properties of regular polygons of coupled microring resonators

Properties of regular polygons of coupled microringresonators

Ioannis Chremmos* and Nikolaos UzunogluMicrowave and Fiber Optics Laboratory, School of Electrical and Computer Engineering,

National Technical University of Athens, 9 Iroon Polytechniou Str., Zografos 15780, Athens, Greece

*Corresponding author: [email protected]

Received 13 July 2007; revised 18 September 2007; accepted 18 September 2007;posted 18 September 2007 (Doc. ID 85260); published 25 October 2007

The resonant properties of a closed and symmetric cyclic array of N coupled microring resonators(coupled-microring resonator regular N-gon) are for the first time determined analytically by applying thetransfer matrix approach and Floquet theorem for periodic propagation in cylindrically symmetricstructures. By solving the corresponding eigenvalue problem with the field amplitudes in the rings aseigenvectors, it is shown that, for even or odd N, this photonic molecule possesses 1 � N�2 or 1 � Nresonant frequencies, respectively. The condition for resonances is found to be identical to the familiardispersion equation of the infinite coupled-microring resonator waveguide with a discrete wave vector.This result reveals the so far latent connection between the two optical structures and is based on the factthat, for a regular polygon, the field transfer matrix over two successive rings is independent of thepolygon vertex angle. The properties of the resonant modes are discussed in detail using the illustrationof Brillouin band diagrams. Finally, the practical application of a channel-dropping filter based onpolygons with an even number of rings is also analyzed. © 2007 Optical Society of America

OCIS codes: 130.3120, 230.5750.

1. Introduction

Groups of coupled optical cavities are currentlyattracting the increasing interest of the optical com-munity, because of their potential for a diversity ofapplications, such as optical filtering, sensing,waveguiding, and lasing. These composite opticalstructures can hitherto be divided in two general cat-egories: The first includes clusters of mutually cou-pled resonators, termed “photonic molecules,” whichsupport resonant supermodes that depend stronglyon the cluster geometry. The resonators composingthese structures are mainly wavelength-sized micro-spheres [1–6] or planar microdisks [7–14], support-ing spherical or cylindrical whispering gallery (WG)modes or non-WG modes of lower azimuthal order[6,13]. Other implementations with the individualresonators being based on the photonic bandgapmechanism have also been realized, such as groups ofBragg-mirror photonic cavities, which are coupled

through channels [15,16] and photonic crystal defectcavities [17]. A significant portion of the work in thisfield has been devoted to the study of the fundamen-tal structures of “diatomic” molecules, implementedas systems of resonant bispheres [1–5] and pairs ofmicrodisks [10,11,14] or cavities in a photonic crystalenvironment [15,17]. The coupling of resonatorscauses the splitting of their individual modes to bond-ing and antibonding supermodes, which are charac-terized by even or odd parity, respectively, to thesymmetry plane. The extension of the research to“multiatomic” photonic molecules has revealed newfunctionalities, which are based on the mutual cou-pling of resonators arranged in symmetric shapes.For example, symmetric molecules of WG disks canbe tuned to significantly enhance the Q factor of anondegenerate supermode, so that a quasi-singlemode structure composed by individual resonators ofhigh free spectral range can be obtained [8,9]. Thesesymmetric molecules have also been found to bepromising for the implementation of lasers by virtueof their low lasing thresholds [11–13] and tunabilityfor directive field emission patterns [18] or as sensi-

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7730 APPLIED OPTICS � Vol. 46, No. 31 � 1 November 2007

tive optical sensors owing to the strong interaction ofthe supermodes with the environment [9].

The second category includes linear periodic ar-rays of resonators, termed coupled-resonator opticalwaveguides (CROWs) [19], in which the electromag-netic coupling between successive elements is usedto effectively guide light. The different implementa-tions so far include arrays of photonic crystal defectcavities [20], Fabry-Perot cavities [21], microspheres[22,23], microrings and disks coupled serially [24–26]or vertically [27] and nanoparticle chains [28]. Thegroup velocity of light in a CROW is proportional tothe interresonator coupling coefficient and inverselyproportional to the electric cavity size (equivalentlythe cavity round-trip time) and thus can be signifi-cantly reduced, a fact that makes these waveguidesattractive for a number of photonic applications, suchas the design of delay lines [29].

This paper analytically investigates for the firsttime the resonant properties of a closed cyclic array ofN microring resonators forming a regular polygon.Each ring is considered to be single mode (it thereforesupports only two degenerate modes of clockwise andanticlockwise propagation) and is coupled only to itsneighbors. Consequently, the considered structurepossesses the geometry of a symmetric photonic mol-ecule [8], operates however with the photon-hoppingmechanism between successive resonators in aCROW and not with mutual coupling between allresonators. It therefore combines elements from bothmentioned categories and can be viewed equally cor-rectly as a symmetric microring CROW loop or aspecial case of a symmetric microring polygon mole-cule with negligible coupling between nonadjacent“atoms.”

The structure is analyzed through the method oftransfer matrices [24], which reduces the propagationand coupling phenomena in the resonators to linearequations of simple form, following from coupled-mode theory. This approach assumes that the reso-nators are weakly coupled through a coupling regionof very small, ideally infinitesimal length, which isaccurate for rings guiding modes of sufficiently highazimuthal order. Under these assumptions, the anal-ysis is also valid for microdisk resonators with a sin-gle WG mode, or WG microspheres that are coupledin the equatorial plane through the fundamentalspherical WG mode (m � l, according to the standardnotation for spherical WG modes).

The essence of the analysis of the resonant coupled-ring polygons is the application of the Floquet theo-rem, in a form appropriate for cylindrically periodicstructures. This leads to an eigenvalue problem hav-ing the field amplitudes in the rings on resonance aseigenvectors. Its characteristic equation gives thecondition for resonances, which is found to be identi-cal to the familiar dispersion equation of the corre-sponding ring CROW [19,24], i.e., the infinite straightchain of the same microrings, with the single modi-fication of a discrete wave vector that is imposed bythe cylindrical symmetry. This conclusion is an im-portant new result that reveals a fundamental, and

so far latent, connection between the closed symmet-ric resonant loop and the corresponding infinite array(waveguide) of coupled ring resonators. The mathe-matical derivation is based on the critical observationthat, for a regular polygon, the field transfer matrixover two successive rings is independent of the poly-gon vertex angle and therefore equal to the corre-sponding transfer matrix of the ring CROW. For aneven or odd number of rings, the obtained conditionpredicts the existence of 1 � N�2 or 1 � N differentresonant frequencies, distributed along the familiarnarrow frequency band of a CROW, which is con-trolled through the interresonator coupling strength.The field amplitudes in the rings on resonance aredetermined analytically as the corresponding eigen-vectors. By means of the classic picture of Brillouindispersion diagrams for periodic structures, themodal properties and the physical interpretation ofthese resonant modes are elaborated.

Note that the cyclic array of rings has previouslybeen studied in [30] with the aim to study the trans-mittance or reflectance when it is coupled to an ex-ternal waveguide. Under this scope, this analysis wasrestricted to specific numerical examples without de-riving the general resonant properties of an isolatedregular N-gon, which could explain the resonancesappearing in the spectrum of the output power. Sim-ilarly, a relevant analysis of a cyclic ring array in [31]as a slow-light structure did not derive the number ofresonances or an analytic dispersion equation. Theunderlying connection between this structure and theCROW is therefore revealed here for the first time tothe authors’ knowledge. We furthermore verify theresonant behavior of this system by analyzing thechannel-dropping filters, which are formed whenpolygons with an even number of rings are symmet-rically coupled to two parallel waveguides. Similarsymmetric filters have been analyzed through themethod of mode coupling in time [32], without touch-ing the issue of discrete resonances inside the trans-mission band. In the present analysis, apart from theverification of the predicted resonances, a basic con-clusion is the existence of two different profiles for thedropped power, depending on whether even N is alsoa multiple of 4 or not.

2. Analysis

Figure 1(c) shows the propagation and coupling ofwaves in two successive rings in an arbitrary N-gon.According to transfer matrix analysis, the two pro-cesses are respectively expressed through the matrixrelations,

�a2�

a1�� e�j�N 0

0 ej��N��a1

a2�, � b1

a1�� t �j�

�j� t �� b2

a2��,

(1)

where �N � �R�N is the phase acquired after propa-gation over an arc of �N rad, where �N � � � 2��N isthe vertex angle of the polygon, � � �2�R is theround-trip phase, and � is the interring coupling co-

1 November 2007 � Vol. 46, No. 31 � APPLIED OPTICS 7731

efficient, satisfying �2 � t2 � 1 for lossless coupling.Also � � n�c is the propagation constant of the ringmode, where n�� is the mode effective index, and Ris the ring radius. By incorporating the waves enter-ing and exiting the coupling region in each ring intoa single vector variable, as a � �a1 a2�T and b ��b1 b2�T, Eq. (1) gives the compact matrix equationfor the transmission between successive rings:

b � CP��N�a. (2)

Note that we have adopted the convention that thefirst and second element of the defined vectors is,respectively, the wave exiting and entering the cou-pling region [Fig. 1(c)]. In the last equation,

P��N� � �e�j�N 00 ej��N��, C �

1j��1 �t

t �1�, (3)

are the matrices of propagation and coupling pro-cesses, respectively. By applying this rule to the nextring [Figs. 1(a) and 1(b)], we find that c � T · a,where the total transmission matrix over two succes-sive rings is given by

T � CP�� N�CP��N� �2ej��2 sin��2�

�2

�1 �tej�

t �ej� �� I, (4)

where I is the identity matrix. From Eq. (4) it is seenthat matrix T is independent of the polygon vertexangle �N (and therefore the number of rings) anddepends only on the ring round-trip phase � and theinterring coupling strength �. This property is a re-sult of the fact that, in an equilateral polygon, thesum of the arcs propagated by the codirectionalwaves in two successive rings, before being coupled tothe third ring, is 2� [for example, the waves a1 and b1in Fig. 1(a)]. It furthermore allows the application ofthe Floquet theorem, properly formulated for periodicpropagation in a cylindrically symmetric structure.This procedure is in direct analogy to the standardapplication of the Floquet theorem in the infinite ringCROW [24], where one has �N � �, which can bethought of as the limit for N → �. Evidently, due toits mentioned independence of �N, transfer matrix Tholds also for propagation through two successiverings in the ring CROW.

From Figs. 1(a) and 1(b), it is obvious that theFloquet (or Bloch) condition is different for an even orodd number of resonators. When N is odd, two coun-terpropagating waves are excited in each ring (2Nwaves in total in the structure), and the initial waveis reached after N operations of T, i.e., TNa � a.Referring to the regular pentagon of Fig. 1(a), forexample, the closed sequences of waves connectedwith T are: �a, c, e, b�, d�� and �a�, c�, e�, b, d�. WhenN is even, only one wave is excited in each resonator

Fig. 1. Regular (a) pentagon and (b) hexagon of coupled ringresonators. (c) Propagation and coupling of waves in two adjacentrings.


(N waves in total) and the initial wave is reachedafter N�2 operations of T, i.e., TN�2a � a. Referring tothe regular hexagon of Fig. 1(b), for example, theclosed sequences of waves connected with T are:�a, c, e� and �b, d, f�. These arguments imply thatFloquet condition must be written in the form,

Ta � e�jwma, wm ��4m��N, N even2m��N, N odd , (5)

where m is an integer. Equation (5) formulates aneigenvalue problem for matrix T, where the eigen-value �m e�jwm takes N�2 (N even) or N (N odd)different values, obtained for m � 0, 1, . . . , N�2 � 1and m � 0, 1, . . . , N � 1, respectively. The charac-teristic equation |T � e�jwmI| � 0 is the condition forresonances, from which we obtain after some algebra:

sin��m�2� � � cos�wm�2�. (6)

Recalling that ��2 � ��R � n�R�c, Eq. (6) is iden-tical to the familiar dispersion equation of a CROW[19,24] with the discrete phase wm�2 in place of theproduct K� of the Bloch wave vector and lattice pe-riod. After substituting �m � �0 � ��m, where �0 ��0� � 2q� for rings resonating at an angularfrequency � 0 (integer q is the azimuthal order ofthe ring mode), Eq. (6) gives the resonant frequenciesof the polygon as m � 0 � �m where

�m �0

q�

�sin�1� cos�2m��N��, m � 0, 1, . . . , N�2, N even

sin�1� cos�m��N��, m � 0, 1, . . . , N, N odd .

(7)

Note that in order to obtain Eq. (7), the effectiveindex must be considered to be constant n�� n�0� over the corresponding frequency band, whichis a good approximation for weakly coupled rings�� 1� and large azimuthial orders q, which are theconditions for a narrowband CROW [bandwidth� � 2�0��q��]. Also note that under the condition� �� 1, Eq. (7) can be further simplified sincesin�1�x� � x for x �� 1.

Equations (6) and (7) show that the effect of loopinga ring CROW to form a regular N-gon is the discreti-zation of its continuous mode spectrum to a set of1 � N�2 (N even) or 1 � N (N odd) resonant frequen-cies. This result should be physically expected byintuition as N → �, since the condition for resonanceswould then be that the accumulated phase of theBloch wave with propagation constant K, after a fullrotation (propagation over N resonators), must beKN� � 2m�. Here and for the first time to theauthors’ knowledge, we have proved that this allega-tion is true for any finite and even N � 4 (recall theconnection -equivalence- of K� with wm�2) and alsofound its modification for odd N � 3 [second case ofEq. (5)]. It is clarified that, for N � 2, the condition

for resonances in a double-ring resonator system issin��m� � � (resonance split), which evidently meansthat Eq. (6) does not cover the diatomic molecule.

Figures 2(a) and 2(b) show the distribution of thedetermined resonances in a ��m � wm�2 diagram(counterpart of the familiar k � � diagram for peri-odic waveguiding structures) for the case of an even�N � 8� and an odd �N � 5� number of rings, notingalso the corresponding eigenvalues �m. The continu-ous dispersion lines of the corresponding straightCROW with infinite rings, satisfying sin��2� � � cos�K��, have also been plotted. It is interestingto see that, as follows directly from Eqs. (5) and(6), a resonant state at the central frequency 0[��m � 0, wm � �2p � 1��, p integer] exists only for aneven N that is a multiple of 4 �N � 4p�, while therealways exist resonances at the edges of the band�wm � 2p��. At each normalized resonant frequency��m there exist two complex conjugates eigenvalues,�m and �̄m � �m

�1 (e.g., �1 � �j, �3 � j, for N � 8). At

Fig. 2. Brillouin diagram of resonances for a regular (a) octagon�N � 8� and (b) pentagon �N � 5� with interring coupling coefficient� � 0.2. The dashed curves are the continuous dispersion lines ofthe infinite ring CROW.


the edges of the band the eigenvalues are equal to 1,while at the center of the band (when N � 4p) theyare equal to �1. The properties and the physicalinterpretation of these modes are discussed in Sec-tion 3.

3. Properties of the Resonances

To completely define the resonances, the correspond-ing field amplitudes in the rings should be deter-mined. Using any one of the two equations of theeigenvalue problem (5) under the condition (6), theratio of the amplitudes of the fields entering andemerging from the coupling region in any ring on aresonance with ��m � 0 (the case ��m � 0 will beexamined separately) is found to be

a2

a1 �m �

e�j�m�2

t cos��m�2� � sin�wm�2��, (8)

where the upper and lower sign correspond to thesign in Eq. (6). The ratio of field amplitudes in neigh-bor rings is also needed. This is found using any oneof the two equations of matrix Eq. (2) taking also intoaccount the ratio �m of Eq. (8). Since b2�b1 � a2�a1� �m, we have b1�a1 � b2�a2, so that the result can bewritten in the vector equation form,

b � �aej��m�N�wm�2�, (9)

where the relation �N � � � 2��N has also been used.Note that if we choose waves b and c [Figs. 1(a) and1(b)], Eq. (9) is the same except for a negative sign for�m. Therefore, the combination of the two expressionsverifies the Floquet condition c � ae�jwm. In conclu-sion, by normalizing an arbitrarily chosen amplitudeto unity, for example, a1 � 1, Eqs. (8) and (9) deter-mine all waves in the structure, for even or odd N.

Special attention should be given to the case of aresonance at the central frequency ��m � 0�, which,as already mentioned, exists for N � 4p (p integer).In that case the transmission matrix of Eq. (4) be-comes T � �I and has the double eigenvalue � � �1wm � �2p � 1��, p integer] and the two independenteigenvectors a � �1 0�T, �0 1�T. The correspondingfield amplitudes b in the neighbor rings are foundfrom Eq. (2) to be

a � �1 0�T → b � �jej�0�N��1�1 t�T,

a � �0 1�T → b � �jej�0�N��1�t 1�T. (10)

Note that these modes correspond respectively to theclockwise and anticlockwise circulations of power inthe molecule with a condition of full power couplingbetween adjacent rings. As a result, in half of therings the power carried in the direction opposite topropagation is zero, which is the explanation of thezero elements in eigenvectors a. Also, the eigenvalue� � �1 means that c � �a etc., i.e., the fields in ringsn, n � 2, n � 4, . . . are oscillating in antiphase.

It is very interesting to study the interpretation ofthe resonances appearing in the ��m � wm�2 dia-grams (Brillouin diagram) of Fig. 2. Figure 3 showsthe pairs ��m, �m� of eigenvalue and field amplituderatio that correspond to resonances at equally de-tuned frequencies 0 �m. Note that we have alsoused the property �m

��m, wm��m��m, wm� � e�j�m,

obtained from Eqs. (6) and (8). From Fig. 3 we con-clude that the points with the same frequency detun-ing (equivalently ��m) and Bloch phase wm � 2n�(n integer) correspond to the same field amplitudes inall rings [we also remind that the sign in Eq. (9)changes when wm�2 is changed by � so that theratio remains the same]. In an infinite periodic struc-ture, such as a waveguide grating, the actual guidedmode is composed of a linear superposition of allthese Floquet harmonics for n � ��, . . . , �. How-ever, in the present cylindrically symmetric and pe-riodic structure, all these points (noted with * in thefigures) correspond to the same resonant mode thatsatisfies the condition ��2 � ��e

�jwm, 0 � wm � �,where � denotes the ring. At the same frequency(same ��m) the points not noted with * have the Blochphase �wm � 2n�, the pair ��̄m, �̄m

�1�, and corre-spond to a mode satisfying ��2 � ��e

jwm, i.e., rotat-ing in the opposite (anticlockwise) direction. Inanalogy to the CROW problem, note that these reso-nances belong to the continuous CROW dispersionlines with the same and negative slope, which corre-spond to a backward wave. In conclusion, the reso-nances ��m, �m� and ��̄m, �̄m

�1� are degenerate modesrotating clock- and anticlockwise. The magnitude ofthe wave amplitudes in the rings is 1 for the wavespropagating in the direction of rotation (for example,a1, b1, . . . for the clockwise mode) and |�m| for theopposite propagating waves. As seen in Fig. 3, similarconclusions also hold for the resonances at the sym-metric with respect to 0 frequency ��m�. The“forward” modes (those on the continuous lines withpositive slope and noted with ��) now satisfy theFloquet condition ��2 � ��e

�jwm�, with phase

Fig. 3. Pairs ��m, �m� of eigenvalue and field amplitude ratio forresonances with the same frequency detuning from 0.


wm� � 2� � wm �� wm� � 2�� and have the complexconjugate relative amplitude �̄m.

Apart from the forward and backward resonantmodes, it is very interesting to study the modes lo-cated on the standing parts (extrema) of the contin-uous dispersion lines (see Fig. 2) and occur at theedges of the CROW band sin��m�2� � ��. In thiscase the transmission matrix T has the double eig-envalue � � 1 (wm � 2p�, p integer) and only oneeigenvector a � �1 �m�T, where �m � e�j�m�2, follow-ing from Eq. (8). The two cases of Eq. (9) also give thesame value for the ratio of field amplitudes betweenadjacent rings, which is b � �aej�m�N. An immediateconclusion of these relations is that all waves in therings have the same magnitude. Furthermore, usingthese amplitude ratios the following relations can beshown to hold

a1e�j�N�2 � �b2e

�j�N�2 � a2ej��N��2 � �b1e

�j��N��2.(11)

By assuming fictitious lines connecting the center andthe vertices of the polygon, it is easy to see that the lastrelations show that the fields of adjacent rings at thepoints where these lines intersect the rings (see Fig. 2)are in antiphase. Especially in the case of odd N, thefield in each ring is standing, being the superposition oftwo counterpropagating waves with equal amplitude.Taking, for example, a, a�, and since Ta� � a� it isa� � b � �aej�m�N, which furthermore yields a zerototal field at the mentioned intersection points (fieldnodes or analytically a1�e

�j��N��2 � a2ej��N��2 � 0).

The symmetry of the standing modes for even or oddN is shown in Fig. 4. For odd N, Fig. 4(a), the linesmentioned before and the lines connecting the center,and the middles of the sides of the polygon are mirrorplanes of odd field symmetry (electric�magnetic wallsfor TE�TM field). For even N, Fig. 4(b), the mirrorplanes of odd field symmetry are the lines connectingthe center and the middles of the polygon sides. Thefields in the rings are rotating (not standing) and arein antiphase in the neighbor rings (see also the defi-nition of � modes in [13]).

From a theoretical point of view, it is useful torealize that the previous results for the modes ofcoupled-ring polygons are in agreement with thenumber of modes predicted through the symmetryanalysis of this structure or other polygon-shapedoptical resonators based on group theory [8,33]. In-deed, for even N, there are N�2 � 1 resonant frequen-cies inside the band and two resonant frequencies atthe edges (1 � N�2 in total). Each inner resonancecorresponds to two possible modes, since the wave ineach ring may be rotating clockwise or anticlockwise[equivalently, one may consider the sum and differ-ence of these two modes in the same sense that ex-ponential waves exp� jkx� are replaced by sinusoidalwaves sin�cos�kx�]. Furthermore, as mentioned be-fore, each of these modes is double degenerate sincethe CROW wave can rotate clockwise or anticlock-

wise around the polygon. Similarly, each of the res-onances at the band edges (standing modes)corresponds to two possible modes, since again thewave in each ring may be rotating clockwise or anti-clockwise [see Fig. 4(b)]. These modes are standingand not traveling around the polygon �wm � 2p�� andare nondegenerate. In conclusion, there are N � 2double-degenerate modes and four nondegeneratemodes, giving a total of N � 2 modes, in agreementwith the mode numbers given by group theory [8,33].For odd N the field in each ring is always standing

Fig. 4. Field symmetry for the standing modes and (a) oddN � 5 or (b) even N � 6. The dashed lines are planes of zero field.


[Fig. 1(a)] and we therefore have N � 1 double-degenerate modes inside the band and two nondegen-erate modes at the edges, giving in total N � 1 modes.

In Section 4, the coupled-ring resonator polygons,which were so far studied as isolated resonant struc-tures, are examined as part of a standard channel-dropping filter system, which is the simplest way tosee how their resonant properties are manifested un-der coupling to external waveguides. The analysis isrestricted to polygons with an even number of rings,resulting in symmetric optical quadrupoles.

4. Channel-Dropping Filter Application

Figure 5 shows a channel-dropping filter device, ob-tained by coupling a coupled-microring hexagon totwo parallel waveguides. The parameters of the poly-gon are defined as in Section 3. A coupling coefficient�w is assumed at the input and drop points, where thecoupling of waves is expressed by the matrix equa-tions,

f � Dw · adrop, a � Dw · ain, Dw �1

j�w��tw 1

�1 tw�, (12)

where for lossless coupling �w2 � tw

2 � 1. Followingthe convention used previously, the input anddropped field vectors are written as ain � �Tt 1�T,adrop � �Td 0�T, where Tt and Td are the complexamplitudes of transmitted and dropped waves, re-spectively, under an incident waveguide mode of uni-tary amplitude. From Fig. 5, it is evident that sixwaves must be determined in the each of the ringscoupled to the input and drop waveguides. The prop-agation of these waves in the rings is expressed by therelations (six equations in total):

f � A · d � B · e, e2 � d1e�j�N

a � A · b � B · c, c2 � b1e�j�N

,

A � �0 ej��N��2

0 0 �, B � � 0 0e�j��N��2 0�. (13)

The waves in the two rings, which are coupled to theinput and drop waveguide, are connected due to prop-agation through the upper or lower part of thecoupled-ring sequence. This relation is expressed as

d � S · b� � S · D · b, c � S · e� � S · D · e, (14)

where matrix D of interring coupling is defined asmatrix Dw of Eq. (12) with � in place of �w. As deducedfrom Fig. 5, in the case of a hexagon, matrix S is equalto the transmission matrix T over two rings, whichwas defined in Eq. (4). For a general N-gon (even N)it can be shown that S is given by

S ��CP��N�Tp�1, N � 4p

Tp, N � 4p � 2, (15)

where p is an integer and matrices P and Cwere defined in Eq. (3). Note also that in the caseN � 4p the direction of propagation for waves adrop, d,e, f, of the drop waveguide and the adjacent ring isopposite to this shown in Fig. 5.

Equations (12)–(15) are combined to build a14 14 linear system, which is solved for the trans-mitted and dropped field amplitudes Tt and Td, forany even N (the rest unknowns are the two compo-nents of vectors a, b, c, d, e, f). The transmittedpower profile is found by energy conservation, whichin the present lossless structure is written as |Tt|

2

� 1 � |Td|2.

Figures 6–9 show the computed results for thedropped power |Td|

2 for filters with N � 6, 8, 10, 12,versus the difference �� 0 of the ring round-trip phase from the value �0 � ��0� � 2q�, whichcan be viewed as normalized frequency detuning. Thecoupling coefficients �, �w have been considered con-stant in the band of interest, which has a width of�� 4 sin�1�� [the distance from the first to the last

Fig. 5. Channel dropping filter of a regular hexagon symmetri-cally coupled to two waveguides.

Fig. 6. Dropped power |Td|2 for a regular hexagon �N � 6� versus

�� 0, in two cases of ring-waveguide coupling coefficient�w � 0.25 (dashed curve) and 0.35 (solid curve). The interringcoupling coefficient is � � 0.2.


resonance, Eq. (7)]. Note that with frequency inde-pendent coupling coefficients, the power profiles are2�-periodic functions of ��.

The results of Figs. 6–9 for the channel-droppingfilter are in perfect agreement with the resonantproperties of the isolated polygons, which were inde-pendently determined in Section 3. Using Eq. (7), inthe form ��m � 2 sin�1� cos�2m��N��, the positionof all resonances can be verified. For � � 0.2 andN � 6, we have ��m � 0.20, 0.40, which agreewith the position of resonances shown in Fig. 6. ForN � 8 we have ��m � 0, 0.28, 0.40, in agreementwith Fig. 7. Similarly, for N � 10 and N � 12 weobtain ��m � 0.12, 0.32, 0.40 and ��m � 0, 0.20, 0.35, 0.40, which agree with Figs. 8 and 9,respectively.

Finally, it is interesting to note the differentchannel-dropping response for the cases of N � 4pand N � 4p � 2, which were also discriminated in theanalysis, Eq. (15). When N � 4p (Figs. 7 and 9), all

resonances appear as maxima |Td| � 1 in the dropport (equivalently |Tt| � 0, rejective resonances).However, when N � 4p � 2 (Figs. 6 and 8) the powerprofile is more complicated. In this case, only theouter (standing) resonances, i.e., those that are ob-tained from Eq. (7) for m � 0, N�2 and have the samefrequency independent of N ��m � 2 sin�1��, arerejective. The rest of the resonances appear as par-tially rejective peaks ��Td� � 1�, which are splitby a sharp dip of the dropped power ��Td� � 0� onthe resonant frequency, so that these are actuallytransmissive resonances �|Tt| � 1�. The maximumdropped power around these resonances falls as weapproaching the central frequency �� 0�.

Note that in order to verify the ideal resonant be-havior of the polygons the rings have been assumed tobe lossless. Loss can be easily introduced in the anal-ysis of this section by adding an imaginary part tothe ring round-trip phase � � j ln � where � � 1 isthe round-trip amplitude decrease factor. However thedetailed numerical study of the effect of losses will bethe scope of a future work and is not included here.

5. Conclusions

The resonant properties of an equilateral polygon ofN coupled microring resonators were for the first timestudied analytically using the method of transfer ma-trices. It was proved that the condition for resonancesis identical to the familiar CROW dispersion equationwith a discrete wave vector imposed by the cylindri-cal symmetry. Therefore, the connection of the closedcyclic ring array with the corresponding ring CROW,which might so far be intuitively expected only in thelimit N → �, was rigorously proved to be valid for anyfinite N. This was a result of a critical property of thefield transfer matrix through two successive rings,being valid only for regular polygons. It may, how-ever, cast light on the analysis and help the under-standing of similar photonic molecule structures. Themodal properties and physical interpretation of thedetermined resonances were discussed in detail. Fi-

Fig. 7. Dropped power |Td|2 for a regular octagon �N � 8� versus

�� (� � 0.2, �w � 0.25).

Fig. 8. Dropped power |Td|2 for a regular decagon �N � 10�

versus �� (� � 0.2, �w � 0.25).

Fig. 9. Dropped power |Td|2 for a regular dodecagon �N � 12�

versus �� (� � 0.2, �w � 0.25).


nally, the practical case of a symmetric channel-dropping filter, based on polygons with an evennumber of rings, was examined using transfer matri-ces and the response at the drop port indeed verifiedthe expected resonant behavior of the polygons.

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properties of regular polygons of coupled microring resonators

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