IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 1, JANUARY/FEBRUARY 2010 77

Relaxation of Critical Coupling Condition andCharacterization of Coupling-Induced Frequency

Shift in Two-Ring StructuresLandobasa Y. M. A. L. Tobing, Student Member, IEEE, Stevanus Darmawan, Student Member, IEEE,

Desmond R. Lim, Mee-Koy Chin, Senior Member, IEEE, and Ting Mei, Senior Member, IEEE

Abstract—We present a systematic study of two-ring–one-bus(2R1B) and two-ring Mach–Zehnder interferometer (2RMZI). Byexciting the cavity modes in certain ways, it is possible to shapethe transmission spectrum that may be suitable for specific appli-cations, such as relaxation of critical coupling for the case of 2R1Bor finesse enhancement for the case of 2RMZI. The deteriorat-ing effects of resonator self-coupling known as coupling-inducedresonance frequency shift is quantized. The two-ring structuresare demonstrated on a silicon-on-insulator platform where goodagreement is obtained between the experiment and the theory.

Index Terms—Coupled resonator system, critical coupling,microring resonator, silicon-on-insulator (SOI) photonics.

I. INTRODUCTION

R ING resonator is a resonant structure formed by a closed-loop waveguide that has resonance properties dependent

on the coupling strength and its internal cavity round-trip loss.Various applications, such as filters, modulators, switches, bio-chemical sensors, and optical buffer, have been theoreticallyproposed and some have been experimentally demonstratedbased on a simple ring resonator [1]–[8]. The fabrications ofring resonator have been demonstrated in various material plat-forms [9]–[11] with radius down to 1.5 µm [12]. The relativelysmall dimensions and flexibility for various optical function-alities indeed have made ring resonator a strong candidate forphotonic integration [13] in photonic integrated circuit.

Configurations based on multiresonator systems have beenshown to have tailored dispersion and spectral characteristics,which is useful for application-specific purposes. For exam-ple, a nearly dispersion-free and box-like response has been

Manuscript received April 30, 2009; revised June 19, 2009. Current versionpublished February 5, 2010. This work was supported in part by the SingaporeNational Research Foundation (NRF) under Competitive Research Programme(CRP) Award NRF-G-CRP 2007-01 and in part by the Ministry of Education(MOE), Singapore, under Research Grant ARC 16/07.

L. Y. M. A. L. Tobing, S. Darmawan, and T. Mei are with the Depart-ment of Electrical and Electronic Engineering, Nanyang Technological Univer-sity (NTU), Singapore 639798, Singapore (e-mail: ltobing@pmail.ntu.edu.sg;sdarmawan@ntu.edu.sg; etmei@ntu.edu.sg).

D. R. Lim is with the Department of Electrical and Electronic Engineering,Nanyang Technological University (NTU), Singapore 639798, Singapore(e-mail: desmond@ntu.edu.sg).

M.-K. Chin, deceased, was with the Department of Electrical and ElectronicEngineering, Nanyang Technological University (NTU), Singapore 639798,Singapore.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2009.2027890

demonstrated in different resonator array geometries [14]–[20].Optical buffering based on few coupled resonators and cascadeof resonators have been demonstrated both in passive and ac-tive schemes [21]–[24]. Recently, there is a growing interest incoupled cavity systems consisting only of two resonators forits strong resemblance to that in atomic systems [25]–[29]. Theresponse of such a system depends on how the resonators in-teract and the excitation of both cavity modes. For example,when the two resonators only interact via the common op-tical waveguides [27], [28], the coherent interaction betweentwo cavity modes is achieved upon adjusting the spacing be-tween resonators and their relative size to each other, whichgives the same spectrum as that in electromagnetically inducedtransparency (EIT) of the atomic systems. There are, however,many other variants of two-cavity system that has not yet beenexplored.

In this paper, we investigate configurations based on two mu-tually coupled ring resonators, side-coupled to one waveguidebus (two-ring–one-bus, 2R1B) and to Mach–Zehnder interfer-ometer (2RMZI). It is shown that the 2R1B exhibits more re-laxed critical coupling condition in low-loss situation, which, ingeneral is difficult to achieve in one-ring system. For the caseof 2RMZI, it is shown that through the indirect interferencebetween the optical pathways in the MZI section with thosefrom the two-ring structure, it becomes theoretically possible toindependently tailor the asymmetricity of the Fano resonancewithout affecting the cavity finesse (i.e., the ring structures) orchanging the resonance detuning. Furthermore, in getting a morecomplete physical analysis, we investigate the deteriorating ef-fects of resonator self-coupling, known as coupling-induced res-onance frequency shift (CIFS) [30], and quantize the additionalphase shift introduced by it, for the two types of devices.

The two-ring structures were fabricated in silicon-on-insulator (SOI) technology under ePIXnet platform [31], withthe same waveguide dimensions as those in our previous works[32]–[36]. The SOI photonic wire is of 450 nm width and220 nm thickness, while the gap separation between waveg-uides or between waveguide and ring resonator is kept fixed at200 nm.

II. TWO-RING–ONE-BUS SYSTEM

The resonance characteristics of the uncoupled two-ringstructure have been discussed elsewhere [37]. In the 2R1B con-figuration schematically shown in Fig. 1, only the lower ring hasaccess to the optical waveguide. Using the existing formulation

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Fig. 1. Schematic of 2R1B. The inset shows optical pathways from two de-caying modes.

for the one-ring system [37], [38], the through transmission (T)is given by

T =∣∣∣∣

ET

EIN

∣∣∣∣

2

=∣∣∣∣

r1 − a1τ21 exp(−iδ1)1 − a1r1τ21 exp(−iδ1)

∣∣∣∣

2

(1)

where τ21 = [r2 − a2 exp(−iδ2)]/[1 − a2r2 exp(−iδ2)] is theloading factor imposed by ring 2 on ring 1, similar to thetransmittance of the one-ring coupled to one optical waveg-uide (1R1B) system. The δm = 2πneff Lm /λ is the round-tripphase shift, with cavity length Lm and modal effective in-dex neff . The lumped amplitude round-trip loss is defined byam = exp(−αLm /2) and α is the absorption coefficient. Thecoupling coefficients between ring 1 and the bus waveguide, andbetween the rings are denoted as r1 and r2 , respectively.

By rewriting the loading factor in the form |τ21 | exp(iθ21),and defining the effective cavity loss in ring 1 as a = a1 |τ21 |,the transmission can be reduced to the one-ring form

T ≡ |τ |2 =∣∣∣∣

r1 − a exp(−iδ)1 − ar1 exp(−iδ)

∣∣∣∣

2

(2)

where δ = δ1 − θ21 is the phase difference between the twooptical pathways depicted in the right panel of Fig. 1

δ=δ1−[

tan−1(

a2 sin δ2

r2 − a2 cos δ2

)

−tan−1(

a2r2 sin δ2

1 − a2r2 cos δ2

)]

.

(3)The first term is the pathway that resonates in ring 1 (P1)

while the last two terms (=θ21) is the pathway that resonatesin ring 2 (P2). As P1 and P2 interfere destructively (δ/π =odd), the light is not confined inside the two-ring system, andthus, gives maximum transmission. When P1 and P2 interfereconstructively (δ/π = even), the light is localized inside thetwo-ring system and gives minimum transmission. The locationof such interferences depends on the size ratio of the two rings(γ = δ2/δ1).

Fig. 2(a) shows the transmission spectra for various valuesof γ, while Fig. 2(b) shows the finite-difference time-domain(FDTD) calculated fields associated with the split resonances.When the rings are identical (γ = 1), there is an even resonancesplitting and the light is equally confined in both rings, as shownin the first two panels of Fig. 2(b). Light intensity distributioncan be estimated by using the relative intensity buildup B21 ,which is defined as the ratio of intensity in ring 2 with respectto that in ring 1 in the lossless case

B21 =1 − r2

2

1 + r22 − 2r2 cos δ2

. (4)

Fig. 2. (a) Transmission spectra for γ values from 1 to 2, the inset showsthe broad and split resonances near the resonance of the bare rings. (b) Fielddistribution for γ = 1 and γ = 2 as calculated using FDTD. Note that thetwo symmetric resonances have a symmetric (S) and an antisymmetric (AS)transverse field profile at the coupling point. NR: narrow resonance for γ = 2.

By inserting the normalized frequency detuning of the splitresonance δ

(±)1 = δ

(±)2 = ± cos−1(r2) into (4) [37], it can be

shown that B21(δ±1 ) = 1, consistent with the FDTD calculation.The symmetric splitting corresponds to the symmetric (S) andantisymmetric (AS) field mode profiles relative to the intercavitycoupling point. When the resonances of the isolated rings aremore detuned to each other (γ is not an integer), one has a caseof uneven resonance splitting consisting of narrow and broadresonances, which corresponds to the most light confinementin ring 2 and ring 1, respectively. To avoid confusion with theresonance of the isolated rings, we defined P1 resonance asthe resonance at which light is mostly confined in ring 1, andsimilarly P2 resonance at which light is mostly confined in ring2. As the two resonances are farther apart from each other,the transmission at P2 resonance becomes progressively lowerbecause the light circulates longer inside the system and, in turn,enhances the effective cavity loss.

When the two resonances are farthest from each other, i.e.,when ring 2 is antiresonant to ring 1 (γ = 2), one has evenresonance splitting around resonance (δ1/π = even), and a nar-row and symmetric resonance at anti resonance (δ1/π = odd).The resonance splitting is the same as that in γ = 1, but withcirculating light in ring 2 about two times smaller than that inring 1, and can be shown in B21(δ±1 ) = (1 + r2)−1 > 1/2, byinserting the normalized detuning for split resonances to (4) inγ = 2 case. This is again confirmed by FDTD calculations inthe second two panels of Fig. 2(b), which shows the same sym-metric (S) and antisymmetric (AS) modes from the couplingpoint.

The symmetric and very narrow P2 resonance (NR) cor-responds to the highest achievable finesse in 2R1B. Insert-ing δ1 = π(δ2 = 2π) to (4) gives B21 = (1 + r2)/(1 − r2),

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TOBING et al.: RELAXATION OF CRITICAL COUPLING CONDITION AND CHARACTERIZATION OF COUPLING-INDUCED FREQUENCY SHIFT 79

showing that almost all light is confined in ring 2, correspond-ing to the highest available cavity finesse. The effective cavityloss is also enhanced along with the finesse, which could relaxthe critical coupling requirement for a low-loss system that isgenerally difficult to obtain in the one-ring system.

III. RELAXATION OF CRITICAL COUPLING IN 2R1B

In one-ring systems, the buildup of intracavity intensity gen-erally occurs at overcoupling situation, where the couplingstrength is larger than the round-trip loss (r < a). This givesa phase buildup and transmission minima (dip) at resonances.The intensity buildup is at maximum in critical coupling con-dition [39], where the coupling strength is exactly equal to theround-trip loss (r = a), giving discontinuous phase responseat resonance and a very high contrast transmission dip. Fromexperimental perspective, however, both coupling strength andround-trip loss are strongly dependent on fabrication conditionand the round-trip loss normally can only be known after thedevice is fabricated. This means that satisfying critical couplingcondition in low-loss resonators (with a > 0.99, or <1% loss)is particularly difficult because a very weak coupling is needed,which requires a very tight fabrication control.

Fig. 3(a) shows the transmission contours as a function of thecoupling (r) and the round-trip loss (a) within the region T ≤−10 dB, denoting the range of near critical coupling condition.When the loss is moderate (a < 0.99), a contrast of 10 dB isstill possible with racetrack coupling. However, when round-triploss becomes very small (a > 0.99), as in the fabricated ringsdemonstrated to date [9], [19], [40]. Slight deviation from criti-cal coupling may set the resonators in undercoupling situation (r> a), in which the intensity buildup rapidly decreases. Indeed, itis generally desired to have near critically coupled transmissionwith a more relaxed requirement. Although near critically cou-pled rings have been experimentally demonstrated [40]–[42], inpractice, it is still difficult to match these two parameters. This isreflected in many attempts to achieve critically coupled rings invarious ways, for example, in integrating semiconductor opticalamplifier (SOA) into the ring [43], postfabrication trimming inpolymer ring resonator [44], and erbium-doped fiber amplifier(EDFA) in fiber ring [45].

In 2R1B, these requirements are relaxed by resonant enhance-ment of the cavity round-trip loss in the P2 resonance of γ =2 case. In Fig. 3(b), we show the transmission for two low-loss situations, i.e., a1 = 0.99 (αL1 = 10−2) and a1 = 0.998(αL1 = 2 × 10−3), as a function of waveguide–ring (r1) and in-terresonator (r2) coupling coefficients. It is evident that for bothsituations, a contrast of >10 dB can be obtained in a broaderrange of coupling coefficients. The main advantage of the two-ring system is that critical coupling for low-loss ring can beachieved in stronger coupling regime. Obviously, this is notachievable in one-ring systems.

The fabricated 2R1B structures are shown in Fig. 4. Here, thecoupling coefficients (r1 , r2) are defined by racetrack couplerswith respective lengths LC1 and LC2 . The extension parameterLC3 is used to match the desired values of γ. The fabricatedtwo-ring devices are aimed for the finesse-enhancement situ-

Fig. 3. Calculated transmission (in decibels) contours as functions of theround-trip loss and the coupling coefficients for: (a) one-ring and (b) two-ringsystems.

Fig. 4. Fabricated 2R1B devices.

ation. In the 2R1B, more relaxed critical coupling conditionin the moderate coupled system is to be demonstrated at γ =2. Together with these devices, individual ring resonators cou-pled to two bus waveguides (1R2B) are also fabricated to verifycurve-fitting parameters. Independent measurements of 1R2Bstructures show a group index of ∼4.25 and round-trip loss of a∼ 0.995, which then are used in the curve fitting of other struc-tures. The radius of ring 1 is kept at 5 µm. Using the extractedparameters from 1R2B devices, the LC1 and LC2 are designedso that (r1 , r2) is situated within the range of critical couplingregion.

The LC1 is fixed at 5.5 µm (∼36% power coupling), whilethe LC2 is varied from 2 (∼12% power coupling) to 3 µm(∼18% power coupling). The measured transmissions are shownin Fig. 5, where there are narrow resonances (i.e., near 1517 nmand 1530 nm) situated in between two split resonances (i.e.,near 1511, 1524, and 1537 nm) for LC2 = 2 and 3 µm. This isconsistent with the calculated transmissions in Fig. 2, where thesplit resonances and narrow resonances are generally around theresonance (δ1/π = even) and the antiresonance (δ1/π = odd)of ring 1, respectively.

One can see that the transmission contrasts of these narrowresonances are more than 10 dB for 1 µm difference in thecoupling length LC2 . This shows a remarkable relaxation of thecritical coupling condition for such resonators with low round-trip loss ∼0.5% (a ∼ 0.995), which in a one-ring system, wouldrequire 0.5% power coupling which is very difficult to achievein practice. To illustrate the sensitivity of the one-ring system,

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80 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 1, JANUARY/FEBRUARY 2010

Fig. 5. Measured transmission spectra of the 2R1B circuits with the interres-onator coupler length to be (left) 3 µm and (right) 2 µm. The racetrack lengthLC1 is kept constant at 5.5 µm.

Fig. 6. Measured transmission spectra of 1R1B for various round-trip lossesand coupling factors.

1R1B devices are fabricated with different coupling lengths andradii (see Fig. 6). It can be seen that despite various round-triplosses (by varying the radius) or coupling factors, the 1R1B stillhas a relatively low contrast. This illustrates how difficult criticalcoupling condition is satisfied for low-loss one-ring system.

IV. TWO-RING SYSTEM COUPLED TO MACH–ZEHNDER

INTERFEROMETER

In the previous works, we have shown that the inherent lim-itation of bistable switching characteristics in one-ring systemscan be overcome by utilizing asymmetric Fano resonance inthe other two-ring structure [38]. The problem, however, is thatthe finesse and resonance asymmetricity are interrelated [35],where the maximum finesse is obtained in fully symmetricalFano resonance. Thus, it is imperative to further investigatethe two-ring coupled Mach–Zehnder interferometer (2RMZI),for the purpose of achieving the maximum finesse with in-dependent control of resonance asymmetricity. The proposed2RMZI is schematically drawn in Fig. 7, with a static phasebias φB in the lower MZI arm. For comparison, single-ringresonator coupled to Mach–Zehnder interferometer (1RMZI) is

Fig. 7. Proposed 2RMZI.

also presented in order to understand the tradeoff of these twodevices.

The working principle of 2RMZI is based on the interferencebetween pathways P1 and P2 from the upper arm and P3 from thelower MZI arm as illustrated by the inset in Fig. 7. Consideringτ ≡

√T exp(iθ) as the complex amplitude transmission of the

2R1B system, the bar and cross transmission of 2RMZI aremodeled as

TBAR = 14 |τ − exp(iφB)|2 , TCROSS = 1

4 |τ + exp(iφB)|2(5)

where the static phase bias φB signifies the phase of pathway P3.It should be noted that (5) can be used for 1RMZI, by assumingnegligible interresonator coupling (r2 ∼ 1).

Fig. 8 shows the 1RMZI and 2RMZI bar transmission (TBAR )spectra calculated by FDTD simulation. In 1RMZI geometry,the coupler length is varied from 0, 2, 4, and 6 µm with ringradius of 5 µm. In 2RMZI geometry, the coupling between ring1 and the upper MZI arm and the coupling between the rings arethe same (r1 = r2), with radius of ring 2 of 10 µm in order tomake γ = 2. For equal MZI arm lengths, ideally, the transmissionspectrum is symmetric. However, even in FDTD calculations, itcan be seen that the transmission spectra for both geometries areintrinsically asymmetric, which originates from the perturbationwith the neighboring ring that induces additional phase shift, asshown in the inset of Fig. 8(a).

These spectra is then modeled by (5) with lumped phaseoffset φB in the lower arm in order to compensate the phaseperturbation from the upper arm, which are shown to be in ex-cellent agreement with FDTD calculations. It should be notedthat this problem has been observed in ring-coupled MZI struc-tures [46], [47], and we believe this is attributed to the effectknown as CIFS [30], which is actually the second order effectof evanescent coupling.

The lumped phase offset for 1RMZI (φB ∼ 0.05π) is gener-ally smaller than those of 2RMZI structures (φB ∼ 0.44π). Thisis attributed to the fact that there are two CIFS effects in 2RMZI,the first between ring 1 and the upper MZI arm, and the secondbetween both rings. In 1RMZI, however, there is only one CIFSfrom waveguide–ring coupling in the upper arm. It can be seenfrom Fig. 8, that the on-resonance (δ1/π = even) peak trans-mission of 1RMZI structure is less sensitive to loss in contrastto the narrow resonant peak at antiresonance (δ1/π = even) in2RMZI. This is because of the relaxation of the critical cou-pling condition in 2R1B structures shown earlier. The 1RMZI

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TOBING et al.: RELAXATION OF CRITICAL COUPLING CONDITION AND CHARACTERIZATION OF COUPLING-INDUCED FREQUENCY SHIFT 81

Fig. 8. (a) Field distribution of the 2-D FDTD simulation of 2RMZI highlight-ing the CIFS role in producing intrinsic MZI phase imbalance (γ = 2). (b) Bartransmission of 1R- and 2RMZI calculated by 2-D FDTD (fade curves) withfitted theory (dark curves) for various coupler lengths (LC ) and fitted valuesr = r1 = r2 .

Fig. 9. 2RMZI fabricated on SOI platform. The inset shows the 3 dB MMIcoupler with 3.5 µm width, 43 µm length, and 3 dB splitting operating bandwidthof ∼50 nm.

and 2RMZI structures work by interference of pathways fromthe two arms. This means that high-contrast and high-finesseresonance can only be obtained as long as the interference is notquenched. This is the reason why critical coupling needs to beavoided in ring coupled MZI structures.

The fabricated 2RMZI is depicted in Fig. 9, where multimodeinterferometers (MMIs) are used as the 3 dB couplers at bothends of the MZI. The MMI length and width are 43.5 and 3.5 µm,respectively, with 1.7 µm center-to-center separation betweenwaveguide ports. The MMI insertion loss was measured to be∼1.3 dB. The parameters of the fabricated 2RMZI follow thesame convention as the fabricated 2R1B in Fig. 4. The LC1 isvaried from 4, 6, and 8 µm, while the LC2 is fixed at 3 µm. Wedenote these devices as DUT01 to DUT03. The radius of ring

Fig. 10. Measured bar (TBAR ) and cross (TCROSS ) transmissions of 2RMZIfor device-under-test (DUT) 01 to 03. The measured and calculated transmis-sions are shown in fade and dashed lines, respectively. The bold curves representan ideal symmetric resonance at the absence of CIFS. The loss factor is measuredas a1 ∼ 0.995.

1 is fixed at 5 µm, while the radius of ring 2 and the extensionparameter LC3 are adjusted to achieve γ = 2.

Fig. 10 shows the measured bar and cross transmissions of2RMZI. The calculated spectrum over one free spectral range(FSR) is shown in dark lines and in good agreement with mea-sured transmissions. It should be noted that the fitting is onlyperformed for one FSR, because the group index is relativelyconstant within that wavelength range. For larger wavelengthrange, one needs to incorporate the wavelength dependency ofgroup index ng (λ). The deduced r1 values for LC1 = 4, 6, and8 µm, and the deduced r2 values are consistent with those re-sults from the 2R1B structure of the same LC2 . As shown inthe earlier FDTD simulation, the three sets of fabricated 2RMZIdevices exhibit substantial phase imbalance φB between the twoMZI arms. It is possible to estimate the resonance parametersof 2RMZI in the absence of the MZI phase imbalance. Thefinesse of ∼355, ∼270, and ∼199 are to be expected, respec-tively. Furthermore, this corresponds to finesse enhancementfactor of ∼12.4, 9.5, and 7 in comparison to those of one-ringsystems [36].

The lumped phase offset used here does not adequately verifythe existence of CIFS. By coupled mode theory [48], it can beshown that the modified propagation mode along the evanescentcoupling due to self-coupling term is β′ = β + kself , where β′

and β are, respectively, the perturbed and unperturbed propaga-tion modes, and kself is the self-coupling term defined by

kself ≡k0

2neff〈E1 |∆ε2 |E1〉 =

π

neff λ

∫

|E(r)|2 ∆(n2)dr.

The ∆(n2) is the perturbation of the square refractive index ofthe neighboring waveguide (waveguide 2), and E(r) is the nor-malized electric field profile of the perturbed waveguide mode(waveguide 1). The CIFS additional phase shift of couplinglength LC is, therefore,

φCIFS = kselfLC . (6)

Thus, the phase shift in fabricated 2RMZI should be propor-tional to the coupling length LC1 and LC2 . From (6), the phase

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82 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 1, JANUARY/FEBRUARY 2010

Fig. 11. Curve-fitting of 2R1B transmissions. The calculated transmissionsare denoted in dark curves.

offset is then modeled as φB = vδ1 , with v signifying the addi-tional phase due to CIFS, normalized to the round-trip phase ofring 1(δ1).

By fixing all parameters except γ and φB , the transmissionsof 2R1B and 2RMZI are then refitted around the asymmetricalsplit resonances. The deviation of γ (∆γ/γ) in 2R1B structuresof LC2 of 1, 2, and 3 µm are obtained from curve fitting of thesplit resonances, which are found to be 3.3×10−4 , 6×10−4 , and7×10−4 , respectively, as presented in Fig. 11. The deviation ofγ can be seen from the splitting asymmetricity and the normal-ized detuning of the mean split frequency δC

1 = (1/2)(δ+1 + δ−1 )

from the ideal case δC1 = 2πm. The wavelength is converted to

normalized frequency by δ1 = 2πngLcav1/λ, where ng is thegroup index within one FSR and Lcav1 = 2(πR1 + LC1) is thecavity length of the ring 1. First, deviation of γ is inserted intothe theoretical formulation in (1) to match the asymmetric split-ting in experiments. Then, the mean split frequency is shiftedto those in experiment by slightly adjusting the group indexobtained from 1R2B measurements.

These ∆γ/γ values are then incorporated to (5) to get thev values in 2RMZI accordingly, which gives v = 2.4 × 10−3

(for LC1 = 4µm) and v = 2.8 × 10−3 (for LC1 = 6µm). Allthese values are compatible with the fact that CIFS phase shiftis proportional to the coupling length. It should be noted that thetwo MMIs in 2RMZI also contributes additional phase error tothe obtained v (90◦ phase difference in MMI coupling is only forideal 3 dB coupler). This confounding effect, however, can beeliminated by taking the difference of CIFS phase over differentcoupling lengths (∆φCIFS/∆LC ). The differential CIFS phaseshift for both structures are ∆φCIFS/∆LC ∼ 10−4 to 2.7 ×10−4 (for 2R1B), and ∆φCIFS/∆LC ∼ 2 × 10−4 (for 2RMZI),which are of the same orders. Despite these indicative measures,more measurements are still needed to further quantify CIFS.

V. CONCLUSION

A systematic study of two-ring systems starting from theuncoupled case, 2R1B, evolving into the 2RMZI has been pre-sented. It has been shown that the critical coupling conditionin low-loss resonator, which is practically difficult to achieve

in one-ring systems, can be relaxed by the use of finesse en-hancement in 2R1B. The transmission contrast of nearly 10dB is achieved within 1 µm tolerance of coupling length, incontrast to the low transmission contrast obtained in the fabri-cated one-ring systems, which have different loss and couplingcoefficients.

For the case of 2RMZI, it becomes theoretically possible totailor the asymmetricity of the Fano resonance without affectingthe cavity finesse or changing the resonance wavelength, due tothe separation between the optical pathways in the lower MZIarm and the optical pathways in the two-ring structures. Theeffect of CIFS for the two-ring structures is further investigated,where the differential normalized CIFS phase shift is deduced tobe in the order of ∼10−4 per micrometer length. By quantizingthe CIFS-induced MZI imbalance, fair agreement is obtainedbetween the theory and the experiment.

ACKNOWLEDGMENT

This paper is dedicated to the deceased Prof. Chin Mee Koywho had been a good mentor, friend, and advisor of L. Y. M.Tobing and S. Darmawan. The authors acknowledge ePIXnetplatform for the device fabrication.

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Landobasa Y. M. A. L. Tobing (S’05) received theB.Eng. and M.Sc. degrees in electronic and electri-cal engineering in 2003 and 2004, respectively, fromNanyang Technological University (NTU), Singa-pore, Singapore, where he is currently working to-ward the Ph.D. degree.

His current research interests include slow lightstructure, nonlinear optics, coupled cavity system,and periodic resonator arrays.

Stevanus Darmawan (S’05) received the B.Eng. de-gree in electronic and electrical engineering fromNanyang Technological University (NTU), Singa-pore, in 2003. He just received the Ph.D. degree.

His current research interests include integratedoptical devices based on Mach–Zehnder interferom-eter, optical filters based on coupled ring resonatorarrays, and photonic integrated circuits.

Desmond R. Lim received the Bachelor’s, Master’s,and Ph.D. degrees in electrical engineering and com-puter science from the Massachusetts Institute ofTechnology, Cambridge, in 1994, 1994, and 2000,respectively.

He has been engaged in the field of silicon pho-tonics for 14 years. He is currently a Senior Memberof the Technical Staff, DSO National Laboratories,Singapore, Singapore. He also holds an adjunct ap-pointment at the Nanyang Technological University,Singapore. His current research interests include mi-

crowave photonics and analog silicon photonics. He is the inventor of more thana dozen U.S. patents.

Dr. Lim was included in the Technology Review TR100 list as one of thetop 100 young inventors worldwide. His work on silicon photonics has beenreviewed in Science magazine as well as in the Wall Street Journal. In 2003,he received the National Science and Technology Academy Young ScientistAward.

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84 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 16, NO. 1, JANUARY/FEBRUARY 2010

Mee-Koy Chin (SM’03) (deceased) received the B.S.degree from Massachusetts Institute of Technology,Cambridge, in 1986, and the Ph.D. degree from theUniversity of California, San Diego, in 1992.

He was an Associate Professor of electrical engi-neering at Nanyang Technological University (NTU),Singapore. He joined NTU in 1993 where he foundedthe first Photonics Research Laboratory in Singapore.He then left in 1997 and spent several years in in-dustry in the U.S. before rejoining NTU in 2003.His research interests included integrated optics and

nanophotonics.Dr. Chin was the Chair of the IEEE Photonics Society (formerly known as

the IEEE Laser and Electro-Optics Society) Singapore Chapter.

Ting Mei (M’00–SM’06) is currently with the Schoolof Electrical and Electronic Engineering, NanyangTechnological University, Singapore, Singapore. Hiscurrent research interests include nanophotonics,semiconductor optoelectronics, low-dimensional ma-terials and devices, and nanostructure semiconductormaterials.

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