. INTRODUCTIONince the proposal of the coupled-resonator optical wave-uide (CROW) by Yariv and co-workers a decade ago [1],umerous variants and applications have been considered
n the photonics literature. Historically, after analyticalxpressions for wave propagation in an infinitely longhain have been exposed, the first task was to treat prac-ical devices consisting of a finite number of unit cells2,3]. Among these we single out two examples: the con-inuously tunable 1 byte delay line using a thermo-ptically controlled CROW [4] and the Fabry–PerotROW configuration that uses a defect, embedded within
he delay line, for tunable narrowband filtering [5]. How-ver, beyond these examples the CROW offers several fur-her architectures and functionalities. One of these is theiatomic CROW, reported here for the first time, to ournowledge, where the interring coupling coefficients orand) the effective microresonator perimeter lengths al-ernate, forming a delay line where the unit cells consistf two microrings and two couplers, creating a narrowubsidiary stop band within the CROW passband. Heree examine analytically and numerically the propertiesf this new device and show how its drop port passbandnd group delay (group velocity) spectra can be manipu-ated through small changes in the interring couplings oraveguide refractive indices. We likewise examine the ef-cacy of simple double-resonator matching terminations,ispersion compensation capabilities, and the effect of theesonator loss on the device performance.
. ANALYSIShe drop port intensity spectrum of the uniform finiteROW is characterized by passbands and stop bands al-
ernating over the free spectral range FSR=c / �ngL�,here L is the perimeter length of the resonator and ng is
he group index of the waveguide. The passband has asany peaks as there are resonators in the chain, centered
n the resonance wavelength of the (identical) resonatorsnd separated by interring coupling that lifts their degen-racy [6]. As this coupling increases the peaks are pushedurther apart, increasing the width of the passband. Al-hough the peaks are not distributed evenly, the relativein terms of the resonator FSR) width of this passband isroportional to the interring amplitude coupling coeffi-ient. The passbands are separated by drop port stopands whose extinction ratio increases with increasingumber of resonators and decreasing coupling strength.he corresponding normalized (to �0=L /vph) group delaypectra have maxima at the locations of the passbandaxima, sharp peaks at the edges of the passband, and a
elatively small value in the stop band whose width de-reases with increasing interring coupling strength. Inact the normalized group delay spectrum of the CROW isery similar to that of a Bragg grating [7] from which itiffers in that it has an extra port (the through port) tohannel the signal reflected from the grating. As we shallemonstrate, the diatomic configuration modifies thepectrum of the uniform CROW by introducing a narrowubsidiary stop band within the passband, in most casesn the middle.
A schematic outline of the diatomic CROW appears inig. 1, where Ki, Li, and �i �i=1,2� are, respectively, thelternating interring power coupling coefficient, the effec-ive perimeter length, and the propagation constant. Theouplers are assumed to be lossless and characterized byelf- and cross-coupling coefficients t and −j� , respec-
i i
010 Optical Society of America
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I. Chremmos and O. Schwelb Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1243
ively, where ti= �1−Ki�1/2 and �i=Ki1/2. The waveguide
oss is taken into account by a common field attenuationoefficient � and by the complex propagation constantsi=�neff,i /c− j�, where neff,i are the real effective indiceshich can be controlled thermo- or electro-optically [8].
ig. 1. Schematic of the diatomic CROW consisting of ring reso-ators with alternating power coupling coefficients Ki, perimeter
engths Li, and/or propagation constants �i �i=1,2� caused by in-ex difference. The dashed rectangle indicates the diatomic unitell. The waves entering �a ,c� and exiting �b ,d� the unit cell arelso shown. When the number of rings is odd (even) the last rings L1 �L2�, the last coupler is K2 �K1�, and the output is at port 3port 4).
± 11
at+l
tKmfiaf
a
�Sa
he per turn amplitude decrement in a microring resona-or is denoted by �i=exp�−�Li�.
As shown in Fig. 1, the signal enters at port 1 �ain� andxits at port 2 �at�, at the through port, and at the droport �ad�, which is port 3 when the number of resonatorss odd (last ring is L1, last coupler is K2) or port 4 whenhe number of resonators is even (last ring is L2, last cou-ler is K1). When the number of unit cells Nu is an integerhe number of rings is even and the output is at port 4;hen the number of unit cells is Nu+1/2 the number of
esonators is odd and the output is at port 3. At resonancehe average resonator perimeter length is an integralultiple of the effective wavelength, i.e., Li=Nri� /neff,i,here in general the individual mode numbers may note the same for the two rings of the unit cell.The analysis starts with the evaluation of the transferatrix of a unit cell consisting of two resonators and two
ouplers as indicated in Fig. 1, starting on the left withhe resonator L1 and ending on the right with the coupler1. The transfer matrix T relates the corresponding in-
oming �a ,c� and outgoing �b ,d� waves according tod c�T=T�a b�T, where
2 �1+2�, with i=�iLi �i=1,2� be-ng the complex round trip phase of the resonator; and the
superscript refers to transposition. When the transmis-ion through a chain of Nu unit cells is considered, the Tatrix must be raised to the power of Nu. To perform this
peration analytically, it is convenient to diagonalize Through T=PDP−1, where D is the diagonal matrix of theigenvalues of T and P is the matrix whose columns arehe corresponding eigenvectors. The diagonalization pro-edure is equivalent to the determination of the Blochodes of the periodic structure [3,9], which satisfy theloquet condition T�a b�T=ej2k��a b�T, with k being theloch wavevector and 2� being the lattice constant. Theroduct of the two eigenvalues, the inverses of each otherorresponding to counter-propagating waves, is the deter-inant of the reciprocal transfer matrix: det�T�=1. We
nd [9]
D = �e−j2k� 0
0 ej2k��, P = �T12 T12
A+ A−� , �2�
here
ej2k� = H ± �H2 − 1,
A = ej2k� − T ,
H =1
2Tr�T� =
��1 − K1��1 − K2�cos��� − cos ̄
�K1K2
,
nd Tij stands for the elements of T and Tr represents therace of a matrix. When the chain consists of Nu1/2 unit cells (odd number of resonators), TNu must be
eft multiplied by TK2TL1
, where
TK2=
1
j�K2� 1 − �1 − K2
�1 − K2 − 1 �, TL1= �e−j1/2 0
0 ej1/2� ,
�3�o account for the last resonator L1 and the last coupler2. Independently of the number of resonators, the resultust also be right multiplied by TK1
to account for therst coupler K1 to the input waveguide. The input �ain ,at�nd output �ad ,0� waves are connected through the trans-er matrix M,
�ad
0 � = M�ain
at�,
where M =�TK2TL1
TNuTK1, for Nu + 1/2
TNuTK1, for Nu,
�4�nd TNu=PDNuP−1, with DNu=diag�e−jk2Nu� ejk2Nu��.Solving Eq. (4) for the transmitted �at� and dropped
ad� waves, the scattering parameters S21=at /ain and41=ad /ain (or S31 when the number of resonators is odd)re determined as
w
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FNdt
3STnlucm
1244 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 I. Chremmos and O. Schwelb
ij stands for the elements of M, and the reciprocity prop-rty det�M�=1 has been used. Assuming the input at port, the group delay at port j is computed from the expres-ion
�j1 = −��j1
��= − Im� 1
Sj1
�Sj1
���, j = 2,3,4 �8�
here �j1 is the phase of Sj1. If this delay is associatedith the minimum physical waveguide length from the
nput to the drop port, i.e., NuL for even number of rings,e obtain the equivalent group velocity as vg=NuL /�41Nuvph�0 /�41 and the corresponding effective group indexs
ng,CROW =c
vg=
c�41
NuL= −
c
NuLIm� 1
S41
�S41
�� . �9�
or odd number of resonators Nu must be replaced withu+1/2 and �41 with �31. When divided by the group in-ex of the waveguides, Eq. (9) represents the slowing fac-or of the diatomic CROW.
. INTENSITY AND GROUP DELAYPECTRAhe intensity spectrum is subject to the real or complexature of the eigenvalues e±jk2�. From Eq. (2) and for loss-
ess resonators it follows that, when �H� 1, the eigenval-es are complex conjugates with unity magnitude, theorresponding Bloch wavevector k is real, and the Blochodes propagate without attenuation. The edge frequen-
ies of this condition determine the passband of the cor-esponding infinite CROW, characterized by the disper-ion equation ej2k�+e−j2k�=2H, i.e.,
cos�2k�� =��1 − K1��1 − K2� cos��� − cos ̄
�K1K2
. �10�
n a finite structure the passband is characterized by thereviously mentioned peaks of the dropped intensity andts width approaches that of the infinite CROW as theumber of unit cells increases. Outside the passband, theigenvalues are real ��H��1�, the Bloch wavevector isure imaginary, and the dropped intensity is weak.A special configuration of the diatomic CROW is a
hain of identical resonators with alternating coupling co-fficients and identical round trip phases 1=2=2��Nr�f /FSR�, where �f= f− f0 is the detuning from the com-on resonance frequency f0 and Nr is the azimuthal modeumber. In this case, from Eq. (10) it follows that the fre-uencies of the passband, normalized to the FSR, can bexpressed as
1
2��sin−1��K1� − sin−1��K2��
��f�
FSR
1
2��sin−1��K1� + sin−1��K2��. �11�
quation (11) indicates that there are two passbands,ymmetric with respect to f0, whose widths are the same:fPB=sin−1��Kmin�FSR /�, where Kmin=min�K1 ,K2�.hese passbands are separated by the subsidiary stopand of the diatomic CROW, centered on f0, whose widths
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Fm=
I. Chremmos and O. Schwelb Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1245
�fSB = FSR1
��sin−1��K1� − sin−1��K2��. �12�
e observe also that the diatomic molecule of the chainas a pair of resonances at f0±sin−1��Kmax�FSR / �2��, cor-esponding to odd/even supermodes. Cascading these mol-cules creates two passbands of width �fPB around theseplit resonances. In a uniform CROW the width of theassband is twice as what is given above, because theound trip phase of a doublet of strongly coupled rings iswice that of a single ring. As K1→K2 the width of theubsidiary stop band vanishes. This subsidiary stop bandn the middle of the passband is the salient feature of theiatomic CROW.A simulation of the drop port passband of a lossless di-
tomic CROW having six unit cells with identical resona-ors and alternating coupling coefficients appears in Fig., showing the port 4 intensity disturbance in the middlef the passband, caused by alternating coupling coeffi-ients. Here K1=0.2 is fixed while K2 covers the rangerom 0.05 to 0.4, and the relative detuning just scans theassband of the Nu=6 diatomic CROW. Observe the de-eloping depression as K2 becomes larger than K1 and theeveloping twin depression as K2 decreases below thealue of K1. Within each of the adjacent passbands, therere as many intensity peaks as the number of molecules.or K2�K1 there are Nu−1 molecules, so the two innereaks of the uniform CROW passband converge at f0 toorm a ridge that is eventually suppressed as K2 dropsurther below K1. A similar simulation for an alternatingoupling coefficient diatomic CROW with an odd numberf identical resonators (Nu+1/2 unit cells) appears inig. 3. The structure can be viewed as a chain of Nu mol-cules, coupled to the input and output waveguideshrough the remaining resonator. The drop port intensityf the corresponding uniform CROW has 2Nu+1 peaks,u of which are symmetrically detuned with the one re-aining at f0. Therefore, as K2 deviates from K1 toward
igher or lower values it is expected that the central peakill be depressed so that, for sufficiently large difference
K2−K1�, there will remain only 2Nu peaks. This is veri-ed in Fig. 3 showing the evolution of the drop port inten-ity with 13 identical resonators, a fixed K1=0.2, and aarying K2. Observe that, with increasing K2�K1, thewo passbands detune from f0 maintaining an approxi-
ig. 2. (Color online) Drop port intensity as a function of nor-alized detuning and K2 for a lossless diatomic CROW with Nu6 (12 resonators) and a fixed K =0.2.
1
ately fixed bandwidth which is determined by theeaker fixed coupling coefficient K1. On the other hand,
or decreasing K2�K1, the passbands become narrower,ith their central frequencies remaining fixed, becauseow the stronger K1 determines the split resonances ofhe molecules. Note that all our simulations are based onicroring resonators designed to operate at �0=1.55 �m
enter wavelength.Let us now consider a diatomic CROW with K1=K2 and
lternating optical perimeter lengths neff,iLi �i=1,2�. Wessume the optical lengths perturbed by ��1 so that theesulting round trip phases become 1,2=2��1±���Nr�f /FSR�. From the latter expression it follows that theerturbation shifts the resonance frequencies of the reso-ators to f0,i� f0FSR�Nr�. Here again we turn to theigenvalues ej2k� of the transfer matrix of the unit cell tond the new transition frequencies of the passbands sat-
sfying �H� 1 [see Eq. (10)]. Unfortunately, for alternat-ng optical lengths this transcendental inequality cannote resolved for the detuning �f as in Eq. (11); however, formall perturbations � is practically constant over a fewSR’s, allowing one to obtain an excellent approximation
o the analytical results,
1
�sin−1��1 − K�sin��Nr���� � �f
FSR�
1
�sin−1��K + �1 − K�sin2��Nr���. �13�
ue to the symmetric perturbation of the resonators, cor-esponding to Nr�1±��, the subsidiary stop band of width
�fSB = FSR2
�sin−1��1 − K�sin��Nr���� �14�
ests at the center of the passband. When the effective pe-imeter length perturbation is not chosen symmetrically,he width of the subsidiary stop band is determined byhe magnitude of the perturbation Nr��1−�2�, while theenter of the subsidiary stop band is offset by the averageSR�Nr��1+�2� /2. This provides a flexibility in selectingoth the width and the location of the subsidiary stopand, a flexibility not available with the coupling strengtherturbation. Observe that the width of the subsidiarytop band and the passband sections on the either side de-
ig. 3. (Color online) Drop port intensity as a function of nor-alized detuning and K2 for a lossless diatomic CROW with Nu6.5 (13 resonators) and a fixed K1=0.2.
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1246 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 I. Chremmos and O. Schwelb
end on the Nr� product, so for larger resonators, i.e.,maller FSR’s, a smaller perturbation is required.
Simulated examples of these results are presented inigs. 4 and 5. Figure 4 plots the drop port intensity of a
ossless diatomic CROW of 13 identical resonators �Nu6.5� and uniform coupling coefficients K=0.2, as a func-
ion of the detuning and mode number perturbation in theange −0.05 Nr� 0.05, indicating a redistribution ofhe ripples on the two sides of the subsidiary stop band,ut retaining the location of the stop band center. In con-rast, Fig. 5 illustrates the gradual offset of the subsidiarytop band center in a 12-resonator lossless device as a re-ult of a linear shift of the average perturbation −0.1Nr��1+�2� /2 0.1, while the width of the stop band is
eld constant as a result of a differential perturbationxed at Nr��1−�2�=0.1.We wish to remark on some symmetry properties that
oncern diatomic CROW devices of finite lengths. In a di-tomic CROW with an odd number of identical resona-ors, interchanging K1 and K2 does not affect the droport spectrum because the left-right reversal of the chainaused by the interchange does not affect the symmetrical31=S13 scattering parameter. The magnitude of the
hrough port parameter �S21�, however, remains unaf-ected only in the ideal loss-free case, as a result ofS21�2=1− �S31�2 that follows from the losslessness condi-ion, �i=1
4 �Si1�2=1, taking into account that S11=0 and41=0 due to the unidirectional propagation in the reso-ators. On the other hand, when the number of resona-ors is even, the structure is bilaterally symmetric; there-ore interchanging K1 and K2 leads to a completelyifferent device affecting both S41 and S21. The same ar-ument leads to the inverse situation for a diatomicROW consisting of rings with alternating optical
engths, neff,iLi, and uniform coupling coefficients �K1K2�: interchanging the lengths has no effect on S41 for aevice with even number of resonators, while it affectsoth S31 and S21 for odd number of resonators. We alsoention the functional flexibility offered by perturbing
oth the coupling strengths and the resonator parameterst the cost of a heightened fabrication complexity, a devel-pment we examined but left unreported.
The effect of increasing coupling strength on the sub-idiary stop band extinction and the effect of increasing
ig. 4. (Color online) Drop port intensity as a function of nor-alized detuning for a lossless diatomic CROW of 13 resonators
ersus the mode number perturbation Nr� �K1=K2=0.2� indicat-ng a shift in the distribution of the ripples but retaining the lo-ation of the stop band center.
ndex perturbation on the group delay characteristics areemonstrated on a ten unit cell (20-resonator) lossy di-tomic CROW in Fig. 6. Figure 6(a) demonstrates that in
diatomic CROW with alternating effective resonatorengths, increasing coupling strength decreases the ex-inction and the bandwidth of the subsidiary stop bandreated by the resonator perturbation, in this case the in-ex modulation. Figure 6(b) shows that increasing � deep-ns the central minimum, increases the width, and raiseshe edge peaks of the group delay spectrum. Although nothown in Fig. 6(b), the same effects are recorded when for
fixed � the common coupling coefficient is decreased.ote the remarkable analogy between Figs. 6 and 7 of [7]
I41 of our Fig. 6(a) corresponds to 1−R in decibels of Fig.of [7]) characterizing Bragg gratings.
. DISPERSION COMPENSATIONhe diatomic CROW exhibits a region of K1 and K2 valueshere the drop port intensity is maximized and both this
ntensity and the group delay are relatively flat over aide range of wavelengths. This can be gleaned from Fig., but a more quantitative assessment is provided in Fig.(a) showing the relative output intensity of a four-esonator lossless diatomic CROW using a fixed K1=0.54nd a variable K2, and in Fig. 7(b) where the correspond-ng group delay spectra are plotted for four K2 valuesanging from 0.22 to 0.28. As is usually the case, the criti-al coupling to obtain a maximally flat transmission doesot occur at the same value required for a maximally flatroup delay, although the two are rather close; K2=0.28nd K2=0.22. The dispersion computed from
Dij =1
NuL
��ij
��= −
2�c
�2NuL
��ij
��, �15�
orresponding to the central section of Fig. 7(b), is plottedn Fig. 8(a), indicating that K2 can effectively control thelope of the dispersion. These characteristics were foundo be unaffected by as much as 8 dB/cm waveguide at-enuation, while the resonant insertion loss of the devicet K2=0.15 was only slightly above 2 dB. We found thathe diatomic CROW provides a much greater flexibility ofispersion control than its uniform counterpart. Figure
ig. 5. (Color online) Drop port intensity as a function of nor-alized detuning for a lossless diatomic CROW of 12 resonators
ersus the average mode number perturbation Nr��1+�2� /2 hold-ng the differential perturbation fixed at Nr��1−�2�=0.1 �K1=K20.2�. Note the gradual shift of the stop band center in contrast
o the result of Fig. 4.
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Fd(rK
I. Chremmos and O. Schwelb Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1247
(b) plots the dispersion characteristics of an eight-esonator diatomic CROW with an effective resonatorength perturbation given by Nr1=Nr�1+��, Nr2=Nr�1��, �=1�10−3, and two-ring anti-reflection (AR) match-
ng �Ka=0.92, Kb=0.72, Kc=0.45� applied to both endsf the chain. As the straight dotted line indicates, the dis-ersion is almost linear over 10% of the FSR. The appliedoss does not affect the performance.
. SWITCHING IN A DIATOMIC CROWe describe a practical scheme for the output intensity
nd group delay switching in a diatomic CROW of evenumbered resonators. Keeping one set of the coupling co-fficients fixed we change the value of the other set. An
ig. 6. (Color online) (a) Drop port intensity of a ten unit cell20-resonator) lossy diatomic CROW where K1=K2 and Nr1Nr�1+��, Nr2=Nr�1−�� with Nr=40 and �=5�10−3. The param-ter is the common coupling coefficient which increases from 0.5o 0.9 in steps of 0.1. Increasing coupling strength decreases thextinction and the width of the dip. (b) Relative group delay athe drop port of the same diatomic CROW as in (a), but here thearameter is �, increasing from 1�10−3 to 5�10−3 in steps of 110−3, while the common coupling strength is fixed at 0.9.arger � corresponds to a deeper minimum. When � is kept con-tant but K1=K2 decreases, the width of this subsidiary stopand increases, its minimum descends, and the peaks markingts edges rise sharply.
xample is given in Fig. 9 where an approximately 23 dBxtinction is reached in a five unit cell (ten-resonator) de-ice keeping K1=0.14 fixed while switching K2 from 0.077o 0.25. This device was assumed to have a per turn dec-ement of �=0.99873 caused by a waveguide attenuationoefficient of �=8 dB/cm. With a reduced complexity inind, we simulated a four-resonator �Nu=2� device and
ound similar switching characteristics, albeit for a differ-nt set of K2 values.
Switching effect can also be demonstrated in diatomicROWs built with identical couplers and alternating ef-
ective indices. Figures 10(a) and 10(b) show the outputntensities and the drop port group delay of a 20-esonator CROW without index perturbation, using K1K2=0.8. The midband ripple of the drop port output for
he unperturbed lossless CROW, 10 log10�K��1 dB, isnly slightly affected by the �=0.99873 round trip loss.he baseline of the normalized group delay is at �41/�0Nu. Perturbing the effective indices by �=4�10−3, cor-esponding characteristics seen in Figs. 10(c) and 10(d)re obtained indicating a deeply suppressed subsidiarytop band and a significant drop in the center-of-bandroup delay. The depth of the passband ripple is not af-
ig. 7. (Color online) (a) Surface plot of lossless four-resonatoriatomic CROW where K1=0.54. Critical coupling is at K2=0.28.b) Normalized group delay of the same CROW for four K2 valuesanging from 0.22 to 0.28 in steps of 0.02. Critical coupling is at2=0.22.
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1248 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 I. Chremmos and O. Schwelb
ected by Nu; however it increases when K1=K2 is re-uced. The reduction in the coupling strength also signifi-antly increases the extinction ratio caused by the sameerturbation. We conclude that to get a sufficiently largextinction with a small perturbation, say �=4�10−4, a re-uced coupling coefficient, in this case K1=K2=0.1, muste applied. Alternatively, using materials with relativelyarge thermo-optic coefficients such as silicon (1.86
10−4/K [10]) would reduce the thermal requirement tobtain the necessary level of perturbation.
. MATCHING AND APODIZATIONhe passband of the uniform lossless CROW has an am-litude ripple at the center of the passband given byipple�f0�dB=20 log10��K�. The magnitude of the ripple in-reases toward the passband edges but does not dependn the number of resonators in the chain. The ripple isaused by the Fabry–Perot type resonances developinglong the chain between the mismatched terminals.
ig. 8. (Color online) (a) Group delay dispersion of a four-esonator diatomic CROW as a function of detuning; K1 is fixedt 0.54. These plots apply to both a lossless as well as to a lossyevice with a waveguide attenuation of �=8 dB/cm�=0.99873�. (b) Group delay dispersion of a diatomic CROWith K1=K2=0.3, an effective perimeter perturbed according tor1=Nr�1+��, Nr2=Nr�1−��, �=1�10−3, and two-ring AR struc-
ures at both ends of an eight-resonator chain using matchingoupling coefficients Ka=0.92, Kb=0.72, and Kc=0.45. The slopef the red dotted line is −0.9�10−15.
ethods to reduce this ripple, similar to those used toatch Bragg gratings to the external circuitry, have been
eveloped [11–13]. Here we adopted a simple two-ring ARatching structure appended symmetrically to both ends
f the diatomic CROW. We demonstrated that this ARtructure is equally effective when used with alternating
or with alternating Lneff diatomic chains.Figure 11 shows a comparison between a lossless uni-
orm CROW of 70 resonators without AR matching, theame CROW with two-ring AR structures terminating itsnds, and the same chain with alternating coupling coef-cients. Figure 11(a) plots the drop port intensities, whileig. 11(b) depicts the corresponding group delay spectra.he coupling strengths of the uniform CROW were set to=0.4, yielding a 4 dB ripple, while those of the diatomicROW were chosen to be K1=0.42 and K2=0.36. A similarerturbation on a matched diatomic CROW with a uni-orm coupling strength �K=0.4� but with alternating ef-ective indices, Nr1=Nr�1+��, Nr2=Nr�1−��, and �=3
10−4, produced essentially the same result.The AR structure used in these simulation consisted of
wo identical Nr=40 microrings, coupled to the externalaveguide, to each other, and to the first resonator of theROW through coupling coefficients K =0.92, K =0.62,
ig. 9. (Color online) Switching in diatomic CROW with tendentical resonators and alternating coupling coefficients, where
1=0.14 in both switch settings. (a) Solid lines: I21; dotted lines:41. (b) Normalized group velocities. Loss of up to �=0.99873�=8 dB/cm� has minimal effect on the characteristics.
a b
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8Awtts
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I. Chremmos and O. Schwelb Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1249
nd Kc=0.44 respectively. This AR device reduced the 4B midband ripple in I41 of the uniform CROW to ap-roximately 0.015 dB. Similar results were obtained us-ng a diatomic CROW of only 30 resonators, but of largerK1−K2� difference. The two-ring AR structure could al-ays be tuned to yield a �0.015 dB ripple, except near
he outer edges of the passband.
. THE EFFECT OF LOSSe already remarked on the effect of loss in some of the
isplayed diagrams which have been computed assuming=0.99873 per turn loss caused by waveguide attenua-
ion and assuming lossless coupling. The waveguide at-enuation includes loss due to radiation caused by finiteending radii, scattering caused by inclusions and guideall surface roughness, and material absorption. Al-
hough the coupler loss has been neglected this can beeadily corrected by adopting a coupling matrix where t2
�2�1. Nevertheless, we prefer to lump the losses attrib-ted to the couplers together with those of theaveguides. Note that imperfect couplers suffer also fromolarization conversion and introduce a limited amount ofifferential loss between even and odd waveguide modes14].
Loss depresses the passband maxima by an amounthat depends not only on �, i.e., on � and L, but also on
ig. 10. (Color online) Switching in lossy ��=0.99873, �=8 dBonsisting of 20 resonators �Nr=40�. (a),(b) The sweep covers thdjacent stop bands. (c),(d) A subsidiary stop band develops in th=4�10−3 to cause Nr1=Nr�1+�� and Nr2=Nr�1−��. The solid lin
he number of resonators and on coupling strengths. Con-ider also that using an explicit waveguide attenuationoefficient � allows the assumption of a real refractive in-ex neff. However, while the number of resonators, theoupling strengths, and the resonator parameters have aefining influence on both the transmission and the groupelay of the diatomic CROW, these characteristics are re-arkably insensitive to the waveguide loss. We found, for
xample, that the maxima of the passband peaks droppedy only 0.2 dB as a result of �=10 dB/cm ��=0.99842�aveguide attenuation in a diatomic CROW of Nu=6 (12
esonators), and only by 0.5 dB when the number of unitells was raised to 20 (40 resonators). The locations of theand-edge frequencies, approximated by Eqs. (11) and13), are insensitive to loss. The same amount of wave-uide loss raised slightly the baseline and reduced incre-entally the excursions of the group delay characteris-
ics, bracketing the valley caused by the diatomicolecules in Figs. 10(b) and 11(b).
. FABRICATION ISSUESlthough we did not fabricate diatomic CROW devices,e wish to comment on possible strategies on how to ob-
ain the coupling and effective resonator size distribu-ions discussed above. The modulation of the couplingtrength, when the resonators are of the same size, can be
iatomic CROW with identical coupling coefficients �K1=K2=0.8�,band situated between approximately �f /FSR= ±0.37 plus thedle of the passband when the effective indices are perturbed byresent I21.
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1250 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 I. Chremmos and O. Schwelb
btained through a fractional shift of the centers of alter-ate resonators along the axis of the CROW. This processroduces alternate gap widths, consequently alternatingoupling strengths along the chain. A critical point withhis technique is the small size of the shift required toroduce small index perturbations. Another technique toroduce alternating coupling coefficients is suggested byhe laterally displaced racetrack resonators recently pro-osed in [15]. In this case the racetrack resonators areairwise offset, as shown in Fig. 12, to produce the de-ired alternating coupling strength. Considering that of-en only a very small amount of coupling strength pertur-ation is necessary to produce the required diatoms, it iseasible to achieve this effect also by applying an overlayn, or irradiate, every second coupling gap [16].
The production of a chain of resonators of alternatingffective perimeters, Lneff, requires the fabrication of twoets of ring sizes or two sets of effective indices. In therst case one can avoid making a chain of alternate diam-
ig. 11. (Color online) (a),(b) Drop port intensity and normal-zed group delay of a uniform CROW (K=0.4, light dashed-dottedine) of a uniform CROW with two-ring AR structures (solid line),nd a diatomic CROW with two-ring AR structures (K1=0.36,2=0.42, dotted line). The CROW consists of 70 lossless resona-
ors �Nu=35� and the coupling coefficients of the AR structure area=0.92, Kb=0.62, and Kc=0.44. When K1 and K2 are inter-
hanged the sidelobe excursions increase, i.e., the AR deviceust be re-optimized. The baseline of the dashed-dotted plot is
41/�0=Nu, and the midband ripple of the matched I41 curve is.015 dB.
ter microrings by opting for a radius-of-curvature main-aining (and also total width maintaining) ring-racetrack-ing scheme. In the second case one is faced with theelicate process of perturbing neff of every second elementf the chain. Since a fractional change of �neff /neff�110−3 is most likely sufficient, the thermo- or electro-
ptic index modulation is the obvious choice [8,17]. Sev-ral of our simulated applications call for a diatomic chainf only a very few unit cells. We foresee, therefore, an ap-lication here of a lithographic stepper process.Finally note that the concept of the diatomic CROW is
lso applicable to resonators of different kinds, such asesonators created by defects in photonic crystals or cas-aded dielectric layers (Bragg reflectors). All our conclu-ions are applicable to these structures as well providedhat the effective cavity lengths and the coupling coeffi-ients are properly interpreted.
. CONCLUSIONnalytical and numerical results of a novel, finite length,nd periodic photonic circuit have been presented whichonsists of direct (or series) coupled microresonators, butiffers from the well known CROW in that it comprises ofwo dissimilar sets of resonators, i.e., a diatomic unit cell,mparting significant design flexibility to the device, in-luding the insertion of a narrow subsidiary stop band ofariable width and location within the original passband.n analysis solving for the new band structure, including
he width of the subsidiary stop band shaped by the prop-rties of the diatomic unit cell, and copious simulationslucidating the interactions between the design param-ters are given. The computed group delay (group veloc-ty) spectra indicate their insensitivity to the resonatoross and the efficacy of simple twin-resonator AR termina-ions in reducing the passband ripple. The utility of theevice as a switch and comparisons between the proper-ies of short and long diatomic chains are also illustrated.
EFERENCES1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-
resonator optical waveguide: a proposal and analysis,” Opt.Lett. 24, 711–713 (1999).
2. J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang,and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103(2004).
3. M. Cherchi, “Bloch analysis of finite periodic microringchains,” Appl. Phys. B 80, 109–113 (2005).
4. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli,“Continuously tunable 1 byte delay in coupled-resonator op-tical waveguides,” Opt. Lett. 33, 2389–2391 (2008).
5. Y. Landobasa and M. Chin, “Defect modes in micro-ringresonator arrays,” Opt. Express 13, 7800–7815 (2005).
6. D. D. Smith, H. Chang, and K. A. Fuller, “Whispering-
ig. 12. Lateral offset technique proposed in [15] to achieve cou-ling strength perturbation to fabricate the diatomic CROW.
1
1
1
1
1
1
1
1
I. Chremmos and O. Schwelb Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1251
gallery mode splitting in coupled microresonators,” J. Opt.Soc. Am. B 20, 1967–1974 (2003).
7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15, 1277–1294 (1997).
8. S. J. Emelett and R. Soref, “Design and simulation of siliconmicroring optical routing switches,” J. Lightwave Technol.23, 1800–1807 (2005).
9. P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Sec.6.
0. A. Densmore, S. Janz, R. Ma, J. H. Schmid, D.-X. Xu, A.Delâge, J. Lapointe, M. Vachon, and P. Cheben, “Compactand low power thermo-optic switch using folded siliconwaveguides,” Opt. Express 17, 10457–10465 (2009).
1. P. Chak and J. E. Sipe, “Minimizing finite-size effects in ar-tificial resonance tunneling structures,” Opt. Lett. 31,2568–2570 (2006).
2. J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel,“Apodized coupled resonator waveguides,” Opt. Express 15,10196–10206 (2007).
3. M. Sumetsky and B. J. Eggleton, “Modeling and optimiza-tion of complex photonic resonant cavity circuits,” Opt. Ex-press 11, 381–391 (2003).
4. G. Cusmai, F. Morichetti, P. Rosotti, R. Costa, and A. Mel-loni, “Circuit-oriented modeling of ring-resonators,” Opt.Quantum Electron. 37, 343–358 (2005).
5. J. D. Doménech, P. Muñoz, and J. Capmany, “The longitu-dinal offset technique for apodization of coupled resonatoroptical waveguide devices: concept and fabrication toler-ance analysis,” Opt. Express 17, 21050–21059 (2009).
6. G. N. Nielson, D. Seneviratne, F. Lopez-Royo, P. T. Rakich,Y. Avrahami, M. R. Watts, H. A. Haus, H. L. Tuller, and G.Barbastathis, “Integrated wavelength-selective opticalMEMS switching using ring resonator filters,” IEEE Pho-ton. Technol. Lett. 17, 1190–1192 (2005).
7. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson,“Micrometre-scale silicon electro-optic modulator,” Nature435, 325–327 (2005).
