Tooth-shaped grating-assisted structure for compactmultimode interference coupler

Bidyut Deka1,2 and Partha Pratim Sahu1,3

1Department of Electronics & Communication Engineering, Tezpur University,Tezpur-784028, Assam, India

2e-mail: deka.bidyut@gmail.com3e-mail: pps@tezu.ernet.in

Received 8 April 2011; revised 4 August 2011; accepted 9 August 2011;posted 10 August 2011 (Doc. ID 145654); published 23 August 2011

A compact tooth-shaped grating-assisted structure has been proposed for a multimode interference(MMI) coupler and studied theoretically with silica waveguides with silicon oxynitride cores by usinga mathematical model based on the sinusoidal mode simple effective index method. The dependenceof the access waveguide length and beat length of the grating-assisted MMI (GA-MMI) structure onaccess waveguide separation (h) is shown. The total length of the GA-MMI coupler is less than that ofa grating-assisted two-mode interference (GA-TMI) coupler (h ¼ 0) for an access waveguide S bendingloss of 0:2dB. It is also seen that the effect of fabrication tolerance on power imbalance of a GA-MMIcoupler is almost close to that of a GA-TMI coupler. © 2011 Optical Society of AmericaOCIS codes: 130.2790, 230.3120, 250.5300, 230.7390, 350.2770.

1. Introduction

Compact waveguide device components have be-come essential for large-scale integrated opticaldevices such as wavelength division multiplexers/demultiplexers, optical matrix switches, etc., for all-optical networks. The basic components are a two-mode interference (TMI) coupler [1,2], a multimodeinterference (MMI) coupler [3–5], and a directionalcoupler with a small coupling gap [6]. Recently,grating-assisted structure has been introduced in theapplication of TMI [7] couplers in integrated opticsfor its compactness, which is very much required forlarge-scale integration of photonic integrated devices[1–7]. Apart from the above property, it has polariza-tion insensitiveness and requires fewer fabricationsteps. MMI-based devices have become attractivedue to lower access waveguide bending losses thanTMI couplers, and, apart from that, it has propertiessuch as tolerances to the fabrication parameters,polarization insensitiveness, etc. In this direction,

the finite-difference time-domain (FDTD) methodhas already been used by previous authors for studyof tooth-shaped grating-assisted TMI (GA-TMI)couplers, offering little inside the mode propagationanalysis showing polarization insensitiveness,fabrication tolerances, etc. [8,9]. It is seen that veryfew studies are made for grating-assisted MMI(GA-MMI) couplers.

In this paper, we have proposed and studied atooth-shaped grating-assisted structure for an MMIcoupler with a mathematical model based on sinusoi-dal modes using the sinusoidal mode simple effectiveindex method (SM-SEIM) [10–12], which providesthe detailed study of modal power in the tooth-shaped grating-assisted structure of a compact2 × 2 MMI coupler. The dependence of access wave-guide length on h with a fixed value of the S bendingloss for a GA-MMI structure and a tooth-shapedGA-TMI structure is shown. The effect of fabricationtolerance on power imbalance of a GA-MMI coupleris also mentioned. The fabrications of such compactand ultracompact devices with high index contrast(Δn) ∼2–16% are feasible using a silicon on insulator(SOI) waveguide, SiO2=silicon oxynitride (SiON)

0003-6935/11/25E193-07$15.00/0© 2011 Optical Society of America

1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS E193

[13], and InP/GaAsInP [6,14] material. Although abroad range of Δn can be realized using these mate-rials, SOI material has a higher fiber to chip loss(∼23dB=facet) [7] and propagation loss (∼3dB=cm)[7] than SiO2=SiON material [7,15]. In this paper,we have studied tooth-shaped grating-assisted geo-metry in TMI/MMI couplers based on silica wave-guides with SiON cores. We have compared theresults of the structure with conventional MMI cou-plers (without grating) and tooth-shaped GA-TMIcouplers.

2. Design and Principle

Figure 1(a) shows the top view of a 2 × 2 compacttooth-shaped GA-MMI coupler with a channel wave-guide consisting of two single-mode input S bentaccess waveguides of core width a and thickness b(Waveguide-1 and Waveguide-2), two single-modeoutput S bent access waveguides of the same coresize (Waveguide-3 and Waveguide-4), and a couplingregion of length L with alternating guiding width(Wm ¼ 2aþ h, h ¼ the gap between two input accesswaveguides) and grating width (Wg ¼ Wm þ 2ΔW,where ΔW ¼ tooth width of grating). The couplingregion consists of N total numbers of grating periods(Λ ¼ lm þ lg, where lm and lg are the length of guidingwidth (K ¼ m) and grating width (K ¼ g) in eachgrating period, respectively, which are also discussedlater in this section). n1 is the refractive index of thecore and n2 is the refractive index of the claddingregion. When the input power P1 is launched into thelowermost input S bent access waveguide, the outputpowers P3 and P4 are obtained as the bar state andthe cross state, respectively.

When the input signal mode field of propagationconstant βiðλÞ is incident through the input single-mode S bent access waveguide (Waveguide-2), multi-ple modes are excited in the GA-MMI region. At theend of the GA-MMI region based on relative phasedifference between these modes in both guidedand grating region, the light power is coupled intotwo single-mode S bent output access waveguides(Waveguide-3 and Waveguide-4). Since the funda-mental and first-order modes carry most of the signalpower, the beat length (defined as the length for πphase shift) for the MMI couplers assisted with totalN numbers of grating period is obtained as

Lπ ¼ ½ðN þ 1Þlm þNlg�¼ π

½ðβm00 − βm01Þ þ ðβg00 − βg01Þ�; ð1Þ

where βm00, βm01, βg00, and βg01 are the propagation con-

stants of fundamental and first-order modes in theguiding region and grating region, respectively.

Figure 1(b) shows a two-dimensional (2D) cross-sectional schematic view of the 2 × 2 tooth-shapedGA-MMI coupler of Fig. 1(a). As the GA-MMI cou-pling regions in the transverse dimensions (alongthe Y axis) are smaller (a minimum of two times, asmentioned later in Section 3) than the lateraldimensions (along the X axis) and have the sametransverse behavior everywhere of the GA-MMIcoupling region (in the XZ plane), it is justifiably as-sumed that the waveguide structure is to be single-mode in the transverse dimension (it is verified laterin Section 3 with beam propagation method (BPM)results obtained from optiBPM software). So themode fields in GA-MMI couplers can be representedin 2D. The input field profile Hðx; 0Þ launched into atooth-shaped GA-MMI coupler (z ¼ 0) is composed ofmode field distribution of all modes excited in theGA-MMI region, and in 2D approximation, can beexpressed as

Hðx; 0Þ ¼Xr−1i¼0

biHiðxÞ; ð2Þ

where bi is the field contribution coefficient of atooth-shaped GA-MMI coupler for the ith-order modeand HiðxÞ is the mode field distribution of the ith-order mode at z ¼ 0.

The composite field profile at a distance z insidethe GA-MMI region can be represented in 2D approx-imation as a summation of all the guided modes.

Hðx; zÞ ¼Xr−1i¼0

Hiðx; zÞ

¼Xr−1i¼0

biHiðxÞ exp½jðβ0 − βiÞz�; ð3Þ

where i ¼ 0; 1; 2;…; ðr − 1Þ denotes the order ofguided modes, β0 is the propagation constant of

Fig. 1. Schematic diagram of a 2 × 2 tooth-shaped GA-MMIcoupler. (a) Three-dimensional top view. (b) 2D cross-sectional viewwith the x and z axes.

E194 APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

zeroth- order (fundamental mode), and βi representsthe propagation constant for the ith-order mode.

Since the width of the access waveguide (a ≈

1:5 μm) is required to be small for single-mode opera-tion of the access waveguide by keeping the normal-ization frequency V ≈ 2:3, the lateral penetration ofthe mode field outside the waveguide is negligiblefor the lateral high-index contrast. Thus, the inputmode field profile HiðxÞ for the ith mode can be ap-proximated for the tooth-shaped GA-MMI region as

HiðxÞ ¼ sin�ðiþ 1Þ πx

Wg

�: ð4Þ

At the end of the tooth-shaped GA-MMI coupling sec-tion, optical power is either transferred to the outputS bent access waveguide or lost out of the end of thetooth-shaped GA-MMI waveguide. The mode fieldof output access waveguides is contributed by allguided modes propagated in the GA-MMI region.The mode fields at the third S bent access waveguidecan be written as

H3ðx;LÞ ¼Xr−1i¼0

H3;iðx;LÞ

¼Xr−1i¼0

c3;iHiðxÞ exp½jðβ0 − βiÞL�; ð5Þ

where L ¼ ½ðN þ 1Þlm þNlg� and c3;i ¼ffiffiffiffiffiffiffiffiC3;i

p ¼ theith-order mode’s contribution coefficient (forWaveguide-3), which can be calculated by using amathematical model based on sinusoidal modes andthe SM-SEIM [12] as

C3;i

C0≈

π216b2k2ðn2

1 − n22Þ

× expf−hkðn2eff − n2

2Þ1=2g

× ½expfbkðn21 − n2

2Þ1=2g− expf−bkðn2

1 − n22Þ1=2g�; ð6Þ

where

C0 ¼ 0:4FC

×ðn2

1 − n2eff Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2eff − n2

2

qneff ðn2

1 − n23Þ�Wm þ 2

k0ffiffiffiffiffiffiffiffiffiffiffin2eff−n

22

p� ; ð7Þ

Fc ¼3ð1þ 0:2hÞ

f13:5þ 185ðβ00m − βimÞgh; ð8Þ

neff ¼ βKi� λ2π

�; K ¼ m; g: ð9Þ

The contributed power to the third S bent accesswaveguide by the ith-order mode is given by

Pi3 ¼ jH3;iðx;LÞj2: ð10Þ

Normalized power coupled to Waveguide-3 (as a barstate) for the tooth-shaped GA-MMI coupler can beapproximated as

P3;iðx;LÞP1;iðx; oÞ

¼

���Pr−1i¼0 H3;iðx;LÞ

���2���Pr−1i¼0 H1;iðx; 0Þ

���2 ≈

Xr−1i¼0

C3;iH2i ðxÞ

þXr−1i¼0

Xr−1j¼1þi

2642 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C3;iC3;j

qHiðxÞHjðxÞ

× cos

8><>:

Xr−1i¼0;j¼iþ1K¼m;g

½ðN þ qKÞðβKi − βKj ÞlK �9>=>;375; ð11Þ

where i; j ¼ 0; 1; 2; 3;…; ðr − 1Þ are the order ofthe modes, provided j > i; qK ¼ 0; 1 for thegrating region (K ¼ m) and guided region (K ¼ g),respectively; N ¼ number of grating periods; C3;i,C3;j ¼ contribution coefficients (measure of fieldcontribution of the ith- and jth-order modes tolower output access waveguides); and βi, βj ¼propagation constants for the ith- and jth-ordermodes, which are determined from dispersiveequations [16]. The length of the guiding width(lm) and grating width (lg) can be determined usingEq. (12) [8,9].

lK ¼ λ4neffðj;KÞ

; K ¼ m; g: ð12Þ

Normalized power coupled to Waveguide-4 (as across state) for the tooth-shaped GA-MMI couplercan be approximated as

P4;iðx;LÞP1;iðx; oÞ

¼

���Pr−1i¼0 H4;iðx;LÞ

���2���Pr−1i¼0 H1;iðx; 0Þ

���2 ≈

Xr−1i¼0

C4;iH2i ðxÞ

þXr−1i¼0

Xr−1j¼1þi

2642 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C4;iC4;j

qHiðxÞHjðxÞ

× cos

8><>:

Xr−1i¼0;j¼iþ1K¼m;g

½ðN þ qKÞðβKi − βKj ÞlK �9>=>;375; ð13Þ

where C4;i ¼ c24;i and c4;i ¼ the contributioncoefficient of the ith mode, which can be calculated

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by using a mathematical model based on SM-SEIM[12], for output access Waveguide-4.

The transition length (LT) of the S bent accesswaveguide (along the z direction) from Fig. 1(b)can be obtained as [7]

LT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�H −

h2

��4Rþ h

2−H

�;

sð14Þ

where R, H, and h are bending radius, height,and coupling gap between two access waveguides,respectively. The bending loss (TS) in S bent accesswaveguides can be approximated as [15,17]

TS ¼ 4:343αS; ð15Þ

where α ¼ the loss coefficient that mainly dependson bending radius R and

S ¼ 2R cos−1�1 −

ðH − h=2Þ2R

�: ð16Þ

3. Results and Discussions

Figure 2 shows the normalized coupling power distri-bution for the bar coupling (P3=P1) state and thecross coupling (P4=P1) state versus the number ofgratings (N) obtained by using Eqs. (1) and (11)–(13)for different waveguide separation gaps, h ¼ 0:0,3.0, 4.0, 5.0, and 6:0 μm for the tooth-shapedGA-MMI coupler with ΔW ¼ 0:05 μm, a ¼ 1:5 μm,b ¼ 1:5 μm, lm ≈ 0:26 μm, lg ≈ 0:26 μm,Δn ¼ 5%, clad-ding index ≈ 1:45, and wavelength (λ) ≈ 1:55 μm. InFig. 2, h ¼ 0:0 μm corresponds to the TMI coupler.It is seen from the figure that the peak cross-couplingpower (P4=P1) is obtained at beat lengths where the

N values are 41, 70, 85, 105, and 134 for h ¼ 0:0, 3.0,4.0, 5.0, and 6:0 μm, respectively. So the beat lengthsobtained using Eq. (1) are ∼22:2, 36.0, 40.0, 57.8, and70:5 μm for h ¼ 0:0, 3.0, 4.0, 5.0, and 6:0 μm, respec-tively. The increase of beat length with the increaseof h is mainly due to excitation of higher-order modes(apart from lower-order modes) having less couplingefficiency as these modes are partly transferred. It isseen (not mentioned in the figure) that the number ofmodes excited in GA-TMI and GA-MMI couplers forh ¼ 0:0, 3.0, 4.0, 5.0, and 6:0 μm is two, four, five, andsix, respectively. It is evident from the figure that ash decreases, the peak normalized cross-couplingpower decreases. This is due to increase of the radia-tion losses at the bending portion of the input/outputaccess waveguides with the decrease of h, which isevident from Eqs. (15) and (16), respectively. It is alsoseen (not shown in the figure) that the polarizationdependence of the GA-MMI coupler is almost equiva-lent to the GA-TMI coupler, but is slightly more thanconventional MMI/TMI couplers because the numberof waveguide parameters in the grating-assisted geo-metry is more than that of conventional structures.

We have also compared coupling power distri-bution (obtained by using the SM-SEIM) with lightwave propagation results obtained by usingOptiBPM software (version 9.0) based on the FDTDmethod. Figure 3 shows the light wave propagationat the cross-coupling point (at the beat length) for(a) the tooth-shaped GA-TMI coupler with ΔW ¼0:05 μm, h ¼ 0 μm; (b) the conventional TMI couplerwith ΔW ¼ 0 μm, h ¼ 0 μm; (c) the tooth-shapedGA-MMI coupler with ΔW ¼ 0:05 μm, h ≠ 0 μm; and(d) the conventional MMI coupler with ΔW ¼ 0 μm,h ≠ 0 μm. It is seen that the beat length for theGA-TMI, GA-MMI, conventional TMI coupler, andconventional MMI coupler is obtained as ∼22:3,36.2, 45.6, and 73:5 μm, respectively, which arealmost close to that obtained with the SM-SEIMmethod. It is also evident from the figures that thepropagation loss in the GA-TMI/GA-MMI regionis slightly more than that in the conventional TMI/MMI coupler due to multiple reflections in thegrating region.

Figure 4 shows the beat length (Lπ) versus indexcontrast (Δn) of the tooth-shaped GA-MMI couplerfor ΔW ¼ 0:05, 0.1, and 0:25 μm, and the conven-tional MMI coupler (ΔW ¼ 0 μm) with a ¼ 1:5 μm,b ¼ 1:5 μm, h ¼ 3:0 μm, Wmð∼2aþ hÞ ¼ 6:0 μm, clad-ding index ≈ 1:45, and wavelength ≈ 1:55 μm.

It is observed from the plot that as the indexcontrast (Δn) increases, the beat length decreases,and it slowly decreases for Δn > 5%. The variationof the beat lengths with Δn for ΔW ¼ 0:05 μm arealmost close to that for ΔW ¼ 0:1 and 0:25 μm, butthe beat length for a conventional MMI coupler isapproximately two times higher than that for atooth-shaped GA-MMI coupler (ΔW ≠ 0 μm). So wehave chosen Δn ¼ 5% and ΔW ¼ 0:05 μm for furtherstudy. It is seen that forΔn ¼ 5% andΔW ¼ 0:05 μm,the beat length of tooth-shaped GA-MMI couplers is

Fig. 2. Normalized coupling power distribution of tooth-shapedGA-MMI couplers for h ¼ 0:0, 3.0, 4.0, 5.0, and 6:0 μm withΔW ¼ 0:05 μm, a ¼ 1:5 μm, b ¼ 1:5 μm, cladding index ≈ 1:45,Δn ¼ 5%, and λ ≈ 1:55 μm.

E196 APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

∼50% lower than conventional MMI couplers. Theless beat length in a GA-MMI coupler than that ofa conventional MMI coupler is due to the multiplereflections that occurred in the tooth-shaped gratingregion.

The dependence of transition length LT and beatlength Lπ on h of the MMI structure with tooth-shaped grating is studied by considering a fixed Sbending loss TS of 0:2dB with Eqs. (1), (14), and (15),as shown in Fig. 5. In the figure, h ¼ 0 μm corre-sponds to a TMI coupler with tooth-shaped grating,while LT and Lπ are obtained as 132 and 22:2 μm,respectively, for same S bending loss. The total devicelength Ltot is obtained as 2LT þ Lπ ¼ 286:2 μm. It isseen from the figure that for the tooth-shapedGA-MMI coupler, beat length increases with the

increase of h, whereas the transition length LTdecreases with h for the same TS.

The optimum value of h is obtained at the crossingpoint of the curves (LT versus h and Lπ versus h) as∼4 μm, at which the value of LT and beat lengthLπ are ∼114:5 and 40 μm, respectively. The totaldevice length Ltot of the MMI coupler with a tooth-shaped grating is obtained as 2LT þ Lπ ¼ 269 μm,which is 17 μm less than that of the tooth-shapedgrating-based TMI coupler. The figure also showsthe dependence of transition length (LT) and beatlength (Lπ) on h of the MMI region of the proposedstructures by considering a fixed S bending loss

Fig. 3. (Color online) Beam propagation results at a cross-coupling point obtained by using OptiBPM software for (a) a tooth-shapedGA-TMI coupler, (b) a conventional TMI coupler, (c) a tooth-shaped GA-MMI coupler, and (d) a conventional MMI coupler with a ¼1:5 μm, b ¼ 1:5 μm, cladding index ≈ 1:45, Δn ¼ 5%, and λ ≈ 1:55 μm.

Fig. 4. Beat length (Lπ) versus index contrast (Δn) of tooth-shaped GA-MMI couplers with ΔW ¼ 0:05, 0.1, and 0:25 μm,and a conventional MMI coupler (ΔW ¼ 0 μm).

Fig. 5. Transition length (LT ) and beat length (Lπ) versus wave-guide separation gap (h) variation of tooth-shaped a GA-MMIcoupler (solid line) and a conventional MMI coupler (dotted line)with a ¼ 1:5 μm, b ¼ 1:5 μm, index contrast ≈ 5%, and claddingindex ≈ 1:45.

1 September 2011 / Vol. 50, No. 25 / APPLIED OPTICS E197

(TS) of 0:2dB. It is seen that the beat length of aconventional MMI coupler is two times larger thanthat of a tooth-shaped GA-MMI coupler.

In N ×N photonic matrix switching applications,it is required to keep a maximum access waveguidebending loss of 0:1dB due to large scale integration[18]. So we have studied the reduction of bending losswith the increase of h for a tooth-shaped grating-based MMI coupler with a ¼ 1:5 μm, b ¼ 1:5 μm,index contrast ≈ 5%, and cladding index ≈ 1:45, asshown in Fig. 6. The figure also shows the varia-tion of beat length with h. It is seen that as h in-creases, beat length increases, whereas bending lossdecreases with the increase of h, and the optimumvalue of h is obtained at the crossing point of thecurves (bending loss versus h and Lπ versus h) at∼4 μm, at which the value of the bending loss andbeat length Lπ are ∼0:1dB and 40 μm, respectively.

Since it may not be possible for accurate fabrica-tion of a device structure with exact designed param-eters, it is necessary to study its performancedegradation with a small unwanted variation of thewaveguide parameters. In this paper, the effect offabrication tolerances (δw) of MMI width on thepower imbalance of tooth-shaped GA-MMI couplersand conventional MMI couplers (ΔW ¼ 0 μm) hasbeen studied.

Figure 7 shows the plot for power imbalance½¼ 10 log10ðP3=P4Þ� versus fabrication tolerances(�δw) of the tooth-shaped GA-MMI width with h ≈

3:0 μm, a ¼ 1:5 μm, b ¼ 1:5 μm, index contrast ≈ 5%,cladding index ≈ 1:45, and λ ≈ 1:55 μm. It is seen thatthe power imbalance increases with �δw symmetri-cally for both the structures and the increase of thepower imbalance for the tooth-shaped GA-MMI cou-pler is slightly more than that of the conventionalMMI coupler due to having more device parametersin the tooth-shaped GA-MMI coupler. The figure alsoshows the variation of the power imbalance with�δwfor the GA-TMI coupler and the curve for the same isalmost close to that of the GA-MMI coupler. It is alsonecessary to study the dependence of the powerimbalance on wavelength for the conventional MMIcoupler and tooth-shaped GA-MMI coupler.

Figure 8 shows the power imbalance versus wave-length for a ≈ 1:5 μm, b ≈ 1:5 μm, h ≈ 4:0 μm, indexcontrast ≈ 5%, and cladding index ≈ 1:45. In thefigure, the solid line indicates the curve for a3dB tooth-shaped GA-MMI coupler of couplinglength ≈ 20 μm and the dotted line shows the curvefor a 3dB conventional MMI coupler of couplinglength ≈ 41 μm.

It is seen from the plot that in both cases minimumpower imbalance is obtained at λ ≈ 1:55 μm and itis almost symmetrically increased on both sides ofλ ≈ 1:55 μm. The increase of the power imbalancefor the tooth-shaped GA-MMI coupler is sharp incomparison to the conventional MMI coupler. Thedashed line in the figure represents the variationof the power imbalance versus wavelength for theGA-TMI coupler and the curve for the same is almostsuperposed to that of the GA-MMI coupler. So thedependence of the power imbalance on fabrication

Fig. 6. Bending loss (TS) and beat length (Lπ) versus waveguideseparation gap (h) variation for tooth-shaped a GA-MMI couplerwith a ¼ 1:5 μm, b ¼ 1:5 μm, index contrast ≈ 5%, and claddingindex ≈ 1:45.

Fig. 7. Power imbalance characteristics versus width tolerances(δw) for a tooth-shaped GA-MMI coupler, a tooth-shaped GA-TMIcoupler, and a conventional MMI coupler with index contrast ≈ 5%,cladding index ≈ 1:45, h ≈ 3:0 μm, a ¼ 1:5 μm, b ¼ 1:5 μm, andλ ≈ 1:55 μm.

Fig. 8. Power imbalance characteristics versus wavelength varia-tion for a tooth-shaped GA-MMI coupler (solid line), a tooth-shaped GA-TMI coupler (dashed line), and a conventional MMIcoupler (dotted line) with a ¼ 1:5 μm, b ¼ 1:5 μm, h ≈ 3:0 μm, indexcontrast ≈ 5%, and cladding index ≈ 1:45.

E198 APPLIED OPTICS / Vol. 50, No. 25 / 1 September 2011

tolerance and wavelength for the GA-MMI coupler isalmost same as that for the GA-TMI coupler.

4. Conclusion

A compact 2 × 2 tooth-shaped GA-MMI coupler hasbeen studied for what we believe to be the first timeby using a mathematical model based on SM-SEIM.It is seen that the beat length of the tooth-shapedGA-MMI with an access waveguide separation h ¼4:0 μm is 40 μm, which is ∼50% less than that of aconventional MMI coupler with the same value ofh. We have also studied the dependence of the accesswaveguide length on h with a fixed value of S bend-ing loss for a GA-MMI coupler and compared it withthat of a tooth-shaped GA-TMI coupler. It is foundthat the device length including access waveguidelength of the GA-MMI coupler is less than that of theGA-TMI coupler for a fixed value of access waveguidebending loss. Although the effect of fabrication toler-ance on the power imbalance of the GA-MMI coupleris more than that of a conventional MMI coupler, it isalmost same as that for the GA-TMI coupler.

References1. A. Neyer, “Integrated optical multichannel wavelength

multiplexer for monomode systems,” Electron. Lett. 20,744–746 (1984).

2. A. K. Das and P. P. Sahu, “Compact integrated optical devicesusing high index contrast waveguides,” in Proceedings ofthe IEEE Wireless and Optical Communication NetworkConference (IEEE, 2006), pp. 1–5.

3. P. P. Sahu, “Compact multimode interference coupler withtapered waveguide geometry,” Opt. Commun. 277, 295–301(2007).

4. P. P. Sahu, “A tapered structure for compact multimodeinterference coupler,” IEEE Photon. Technol. Lett. 20,638–640 (2008).

5. L. B. Soldano and E. C. M. Pennings, “Optical multi-modeinterference devices based on self-imaging: principles andapplications,” J. Lightwave Technol. 13, 615–627 (1995).

6. M. K. Chin, C. W. Lee, and S. Darrnawan, “High indexcontrast waveguides and devices,” Appl. Opt. 44, 3077–3086(2005).

7. P. P. Sahu, “A compact optical multiplexer using siliconnano waveguides,” IEEE J. Sel. Top. Quantum Electron. 15,1537–1541 (2009).

8. T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C.Chen, “A novel wavelength-division multiplexer using gratingassisted two-mode interference,” IEEE Photon. Technol. Lett.16, 2251–2253 (2004).

9. T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, andM. H. Cha, “A novel ultra compact two-mode-interferencewavelength division multiplexer for 1:5 μm operation,” IEEEJ. Quantum Electron. 41, 741–746 (2005).

10. K. S. Chiang, “Effective index method for the analysis ofoptical waveguide couplers and arrays: an asymptotic theory,”J. Lightwave Technol. 9, 62–72 (1991).

11. K. S. Chiang, “Analysis of the effective-index method for thevector modes of rectangular-core dielectric waveguides,” IEEETrans. Microwave Theory Tech. 44, 692–700 (1996).

12. B. Deka and P. P. Sahu, “Transformation relationship ofdirectional coupler with multimode interference coupler andtwo mode interference coupler,” J. Opt. 38, 75–87 (2009).

13. F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton,“Bent waveguide couplers for demultiplexing of arbitrarybroadly separated wavelengths using two mode interference,”IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).

14. M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compactmultimode interference 3dB coupler with strong lateralconfinement by deep dry etching,” IEEE Photon. Technol.Lett. 12, 492–494 (2000).

15. P. P. Sahu, “Silicon oxinitride: a material for compactwaveguide device,” Indian J. Phys. 82, 265–272 (2008).

16. H. Nishihara, M. Haruna, and T. Suhara, Optical IntegratedCircuits (McGraw-Hill, 1989).

17. P. P. Sahu, “A double S-bend geometry with lateral offsetfor compact two mode interference coupler,” J. LightwaveTechnol. 29, 2064–2068 (2011).

18. R. Kashahara, M. Yanagisawa, T. Goh, A. Sugita, A. Himeno,M. Yasu, and S. Matsui, “New structures of silica-based planarlight wave circuits for low power thermooptic switch andits application to 8 × 8 optical matrix switch,” J. LightwaveTechnol. 20, 993–1000 (2002).

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