = rA to r = rB HA

HB

=

y1− y2

y2 y1+

EA

−EB

Field computation



HB

=

y1− y2

y2 y1+

EA

−EB

yields HB = YBEB

Field computation



HB

=

y1− y2

y2 y1+

EA

−EB

yields HB = YBEB

with the recurrence YB = y2

(y1− − YA

)−1y2 − y1+

Field computation



HB

=

y1− y2

y2 y1+

EA

−EB

yields HB = YBEB


(y1− − YA

)−1y2 − y1+

In cartesian coordinates matrices y1,2 dependonly on the layer thickness

Field computation



HB

=

y1− y2

y2 y1+

EA

−EB

yields HB = YBEB


(y1− − YA

)−1y2 − y1+

In cartesian coordinates matrices y1,2 dependonly on the layer thickness→ now y1,2 depend on the radii rA, rB→ now y1,2 are computed using cylinder functions

Determination of the radiation losses

Inverse transformation to spatial domain−→ characteristic equation

Z(neff ) HA = 0



Z(neff ) HA = 0

Solution

det(Z(neff )) = 0

by a search for zeros in the complex plane C



Z(neff ) HA = 0

Solution

det(Z(neff )) = 0

by a search for zeros in the complex plane C

−→ Radiation losses (dB/90◦):

L = Im(neff ) · R π10

ln 10

Geometry of a Bent Rib Waveguide Sensor

n 3

n 1 n 2

w

t

h 2

h 3

R

Si 3N4

Si O2

Si

gas

OIWS108A�

Geometry of a Bent Rib Waveguide Sensor

n 3

n 1 n 2

w

t

h 2

h 3

R

Si 3N4

Si O2

Si

gas

OIWS108A�

w = 3÷ 5µm

n1 = 1.989 t = 0.1÷ 0.3µm

n2 = 1.456 h2 = 5µm

n3 = 3.5

Sensitivity

1.5 2 2.5 30

1

2

3

4

5

6

7

8

9

Sen

sitiv

ity in

%

w [µm]

Sensitivity depending on thickness t for w = 3.0 µm

Distribution of the Radial Electric Field

−4 −3 −2 −1 0 1 2 3 4 5 6 7−5

−4

−3

−2

−1

0

1

z [µ

m]

dr [µm]

Fundamental mode, w = 3.0 µm


−4 −3 −2 −1 0 1 2 3 4 5 6 7−5

−4

−3

−2

−1

0

1

z [µ

m]

dr [µm]

First higher order mode, w = 3.50 µm


−4 −3 −2 −1 0 1 2 3 4 5 6 7−5

−4

−3

−2

−1

0

1

z [µ

m]

dr [µm]

First higher order mode, w = 3.13 µm

Conclusion: Advantages and Disadvantages

The presented model yields most accurate results for• propagation constant

• radiation loss



• radiation loss

• sensitivity

• evanescent field



• radiation loss

• sensitivity


Comparison at the TU Delft for a polarization converter:

MoL is superior to the Effective Index Method (EIM)and Finite Element Method (FEM)



• radiation loss

• sensitivity


Comparison at the TU Delft for a polarization converter:

MoL is superior to the Effective Index Method (EIM)and Finite Element Method (FEM)

Disadvantage

• only applicable to strictly rotational structures

⇒ Extension: Combination with other numerical methods

2 modelling of ridge waveguide bendstexpower.sourceforge.net/gallery/methodoflines.pdf · bessel...

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