# Determination of single mode condition in dielectric rib waveguide with large cross section by finite element analysis

**Abstract**

The single mode condition in large cross section rib waveguides is of great interest because almost every kind of active and
passive integrated optoelectronic device or sensor is designed to sustain only the fundamental mode of propagation for better
matching with optical fibers. In this paper we present a criterion to determine the single mode condition for a large cross
section rib waveguides, by comparison between the numerical solutions found with Neumann boundary conditions and Dirichlet
boundaries conditions applied when solving the eigenvalues problem.

# Figures

Determination of Single Mode Condition

in Dielectric Rib Waveguides with Large Cross

Section by Finite Element Analysis

M. De Laurentis, A. Irace, G. Breglio

Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni

Università degli Studi di Napoli “Federico II”

Via Claudio, 21I-80125 Naples, ITALY

e-mail: a.irace@unina.it

The single mode condition in large cross section rib

waveguides is of great interest because almost every

kind of active and passive integrated optoelectronic

device or sensor, is designed to sustain only the

fundamental mode of propagation for better matching

with optical fibers.

In this paper we present a criterion to determine the

single mode condition for a large cross section rib

waveguides, by comparison between the numerical

solutions found with Neumann boundary conditions

and Dirichlet boundaries conditions applied when

solving the eigenvalues problem.

I

NTRODUCTION

The main issue when solving the Helmholtz

equation with numerical techniques is that the

numerical solver may find solutions that are not

physical nor related to the geometries of the problem,

but “inspired” by the boundaries conditions. Such

solutions are usually caused by the unavoidable need

to limit the inspection domain to save computational

resources. Sometimes it can be difficult to distinguish

between physical solution and these “spurious”

solutions. Therefore, if we want to investigate the

single-mode condition in rib waveguides, we have

choose a robust criterium to understand weather a

numerical solution is either a guided mode or it is

leaking away from our guiding structure.

The rib waveguide guides modes are supposed to be

well confined nearby the rib region and insensible of

the lateral boundaries, so we suppose the non physical

solutions having larger spatial extension and, for these

reason, they are more sensible to lateral boundary

conditions. Therefore, by changing the rib section

geometrical dimensions, we expect a higher difference

between the eigenvalue of first mode solution found

with Dirichlet boundaries conditions the one found

with Neumann boundaries conditions, when these

solutions become not physical (i.e. the mode is not

longer guided).

S

INGLE MODE CONDITION: FEM ANALYSIS

Along this line of argument, we have developed a

numerical code based on FEMLAB and MATLAB

which, keeping fixed the rib height H, studies the

difference (|n

eff10D

- n

eff10N

|) between the first higher

order mode effective refractive index found with

Dirichlet boundaries conditions (n

eff10D

) and first mode

effective refractive index found with Neumann

boundaries conditions (n

eff10N

), by changing etching

value (i.e. changing the etching complement r, see

Fig.1) for each width-height ratio value, w/H, chosen

between 0.5 and 1.75. This has been done in order to

compare our results with recently published literature

data [1-3].

The typical outcome of this analysis is the plot

reported in the Fig. 2 where we observe, for r<r*, the

quantity |n

eff10D

- n

eff10N

| being essentially negligible,

while for r>r*, the difference

|n

eff10D

- n

eff10N

| increases as expected. The r* value is

what we expect to be the boundary between a single

mode waveguide and a multimode one. In Fig. 3 we

show the comparison between our results, Soref [1]

and Pogossian [2] results.

The analysis, originally performed for TE

polarization, can be extended to the TM case and to

different cross sections in order to evaluate if field

polarization or waveguide geometries affect the single

mode condition as they become comparable to the

wavelength of the propagating field.

R

EFERENCES

[1] R. A. Soref, J. Schimdtchen, K. Peterman, Large single-mode

rib waveguides in GeSi-Si and Si-on-SiO2, Journal of Quantum

electronics, 27 ,8, 1971-1974 (1991)

[2] S. P. Pogossian, L. Vescan, A. Vosonsovici, The single-mode

condition fot semiconductor rib waveguides with large cross

section, Journal of Lightwave technology, 16 ,10, 1951-1955 (1998)

[3] J. Lousteau, D. Furniss, A.B. Seddon, T. M. Benson, P.Sewell,

The single-mode condition fot Silicon-omInsulator optical rib

waveguides with large cross section, Journal of Lightwave

technology, 22,8, 1923 (2004)

11th International Workshop on Computational Electronics

TU Wien, 25-27 May 2006

ISBN 3-901578-16-1

313

Fig. 1. Rib waveguide section. H is the rib height; w the rib

width and r the etching complement.

Fig. 2. Difference between first mode solution found with

Dirichlet boundaries conditions and first mode solution found

with Neumann boundaries conditions. Typically, when this

solutions become not physical (i.e. the mode is not longer

guided) the difference explodes, so we can observe a particular

value of r, r* , so that for r<r*, the quantity |neff10D- neff10N|

being essentially negligible, while for r>r*, the difference

|neff10D- neff10N| increases. The r* value is what we expect

to be the boundary between a single mode waveguide and a

multimode one.

Fig. 3. Comparison between our FEM analysis results (circle),

Soref’s formula [1] and Pogossian et al. results [2]. Above the

curves we define the multi mode region, while below the

single mode region.

314

- CitationsCitations2
- ReferencesReferences12

- "In general , in order to fulfill this requirement, ridge waveguides are employed and designed with different numerical methods such as mode matching method [10] , beam propagation method (BPM) [11], finite element method [12] and effective index method [13]. Design optimization studies on a variety of different material systems including SOI [14], GeSi [10], Si nanocrystal sensitized Er-doped SiO 2 [15] and a range of dielectrics [12] have already been reported. However, these studies do not focus on optimization of single mode a- Al 2 O 3 based waveguide structure and polarization filtering action. "

[Show abstract] [Hide abstract]**ABSTRACT:**Both passive and active, single mode, wavelength and polarization insensitive design of Al2O3 rib waveguides on SiO2 substrate is reported. Influence of the waveguide height, etch depth, waveguide width and operation wavelength to the mode number, mode size, birefringence and polarization sensitivity were analyzed with Beam Propagation Method. Design parameters for targeted properties are computed for waveguide widths ranging from 0 to 10 µm, and for etch depth ranging from 0 to 0.5 µm for fixed waveguide height of 0.5 µm. A design window for a fixed width of 3.5 µm and etch depths between 0.325 to 0.375 µm is identified for single mode, wavelength and polarization insensitive operation of Al2O3 waveguides on thermal oxide. A novel rib TE mode selective filter design is also suggested as an output of the numerical simulations.- [Show abstract] [Hide abstract]
**ABSTRACT:**A z-stretching finite difference method is developed for simulating the paraxial light beam propagation through a lens in a cylindrically symmetric domain. By introducing a domain transformation in the z-direction, we solve the corresponding complex difference equations containing an interface singularity over a computational space for great simplicity and efficiency. A specially designed matrix analysis is constructed to the study the numerical stability. Computational experiments are carried out for demonstrating our results.

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