Advances in spectral methods for optoelectronic design

FACTA UNIVERSITATIS (NI�S)Series: Electronics and Energetics vol. 13, No. 1, April 2000, 73{82

ADVANCES IN SPECTRAL METHODS FOR

OPTOELECTRONIC DESIGN

Ana Vukovic, Steve Greedy, Phillip Sewell,

Trevor M. Benson and Peter C. Kendall

Abstract. Two advances in spectral methods for the fast and accurate

analysis of optical waveguides are reported. The Spectral Index method has

been extended to include arbitrarily shaped rib waveguides in air. Then a Half

Space Radiation Mode method is developed to calculate the facet re ectivity

of 2D waveguides buried at realistic depth. Keywords: optoelectronics, rib

waveguides, facet re ectivity

Key words: Optoelecronics, rib waveguides, facet re ectivity

1. Introduction

With the rapid development of modern optoelectronic circuits there isalso an increased need for analysis techniques which can produce accurateand reliable results. Well proven numerical methods such as the Finite Di�er-ence (FD), Finite Element (FE) and the Beam Propagation Methods (BPM)are very popular since they are generally applicable and produce accurate(benchmark) results [1-3]. However, due to their large memory and timerequirements they often fail within an iterative design environment.

On the other hand, the number of semi-analytical methods is continu-ally increasing. The popularity of these methods stems from their simplicity,ease of implementation and e�ciency, and the accuracy of the results. Lowmemory and time requirements, together with high accuracy, make themindispensable tools when evolving a design. Amongst the �rst methods de-veloped were the E�ective Index (EI) [4] and Weighted Index (WI) methods

Manuscript received March 14, 2000. A version of this paper was presented at the

fourth IEEE Conference on Telecommunications in Modern Satellite, Cables and Broad-

casting Services, TELSIKS'99, October 1999, Ni�s, Serbia.

The authors are with School of Electrical and Electronic Engineering, University of

Nottingham, United Kingdown, e-mail is: [email protected].

73

74 Facta Univ. ser.: Elect. and Energ. vol.13, No.1, April 2000

[5]. Both methods are approximate and e�cient but fail to produce reliableresults for propagation constants and �eld pro�les when the structure is nearcut-o�. The Spectral Index (SI) method [6] is another well established andproven method for solving for the polarised modes of a wide range of op-toelectronic structures based on rib waveguides. Recently, the SI methodhas been enhanced to explicitly include the singular behaviour of the princi-pal �eld component in the vicinity of the re-entrant corners which providesimproved accuracy in calculations of the modal propagation constant [7].

Another well proven technique for analysing optical devices is the FreeSpace Radiation Mode (FSRM) method, originally developed for buriedwaveguide analysis [8]. It has been applied to the analysis of a large varietyof practical waveguides and lasers, allowing for many layers and materialloss and gain [9]. Moreover, the FSRM method has been used to calculatethe facet re ectivity of 1D and 2D waveguides with high accuracy [10,11].The FSRM method assumes that the refractive index di�erence between coreand cladding is small, typically less than 10 %, and then assumes that theradiation modes propagate through a medium of uniform refractive index.

The increasing need to couple general optoelectronic circuits to �bresrequires that waveguides buried at realistic depth from the air-semiconductorboundary are also considered. In such cases the air-semiconductor boundarymakes a strong impact on the �elds in the waveguide, as well as on the facetre ectivity, and can not be neglected.

A novel Half Space Radiation Mode (HSRM) method has been devel-oped for the analysis of shallowly buried waveguides. It has been shownthat propagation constant and the modal �eld pro�les can be signi�cantlya�ected by the presence of the air-boundary [12]. The HSRM method hasalso been extended to calculate the facet re ectivity of shallowly buriedwaveguide structures. Low facet re ectivity is important for the properoperating of optical ampli�ers and when coupling between the guides is re-quired. In the case of shallowly buried waveguides, it is expected that theair- semiconductor boundary and 3D-corner (Fig. 2) will cause increasedmodal re ectivity. Results for facet re ectivity of a 1D waveguide normallyincident on an antire ection (AR) coated facet show that the presence of theair-semiconductor boundary and facet corner can not be neglected. Typi-cally, a semiconductor ampli�er structure with two AR coatings, designedfor �40 dB re ectivity for both TE and TM polarisation, can show an in-crease in re ectivity of up to 10 dB when the waveguide is in the vicinity ofan air boundary [13].

In this paper we will present our latest results from further recent de-velopment of the SI and HSRM methods. The SI method has been extended

A. Vukovi�c et al.: Advances in Spectral Methods for Optoelectronic ... 75

Fig. 1. Modelling oblique rib waveguide with M sections ofconstant height h.

Fig. 2. Side view of a waveguide incident on the facet plane.

to include the practical case of imperfect walls (e.g. the schematic non-rectangular cross- section shown in Fig. 1 which arise from certain manufac-ture processes, both intentionally and unintentionally). Also the extension ofthe HSRM method to account for the e�ect of the air-semiconductor bound-ary and a 3D facet corner (Fig. 2) will be described. Both methods will bedescribed in detail for the case of TE polarisation only. The analysis for theTM polarisation follows similarly.

2. SI Method for Imperfect Rib Geometries

Since the original development of the SI method when it was used tosolve for the propagation constant of the air-clad rib waveguides, the SImethod has been enhanced to include �eld singularities at the re-entrantrib corners [7], identify the radiation spectrum, and has also been appliedto vertical rib couplers and 3D propagation problems. Here we focus on


the extension of the SI method to arbitrarily shaped rib geometries with anair cladding. The versatility of the method primarily lies in its simplicityof formulation and ease of implementation. For the simplest case of a ribguide with guide and cladding layers, the method proceeds in 3 phases: a)application of the concept of e�ective widths by which the �eld penetrationinto the air cladding is modelled [6], b) description of the �eld in the ribin terms of a Fourier series and the �eld in the cladding as a superpositionof plane waves and c) formulation of a transcendental equation for � byapplying the variational boundary condition to match the �eld derivative atthe base of the rib. The method is very e�cient and accurate, producingresults in a matter of seconds.

Consider the general case of arbitrary shaped rib waveguide. The rib issplit into M sections of constant height h and variable width Wi as shownin Fig. 1. The idea of the method is to establish the connection between the�eld at the base of the rib and the �eld at the top of the rib. A transcen-dental equation for the propagation constant � is obtained by enforcing twoconditions: a) the condition that E = 0 at the top of the rib (y =Mh), andb) the continuity of the �elds at the base of the rib (y = 0).

The �eld and its derivative at the base of the rib (y = 0) can be ingeneral case expressed as

E(x; 0) =

NXi=1

V0i

hr 2

W0cos(

n�x

2W0)i

@E(x; 0)

@y=

NXi=1

I0i

hr 2

W0cos(

n�x

2W0)i (1)

or in matrix form as"E(x; 0)@E(x; 0)

@y

#=

"FT

0 0

0 FT

0

#�V 0

I0

�(2)

whereN is the number of terms, the overbar symbol ' ' indicates submatricesand 'T ' indicates the transpose matrix. The �eld and derivative at the nextdiscontinuity (y = h) can be expressed using ABCD matrix notation, i.e.,"

E(x; 0)@E(x; 0)

@y

#=

"FT

0 0

0 FT

0

#�A0 B0

C0 D0

� �V 0

I0

�

=

"FT

0 0

0 FT

0

#"TT

01 0

0 TT

01

#�A0 B0

C0 D0

� �V 0

I0

� (2)


where A0, B0, C0, D0 are diagonal submatrices with elements A0mn =D0mn = cos( rh), B0mn = r sin( rh), C0mn = � sin( rh)= r ,

n = 1; 2; : : : ; N with r =p(k0nr)2 � s2 � �2, where nr is the rib refractive

index. The submatrix T 01 is of order N �N and de�nes the overlapping ofthe �elds in the successive rib sections, with elements as

(T01)i;j =

Z W1

0

dxF0iF0j ; i; j = 1; : : : ; N

Proceeding in the similar manner towards the top end of the rib, we get

"E(x;Mh)@E(x;Mh)

@y

#=

"FT

M 0

0 FT

M

# ��

� �V 0

I0

�(4)

Below the base of the rib it is suitable to de�ne

I0 = Y V 0 (5)

where

Yij =

Z1

0

dsF0i(s)F0j(s)� (s):

F0i;j(s) are Fourier transforms of elements of the submatrix F0 and � (s) isthe plane wave response of the multilayered substrate.

Enforcing the condition E(x;Mh) = 0 and substituting (5) into (4)transcendental equation for � is obtained in the form

j�+ �Y j = 0: (6)

3. HSRM Method for 2D Facet Analysis

The HSRM method deals with the guided mode(s) exactly but approx-imates the radiation modes by assuming that they propagate through amedium of uniform refractive index (nun). The penetration of the opti-cal �eld into the air is modelled by moving the original air-semiconductorboundary to a new one on which the condition E = 0 is set. Thus the �eldsare present only in the half space, hence the name of the method.

Looking at the structure shown in Fig. 2 it can be seen that the �eldin the waveguide (z < 0) is restricted to the half space (y < 0), whereasthe �eld outside the facet (z > 0) is unlimited in the y direction. TheHSRM method solves the problem in Fourier space and applying the 2D


Fourier transform on the �eld in the waveguide will involve convolution,which will signi�cantly degrade the e�ciency of the method. In order toavoid convolution, the 2D Fourier transform is chosen such that along ydirection the Fourier Sine transform (FST) is applied for the �elds insidethe waveguide and the exponential Fourier transform (FT) for the �eldsoutside the facet. Along the x-direction the exponential Fourier transformis applied in both regions. Hence, the electric and magnetic �eld in thewaveguide are

e�(p; s) =ei(p; s) +Rer(p; s) + r(p; s); z < 0 (7)

h�(p; s) =hi(p; s) +Rhr(p; s) + Yrr(p; s): z < 0 (8)

and outside the facet

e+(p; t) =f(p; t); z > 0 (9)

h+(p; t) =Yff(p; t); z > 0 (10)

where p is x-directed Fourier variable and s and t are y-directed Fouriervariables which are in general di�erent.

The �eld in the guide is comprised of incident (ei) and re ected (er)guided modes and the backward scattered radiation spectrum (r). R is themodal re ection coe�cient. The �eld outside the facet is represented as aplane wave spectrum (f). Yr and Yf are admittances, which for the TEpolarisation are

Yr = � 2r + p2

r; Yf =

2f + p2

f;

with r = (nunk0)2 � p2 � s2, f = (k0)

2 � p2 � s2.

The aim is to solve eq.(7-10) for the modal re ection coe�cient R. Thisis done by enforcing the continuity of the electric and magnetic �leds at thefacet plane (z = 0), and by imposing the required orthogonality conditionbetween the radiation and guided modes of the form

Z1

0

ds

Z1

�1

dpr(p; s)hr(p; s) = 0:

However, simply equating eqs.(7) and (9), and eqs.(8) and (10) is notpossible due to the di�erent spectral variables. To get around this problem,we have developed an iterative procedure which starts by estimating a start-ing value for the re ection coe�cient R(0). R(0) is found by solving eq.(7-10)


for a modi�ed structure in which air-semiconductor boundary is extendedalong whole z-range. Thus the spectral variables in both regions (z < 0, andz > 0) are the same and R(0) can be straightforwardly found as

R(0) =

1R0

ds1R�1

dpYf (p; s)ei(p; s)� hi(p; s)

Yf (p; s)� Yr(p; s)hr(p; s)

1R0

ds1R�1

dphr(p; s)� Yf (p; s)er(p; s)

Yf (p; s)� Yr(p; s)hr(p; s)

(11)

An iterative procedure in which the magnetic �eld h(p; t) is reexpressedin terms of p and s variables is then adopted as follows:

f(p; s)IFST! f(x; y)

FFT! f(p; t)

YA(p;t)! h(p; t)

IFFT! h(x; y)

FST! h(p; s)! R(eq:13)

(12)

The re ection coe�cient was obtained using

R =

1R0

ds1R�1

dph(p; s)� hi(p; s)

Yr(p; s)hr(p; s)

1R0

ds1R�1

dphr(p; s)

Yr(p; s)hr(p; s)

(12)

The iterative procedure eq.(12) is repeated until convergence on R isachieved, which in most cases requires no more than 4-5 iterations. Themethod is very fast producing results in a couple of minutes.

4. Results

4.1 SI method

The rib waveguide structure shown in Fig. 1 is analysed. The ribwidth at the base of the rib is W0 = 3 �m and the operating wavelength is� = 1:15 �m. Fig. 3 shows the dependence of the normalised propagationconstant b versus rib height, for di�erent rib wall slopes. It can be seen thatfor small rib heights, the shape of the rib does not in uence the propaga-tion constant, due to the fact that most of the �eld is concentrated in theregion below the rib. However, for the larger rib heights, the rib shape cansigni�cantly in uence the propagation constant and hence the �eld pattern.


Fig. 3. The dependence of the propagation constant b onthe rib height for di�erent rib wall slopes.

4.2 HSRM method

Very low facet re ectivity (< 10�4) is required for the correct operationof semiconductor optical ampli�ers. One of the techniques for reducing facetre ectivity is to place antire ection coatings (AR) at the end of the facet.The �rst structure analysed corresponds to a semiconductor ampli�er withcore and cladding refractive indices of n1 = 3:52 and n2 = 3:2 respectively.The waveguide has width W = 1:5 �m and height H = 0:15 �m and theoperating wavelength is � = 1:3 �m. The facet is coated with one AR coat-ing. Two cases are considered; with the waveguide (1) deeply buried in thesubstrate, (D ! 1), and (2) buried at a depth D = 1 �m. Fig. 4 showscontours of equal re ectivity for the TE polarized mode supported by thewaveguide. Contours of equal re ectivities of �30 and �40 dB are plot-ted against the refractive index of the coating and its normalised thickness(dc=�c), where dc is the coating thickness and �c is the wavelength in thecoating. It can be seen that in order to maintain low re ectivity the ampli-�er structure has to have di�erent coating parameters for the two waveguidegeometries.

The second structure has waveguide dimensions W = 1 �m and H =0:2 �m and refractive indices n1 = 3:517, n2 = 3:17. The facet is coatedwith two AR coatings with parameters nc1 = 1:4431, dc1 = 0:3183 �m,nc2 = 2:5161 and dc2 = 0:1783 �m. Fig. 5 shows the power re ectiv-ity versus wavelength for di�erent buried depths D for both TE and TMpolarisations. It can be seen that with decreasing buried depth D, powerre ectivity increases above the acceptable level (10�4). Thus, when usingshallowly buried guides the AR coating needs to be redesigned.


Fig. 4. The variation of TE mode re ectivity using a singlelayer AR coating for the depth D = 1 �m and D !1.

Fig. 5. Power re ectivity versus wavelength for buried depthsD = 3 �m and D !1, for both TE and TM polarisations.

5. Conclusions

The SI method has been extended to include the case of arbitrary shapedrib geometries. It has been shown that for larger rib heights, the shape ofthe rib can signi�cantly a�ect the modal propagation constant. The HSRMmethod has been developed to calculate facet re ectivity for shallowly buried2D waveguides. It has been shown that in a case of optical ampli�ers, smallburied depth can signi�cantly increase modal re ectivity and can thereforebe detrimental for correct operating.


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Advances in spectral methods for optoelectronic design

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