Simulation of integrated optic devices based on BPM

pp. 653-662

Abstract

653

Simulation of integrated optic devices based on BPM

Lara PELLEGRIN01,2, Teresa VIEIRA l , Antonio TEIXEIRA3,4 Paulo ANDRÉ2,3, Paulo MONTEIR01,3,4

The present work shows how to simulate the propagation of optical fields in integrated optics devices, using the Beam Propagation Method, presenting also some results. Finite difference discretization was performed through a Crank-Nicholson algorithm on isotropie media and the software was developed within the Matlab® environment. Structures like ridge waveguide, Y-junctions and parallel waveguides, are simulated and the results are validated through comparison with previously published ones.

Key words: Optical component, Integrated optics, Wave propagation, Numerical simulation, Finite difference method, Optical waveguide, Optical beam, Wave equation, Simulator program.

Résumé

SIMULATION DE DISPOSITIFS OPTIQUES INTÉGRÉS PAR LA MÉTHODE DE PROPAGATION DE FAISCEAU

L'article montre comment simuler la propagation de champs optiques dans les dispositifs d'optique intégrée, en utilisant la méthode de propagation de faisceau, et présente quelques résultats. La discrétisation par différences finies a été effectuée en appliquant l'algorithme de Crank-Nicholson à des milieux isotropes et le logiciel a été développé dans un environnement Mathlab. Des structures, telles que guide à moulure, jonctions en Y et guides d'ondes parallèles, ont été simulées. Les résultats sont validés par comparaison avec des résultats publiés antérieurement.

Mots clés: Composant optique, Optique intégrée, Propagation onde, Simulation numérique, Méthode différence finie, Guide onde optique, Faisceau optique, Équation onde, Programme simulation.

1. Siemens SA, R. Irmàos Siemens - 1, Alfragide, 2720-093 Amadora, Portugal 2. Departamento de Fisica, Universidade de Aveiro - Campus de Santiago, 3810-193 Aveiro, Portugal 3. Instituto de Telecomunicaçôes - Campus de Santiago, 3810-193 Aveiro, Portugal 4. Departamento de Electrônica, Telecomunicaçôes e Informatica, Universidade de Aveiro - Campus de Santiago, 3810-193 Aveiro, Portugal

1/10 ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007

654 L. PELLEGRINO - SIMULATION OF INTEGRATED OPTIC DEVICES BASED ON BPM

Contents

1. Introduction II. Propagation method

III. Implementation description

IV. Numerical results V. Conclusions

References (7 re!)

1. INTRODUCTION

The term "integrated optics" was first proposed in 1960 by S. Miller [1]. It was established to highlight the similarity between planar optical circuits technology and the wellknown integrated electronic circuits. Among its several advantages in performance, when compared with conventional electric methods and bulk optics, it offers low cost and very large scale integration that are two key benefits for telecom operators.

This technology consists on the propagation of infrared electromagnetic waves in waveguide structures. Hence, a way to understand and optimize the devices is to perform their theoretical simulation. More precisely, the Bearn Propagation Method (BPM) is one of the most efficient tools to solve the wave propagation problems. In this work, an implementation of this method is achieved by means of a software package developed in the Matlab® environment with a graphical interface dedicated to simulation of integrated optics devices.

After a theoretical description, the numerical method and the approximations used to solve the propagation equations are described in Section II. In Section III is described the programming options, with available features and sorne graphical interfaces. In Section IV, applications of the model are shown and discussed for sorne important optical integrated structures. Finally, conclusions about the potential and limitations of the model and its implementation are presented.

II. PROPAGATION METHOD

The study of devices in integrated optics is usually performed with the Bearn Propagation Method (BPM) due to its possibility to de scribe non-uniform structures in propagation analysis. There are various kinds of BPM depending on approximations performed or discretization method used [2].

Considering an isotropic medium, the general formulation of the propagation problem includes polarization, thus the fully vectorial wave equation for the electric field (full vectorial BPM) is [3]:

(1) v X V X Ê - n2 k2 Ê = 0

where the wave number in the free space is k = ru ~, with w, eo and 110 being the signal angular frequency, the vacuum permittivity and the vacuum permeability, respectively, and

ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007 2/10

L. PELLEGRINO - SIMULATION OF IN1EGRATED OITIC DEVICES BASED ON BPM 655

n (x, y, z) is the refractive index of the medium. Considering the transversal components of the electrical field (Êt ), we have:

(2) ',;2 E + n2 k2 E - V (V E + aEz ) = 0 t t t t t az

ln a standard waveguide system, an2/az~o, thus, Eq 2 tums into:

(3) aEz + .l... V . (n2 Et)~O az n2 t

Including Eq. 3 into Eq. 2, and separating field transverse components, it cornes:

(4) V2 E + n2 k2 E + jL (alnn2

E + alnn2 E) = 0

x x ax ax x ay y

(5) V2 E + n2 k2 E + jL ( al n n2

E + al n n2 E) = 0

y y ay ax x ay y

The system of coupled vector wave equations for either the transverse electric (Ex and Ey) or magnetic field can be represented by Eq 6 (where Ux and u

y are the envelope of the trans

versal components of the electrical field):

(6)

1

au azx = Axx Ux + Axy uy

auy _ az - AyX Ux + Ayy uy

Axx and Ayy are operators that define polarization dependence, Axy and Ayx represent the coupling between fields. These complex differential operators (Aij' i, j = x, y) are well known [2, 3]. We obtain the semi-vectorial approach, always valid for 2D formulation, ifAX), = Ayx = O. Equations for electric fields and equations for magnetic fields are decoupled, thus they can be solved independently. Vectorial formulation of the electrical fields becomes more important when the refractive index changes fast over the transverse cross-section or the difference at the index discontinuities is large. However, Equation 6 is only valid under the condition that the refractive index along propagation direction (z direction) varies slowly.

If the waveguides are weakly guiding or the polarization is not a critical topic, then wave propagation can be weIl described by scalar approach (Equation 7, Helmholtz equation wideangle (3D»:

(7) au = J..- (a2U + a

2u + ('Pu + (k2_ 7(2)u)

az 2k ax2 ay2 az2

where k is the wavenumber and k is the average phase variation of the field. Assuming that propagation does not occur only near the propagation axis, the formulation

of field propagation can not be made with the paraxial approximation. Ît then needs a wideangle formulation, called the wide-angle BPM.

3/10 ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007


a Solving the quadratic equation Equation 8, in order to have field propagation along z axis, U b . -a ' weo tam: z

(8) ~~ = ik ( 1 a2 a2

1 + k2 (ax2 + ayZ ) + W -p) - 1) U

In a general formulation with polynomials N m and D n' that allows to increase the order of approximation (Padé Order (m, n) in this case) with complexity of structures, the wide angle equation is

(9)

being P the operator defined as

au _ . k Nm(P) az - 1 Dn(P) U

(10) p=...l(L (2) k2 ax2 + ay2 + (k

2 - k2

)

This approach considers only one way traveling waves. If reftections are important, it is necessary to implement the bidirectional BPM, which takes into account the forward and backward propagating waves. In this method, the coupled equations are [4],

(11) (u+ ) (u+ ) out ln

U~ut = M uin

where u+(x, y, z) is the transmitted field, u-(x, y, z) is the reftected field and M is the transfer matrix. This approach presumes that, on one hand, both fields are known at the input of the structure, and, on the other hand, there are uniform regions along z direction (propagation matrices) and interface zones between uniform ones (interface matrices). The coupling occurs at interface regions and waves are decoupled at uniform sections. The propagation matrices propagate independently of the transmitted and of the reftected fields.

If the material is anisotropic, its refractive index is described by a tensor instead of a sc alar. If the material exhibits non linearity, the refractive index is a function of the optical field intensity l.

In case of linear, isotropic and invariant in time electrical field E(x, y, z), the Helmholtz

equation, Eq 7, de scribes field propagation. When ~2~ «~u the paraxial approximation of the Helmholtz equation can be made, leading to z z

(12) au _ i (a2U a2

az - 2k ax2 + ay~ + (k2

- k2) u)

where x and y are the transverse coordinates and z is the propagation direction. This is the approach implemented in our developed software, but in 2D formulation (x and z).

The finite difference BPM (FD-BPM) Cranck Nicholson (eN) scheme was here used to perform the discretization of Eq. 12. It is a method that combines implicit and explicit methods.

ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007 4/10

L. PELLEGRINO - SIMULATION OF INTEGRATED OPTIC DEVICES BASED ON BPM 657

The partial derivates of the differential equations are replaced by approximations that are Taylor expansions and are solved afterward [5]. The eN scheme has the advantage of converging more quickly than an implicit or explicit method.

The electrical field is numerically ca1culated in parallel plans along z propagation direction. The field in the ab ove plan (with a step of L1z between plans) is used to ca1culate the field distribution in CUITent plan, according to Eq. 13.

& & 2(&)2 (E(xi_l' Zj+ 1) + E(xi _1' z) - 2(&)2 (E (Xi + l' Zj+ 1) + E(xi+ l' z)-

& & k2 -2 (13) (&)2 (E(xi, Zj+ 1) + E(xi, z) + 2 ( (Xi' Zj+ 1) - k ) E(xi, Zj+l) +

L1Z 2 -2 '-k 2 (k (Xi' z) - k ) E(xi, z) - 2) (E(xi , Zj+l) + E(xi, z) = 0

This can be written as a tridiagonal matrix form:

EI(z + &) b -a 0 0 K caO 0 K -1 El(z)

Elz+&) -a b -a 0 a caO E2(z)

E3(z + &) o -a b -a o a c a E3(z)

(14)1 M 1= 0 0 M

EN-iZ + &) M -a b-a 0 M a c a 0 EN_2(Z)

EN_l(Z + &) o -a b -a o a c a EN_I(z)

EN(z + &) 0 o -a b o 0 a c EJz)

where

(15) &

a=--2(&?

b = 2j /;; + 2(~)2 - ~ (k2 (xi' Zj+l) - /;;2) ;

- & & -c = - 2)' k + -- - - (k2 (x. z. ) - k2)

2(i1x)2 2 " J+I

In the developed software, electric field propagation is performed using Eq. 15 and Eq. 16. The refractive index differences between the waveguide and sUIToundings are chosen to be superior to 0.1.

5/10 ANN. TÉLÉCOMMUN., 62, nO 5-6, 2007

658 L. PELLEGRlNO - SIMULATION OF INTEGRATED OPTIC DEVICES BASED ON BPM

III. IMPLEMENTATION DESCRIPTION

To support and understand the effects of propagation on a guided media, a program was implemented in Matlab® environment. A tool to allow an easy and fast way to obtain propagation results in waveguides and other integrated structures with clearly understable input physical parameters is targeted. Graphical interface allows easy access to parameters, devices and required outputs.

To simulate with BPM in a simple way, initial information, such as refractive index distribution, incident field properties (intensity, beam width and position) and resolution (step size) is needed. For simulation purposes, it was assumed a refractive index with a step size distribution.

In the implementation, and due to the different requirements of each, sorne predefined geometries (planar waveguide, rib, ridge, channel, Y coupler or direction al coupler) were separated and made available. An user-defined design option is also possible (allowing definition of up to 3 different sections), as displayed on Figure 1.

FIG 1. - Front panel of the developed application, with available simulated structures.

Panneau avant de l'application développée indiquant les structures simulées disponibles.

ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007 6/10


Each structure requires a set of specific parameters, therefore, a new menu appears, requiring the needed inputs, the scheme of the structure and the type of required output, as displayed on Figure 2.

1'tànWiith ~.-t.enQIh Œ§"J

.··Pion" ~

.~ ..

ileamWol

R!t$OUiOn ~

RIb Optan [2D.l4\! ~E- 3

recllon_ ~

SIi"ate

[2J UpdaIe

CoYe,

aetr"'(tionJrJd!"'~ LBack

FIG 2. - Front panel for a rib waveguide, with input parameters and output options.

Panneau avant correspondant à un guide à nervure indiquant les options d'entrée des paramètres et de sortie.

IV. NUMERICAL RESULTS

In Figure 3, a simulation of the field propagation in a 100 /lm length and 2 /lm width waveguide with refractive index 1.44 is displayed. The substrate has refractive index equal to 1.2, the coyer is considered to be air and the electric field has a 1 /lm beam width and centred with the waveguide.

7/10 ANN. TÉLÉCOMMUN., 62, nO 5-6, 2007

660 L. PELLEGRlNO - SIMULATION OF lNTEGRATED OPTIC DEVICES BASED ON BPM

E

011

0.8

Q.4

D.l

o 5

m

-5 0 ZUam)

FIG 3. - Electric field propagation in the waveguide.

Propagation du champ électrique dans le guide d'onde.

Other important integrated optics devices, namely Y branches (Figure 4) and parallel waveguides (Figure 5), are also displayed, with the same electric field input parameters. More precisely, the 5°Y-junction has a 40)lm length and 1 )lm width, with a difference between refractive indexes of waveguide and substrate of 1. The parallel waveguide has a 100 )lm length, 0.5 )lm guide width and 0.5 )lm inter-guide distance .

. )

«l ... ./i. It l1i l!O a 20 Ji 'II

.a 0 z(jm11

(a) (h)

FIG 4. - Y-junction field propagation (a) 3D view (h) 2D view.

Propagation du champ dans une jonction en Y (a) vue tridimensionnelle; (b) vue bidimensionnelle.

ANN. TÉLÉCOMMUN., 62, nO 5-6, 2007 8/10


E

~ '"'

100

l.{J.Im)

(a) (b)

FIG 5. - ParaUel waveguide coupler field propagation (a) 3D view (b) 2D view.

Propagation du champ dans un coupleur à guides d' ondes parallèles (a) vue tridimensionnelle (b) vue bidimensionnelle.

The propagation results of optical fields presented here are roughly comparable with the ones of other simulation programs like the one of Y. Moreau [6,7].

For sake of simplicity, the code is not presented here, however it simply implements the equations described in Section II. It can be seen that with this tool, a view on the evolution of the field along sorne of the most common structures can be obtained. Sorne of the common phenomena can be spotted, eg.: the field shape and amplitude adjustment to the guide in the first part of any of the structures, more clearly noticed in Figure 3; a single-mode Y junction field evolution shows that it can behave, if correctly designed, as an optical splitter, dividing the input signal into two equal parts (for the symmetric case) spatially separated (Figure 4); two parallel dielectric waveguides, which, according to coupled mode theory, if distance between the two waveguides is conveniently chosen, can work as a any ratio spatial coupler, whose coupling ratio depends on the size of the guides, the energy transfers altemately from one waveguide to the other (Figure 5) by a synchronous coherent coupling between the overlap evanescent tails of the guided modes.

Comparisons of this simulation with experimental optical intensity measurement would still be desirable. Nevertheless, improvements need to be performed in arder to take into account effects that were neglected such as polarization, wide-angle formulation and bidirectionality.

9/10 ANN. TÉLÉCOMMUN., 62, n° 5-6, 2007


V. CONCLUSIONS

The BPM method was presented in an extensive and collective way, su ch that, further implementations of the method based on this work can be achieved with minor effort. A package of software was developed, based on the presented equations and approach, being sorne results presented. These c1early agree with the expected results for each of the presented structures.

Acknowledgments

The authors would like to thank Siemens SA for financial support. The support from FEDER, FCT(oreo project-pocr/CTM/59075/2004) is also gratefully acknowledged).

Manuscrit reçu le 10 janvier 2006 Accepté le 28 juin 2006

REFERENCES

[1] MILLER (S. E.), "Integrated Optics: An Introduction", The Bell System Technical Journal, 48, pp. 2059-2068, 1969.

[2] SCARMOZZINO (R.), A. GOPINATH (A.), R. PREGLA (R.), HELFERT (S.), "Numerical Techniques for Modeling Guided-Wave Photonic Devices", J. Select. Top. Quantum Electron., 6 (1), pp. 150-162,2000.

[3] HUANG (W.), Xu (L.), "Simulation of Three-Dimensional Optical Waveguides by a Full-Vector Bearn Propagation Method", J. Quantum Electron., 29 (10), pp. 2639-2649,1993.

[4] RAO (H.), R. SCARMOZZINO (R.), OSGOOD (R.), "A Bidirectional Bearn Propagation Method for Multiple Dielectric Interfaces", Phot. Tech. Lett., 11 (7), pp. 830-832, 1999.

[5] KAWANO (K.), T. KrrOH (T.), "Introduction to Optical Waveguide Analysis", Ist ed., Wiley, 2001. [6] MOREAU (Y.), BRILLOUET (N.), FERROT (S.), K. KRIBICH (K.), P. COUDRAY (P.), ETIENNE (P.), "Simulation rapi

de et accessible sur Internet de dispositifs d'accès multiple en longueur d'onde: phasars et MMI", Proc.Optix 2001, Marseille (France), p. 26-28, Nov. 2001.

[7] http://www.cem2.univ-montp2.fr/-moreau/jBPM.

ANN. TÉLÉCOMMUN., 62, nO 5-6, 2007 10/10

Simulation of integrated optic devices based on BPM

Documents

Transcript of Simulation of integrated optic devices based on BPM