## Abstract

We present a coupling matrix formalism to investigate the effects of periodic and quasi-periodic orders on the photonic bandgap (PBG) structures of coupled-resonator optical waveguides (CROWs) based on microring resonators. For the periodic order case, size-tuned defects are introduced at periodic locations among the regular rings, which are size-untuned, to form a periodic ordered CROW system. The periodic coupled defects result in multiple localization states that lead to the formation of mini-defect bands and mini-PBGs within the PBG of a defect-free CROW. The position and number of such mini-defect bands depend on the size tuning of the defects. For the quasi-periodic order case, the arrangement of the defects and the regular rings in the ring cascade is an intermediate between periodic order and randomness, thus forming a quasi-periodic ordered CROW system. The effects of quasi-periodicity on the PBG structures are illustrated using the Fibonacci sequences, which result in a single high-Q localized state to appear that gradually transits to a mini-band within a wide photonic stop band as the number of lattice cells increases.

©2009 Optical Society of America

## 1. Introduction

Microring resonator is a compact waveguide realization of Fabry-Perot cavity, with the capacity to generate high-Q resonance in a small modal volume. By coupling various resonators together, many interesting phenomena occur [1–3] and have been used in various applications such as optical filtering [4–7], delay lines [8] and optical buffering [9]. If the number of such coupled rings is large, it can be regarded as a new type of waveguide known as CROW [1], in which light propagates via photons hopping from one tightly confined mode to the neighboring one due to the weak interaction between them. Studies on microring CROWs have focused mainly on the properties of the photonic bands for slowing light [1–3, 8, 9] and optical filtering [4–7], with only some coverage [5–7, 10, 11] on the characteristics of the photonic bandgap (PBG) structures. The PBG structures can generally be modified by breaking the lattice periodicity through the introduction of defects to form *disorder* in the periodic system, which generate localized states within the PBG and this is commonly known as PBG engineering. The introduced defect allows light to be localized at the defect cavity, forming a localized mode or defect mode. When such defect mode is coupled to the propagating mode in a finite structure, a high-Q transmission appears at the corresponding resonant frequency. The nature of light localization via implementing disorder in periodic system has been widely explored in photonic crystals [12, 13] and has been recently looked into in ring resonator arrays [10] where the effect of a single defect on the PBG structures was investigated.

In this contribution, we provide an analysis of the effects of PBG structures of direct ring CROWs [3] in the presence of multiple defects of similar size. Using such defects, which are size-tuned rings, we will illustrate how PBG engineering can be achieved by implementing either periodic or quasi-periodic order in ring CROW systems in contrast to using disorder [10, 13]. The defects are introduced at specific locations in the CROW so that together with the regular rings, which are size-untuned rings, there will be either periodic or quasi-periodic order in the final ring arrangement of the cascade. This is in contrast to traditional ring CROWs [14] whereby periodic order is theoretically implemented by cascade of rings of similar size, without defects. Note that strictly speaking, the normal distribution of defects in real systems is generally neither periodic nor quasi-periodic but random. However, for ease of illustration, we loosely use the term “defects” throughout this theoretical work to refer to the size-tuned rings introduced in the CROW to form either periodic or quasi-periodic order in the ring arrangement.

For the periodic order case, we explore how size tuning the defect in each unit cell affects the PBG structures via modes localization. It will be shown that the coupling between the periodic size-tuned defects results in mode splitting of the single cavity defect mode into multiple distinct localization states that lead to the formation of mini-defect bands and mini-PBGs in what was formerly a single PBG in the original defect-free CROW and varying the coupling strengths creates a PBG at each location of resonance in the photonic bands.

In addition, deterministic aperiodic (AP) order in ring CROW system is also looked into in this work. Traditionally, implementing disorder in a periodic system has been employed to achieve light localization, having its origin in the Anderson localization of electron waves in solid state physics. However, it has been found that wave localization can occur not only in disordered or absolutely random systems but also in deterministic AP ordered ones [15]. Such AP systems behave much like disordered ones but are constructed based on a deterministic procedure and thus possess order without periodicity. In photonics, deterministic AP order has already been looked into in optical multi-layers [16, 17] and photonic crystals [18–20] and in this work, we study the nature of localization using one dimensional deterministic AP order in ring CROWs. There are different classes of deterministic AP order, mainly quasi-periodic (QP) and fractal. The main difference between QP and fractal ordered lattices is that the former exhibits two or more incommensurate periods while the latter do not [21]. To achieve a strong connection with the theme of the work in the first part where the effects of periodic order are studied, we will only analyze deterministic QP order as mathematically, quasi-periodic functions can be classified as almost periodic functions [22]. In our work, QP ring CROW is designed by cascading together rings of two different size, namely the defect and regular rings, according to simple rules based on the Fibonacci series, thereby encoding a one dimensional QP order in the ring arrangement. Note that QP order of higher dimensionality can be generated by using photonic crystals in which *identical* dielectric cylinders are arranged according to specific rules in the absence of defects, as demonstrated in [19, 20], unlike the case of our ring Fibonacci CROW. It will be shown here that the quasi-periodicity of the Fibonacci CROWs can also generate mini-PBGs and localized states in what was formerly a single PBG of a CROW with identical rings. This usage of Fibonacci sequences in the design of ring CROW results in the appearance of a single high-Q localized state that transits to a mini-band within a wide PBG as the number of unit cells increases.

We use the transfer matrix formalism [14] in our analysis which is physically more convenient over the tight binding approximation [1] and the coupled mode theory [4, 23] as it can better represent the physical parameters such as cavity size and can be easily coupled with the Bloch theorem to study the PBG structures of periodic systems. In section 2, we use the transfer matrix formalism to analyze the PBG structures of periodic CROWs in the presence of defect size tuning as well as coupling tuning. The Fibonacci class of quasi-periodic ring CROWs is then studied in section 3. Throughout this paper, all the rings are assumed to be lossless so as to better focus on the transfer matrix analysis. We only discuss the effects of losses in section 4. The transfer matrix method is then compared with the finite-difference time domain simulation in section 5. Finally, we conclude, highlighting important results and possible extension of this work as well as discussing possible applications of the proposed ideas.

## 2. CROW with periodic order

#### 2.1 Coupling matrix formalism

The schematic of the CROW with periodic defects is shown in Fig 1. We label the layout of the unit cells as *n* = 1, 2, 3…N. The component rings in each cell are termed as *intra-rings* labeled A and B, respectively, and the ratio of their cavity lengths is defined as the size tuning factor *γ* = L_{B}/L_{A} = δ_{B}/δ_{A}, where δ is the round-trip phase shift in each ring and contains the frequency dependence. Note that ring A is considered as the regular ring and B as the “defect” ring. The input and output fields to the n^{th} lattice are denoted respectively as (a_{n-1,2}, d_{n+1,1}) and (a_{n,2}, d_{n,1}), where the first number in the subscript refers to the unit cell while the second number refers to the intra-rings (1 for ring A and 2 for ring B). The fields in the bus waveguides follow the conventions as that of the unit cell as shown in Fig 1. Within each unit cell, the input field to the 1^{st} intra-ring is denoted as d_{n,2}, while that for the 2^{nd} intra ring is a_{n,1}. Following the analysis in [23], each coupler is treated as a partially transmitting mirror with real transmittivity *t* and reflectivity *r*. We then denote (r_{intra}, t_{intra}) as the coupling scheme between A and B in each unit cell, and (r_{inter}, t_{inter}) as the inter-unit cell as well as the bus-unit cell coupling. Using the nomenclature in Fig 1, the coupling mechanism from (a_{0,2}, d_{1,1}) to (b_{0,2}, c_{1,1}), is represented as:

where *m*
_{11} = *m*
_{22} = -*r*
_{int er}, *m*
_{12} = *m*
_{21} = *it*
_{int er} , in which i is an imaginary number. The fields in the first intra-resonator in the first unit cell can be written as:

where τ_{p} is the attenuation factor due to losses for one round trip in the p^{th} intra-ring (p = 1 for intra-ring A and 2 for intra-ring B) of each unit cell. Using (1) and (2), it can be derived that:

where *Sp*
_{11} = -*A*/*it _{p}*,

*Sp*

_{12}= -

*r*/

_{p}A*it*,

_{p}*Sp*

_{21}=

*r*/

_{p}B*it*,

_{p}*Sp*

_{22}=

*B*/

*it*, in which

_{p}*A*= √

*τ*exp(-

_{p}*iδ*/2),

_{p}*B*= √

*τ*exp(

_{p}*iδ*/2)and

_{p}*p*= 1 or 2 which denotes for either matrix [S1

_{ij}] or [S2

_{ij}]. For the reflectivity

*r*, when p = 1, then r

_{p}_{p}= r

_{inter}and if p = 2, r

_{p}= r

_{intra}and likewise for transmittivity

*t*. Matrices [S1] and [S2] are the intra-ring translation matrices that relate the complex amplitudes (a, b) between each intra ring while matrix [U

_{p}_{ij}] is the unit cell translation matrix that relates the complex amplitudes (a, b) in one ring of a unit cell to those of the equivalent ring in the next unit cell. Matrix U is unitary if the intra-rings and the couplings are lossless. For N unit cells, the field amplitudes (a

_{N,2}, b

_{N,2}) at the last ring B in the N

^{th}lattice can be obtained by iterating the unit cell translation matrix N times via the Chebyshev polynomials [24]:

where x = cos ψ = ½(U_{11}+U_{22}), U_{ij} is the element of the unit cell translation matrix [U] and C_{N}(*x*) = sin[(*N* + 1)*θ*]/sin*θ* are the Chebyshev polynomials of the second kind.

#### 2.2 Band structures of infinite CROW with periodic defects

The photonic band of an infinite CROW with periodic defects can be derived by implementing the Bloch theorem to the unit cell transfer matrix in (3). For a periodic structure of infinite length, the fields are periodic at the lattice constant Λ:

where *K* is the Bloch wave propagation vector. Combining (5) with (3) gives the requirement *Det*∣[U]-exp(*i*KΛ)*I*∣ = 0, where *I* is the identity matrix. Then, making use of the property of unimodular matrix [U]: U_{11}U_{22}-U_{12}U_{21} = 1 and that cos θ = ½(U_{11}+U_{22}), it can be evaluated that:

The characteristic equation of the eigenvalue problem of (6) for the lossless case can be expressed as:

where r_{u} = r_{inter} · r_{intra} and t_{u} = t_{inter} · t_{intra}. Equation (7) gives the dispersion relation for a CROW of infinite length, with size-tuned periodic defects and is only satisfied if the left most cosine term is between -1 and 1. For values outside this range, *K* is purely imaginary and the corresponding frequency range forms the PBG. We now use (7) to analyze the effects of periodic defects on the PBG structures of CROWs in the absence of losses.

### 2.2.1 Results and discussions

### 2.2.1 (a) Effects of size tuning the periodic defects

In this section, the effects of size tuning the periodic defects on the photonic bands and PBG structures of an infinite CROW with unit cell (**AB**) are looked into. For simplicity we first consider only small integer values of the tuning factor, *γ* = 1, 2, 3,… (i.e., Ring **B** is an integer multiple larger than ring **A**), and subsequently non-integer values. Displayed in Fig 2 are the dispersion diagrams for an infinite CROW of different integer size tuning *γ* of the defect. All rings have identical coupling (t_{inter}
^{2} = t_{intra}
^{2} = 0.7) and are assumed to be lossless. The resonance order *m* is defined as *m* = *f* /∆*f*
_{FSR} = δ/2π, where *f* is the eigen-frequency, ∆*f*
_{FSR} denotes the free spectral range and we refer to *δ _{A}* or

*δ*simply as

_{1}*δ*for the round trip phase shift for the size-untuned ring

**A**throughout this paper. In the absence of losses,

*K*is purely real at the photonic bands, which is indicated by the solid blue line, and is strictly imaginary at the PBGs, represented by the shaded regions in the dispersion diagrams. Initially, when

*γ*= 1 that corresponds to L

_{A}= L

_{B}, the wave propagation takes place within the photonic bands centered at the resonance order of integer

*m*. The PBGs, where wave propagation is forbidden, are centered at half-integer

*m*. Tuning

*γ*to higher integer values simply increases the number of PBGs. For integer values of

*γ*, the number of PBGs within the vicinity of

*δ*= π(2m+1) is also

*γ*. In Fig. 2, we show the full sequence of PBG for

*γ*= 1 to 4 within the range

*m*= 0.5 to 2.5. Observation of the behaviors of these PBGs shows the following general principles: First, due to the equal number of rings A and B, the sequence of PBG is always symmetrically centered at

*m*= 1.5, the half-integer value being required by the

*anti*-resonance condition for ring A. Then, within the sequence, each of the PBG is roughly located at a value of

*m*such that

*mγ*is a half-integer. This is the anti-resonance condition for ring B. For example, for

*γ*= 3, the possible

*m*values within the range 1 to 2 are

*m*= 7/6, 9/6 = 1.5, and 11/6. Similarly, for

*γ*= 4, these values are

*m*= 9/8, 11/8, 13/8 and 15/8. However, due to the mutual interaction between rings A and B, the PBG are not exactly centered at these values but “pulled in” slightly towards the center (

*m*= 1.5), and are not of equal width but those near the center are broader.

This general principle applies also for non-integer *γ*, such as when *γ* ranges between 1.5 and 2.5 (given the way we have defined rings A and B it is not necessary to consider the case *γ*< 1). We would find that the most obvious difference is that the PBGs are no longer distributed evenly about the half-integer *m* points (δ = π(2m+1)), but may shift up or down as *γ* decreases or increases from the integer value. The location of each PBG is still given roughly by the anti-resonance condition *mγ*= *n*+1/2 (where *n* is an integer). The physical picture of the formation of these PBGs will be more clearly illustrated using the case of finite CROW in section 2.3.2.

### 2.2.1 (b) Effects of tuning the intra and inter-lattice coupling strengths

In this section, we look into the effects of tuning the coupling strengths such that *t _{intra}*

^{2}≠

*t*

_{inter}^{2}. For a CROW with rings that have identical coupling (

*t*

_{intra}^{2}=

*t*

_{inter}^{2}=

*t*

^{2}), the width of the primary bands centered at

*δ*= 2mπ is proportional to the interaction,

*t*

^{2}. When the intra-ring and inter-unit cell coupling strengths are different, i.e.

*t*

_{intra}^{2}≠

*t*

_{inter}^{2}, it is found that a gap opens up, forming a PBG in the middle of each primary band at integer resonance order

*m*, even without size tuning, i.e.

*γ*= 1. As shown in Fig 3, when ∣

*t*

_{inter}^{2}-

*t*

_{intra}^{2}∣ = ∣

*F*∣

*increases*, the photonic bands shift in the direction of the arrows and PBG forms and widens within each resonance location of

*δ*= 2πm. The width of the PBG formed is proportional to the absolute magnitude of

*F*and is independent of its sign (i.e whether

*t*

_{intra}^{2}or

*t*

_{inter}^{2}is larger). The sign only dictates how the photonic bands move as ∣

*F*∣ changes. A negative (positive) sign signifies a weaker (stronger)

*t*

_{inter}^{2}relative to

*t*

_{intra}^{2}and is related to an outward (inward) movement of the photonic bands from the locations of the integer resonance order

*m*as ∣F∣ increases. Note that in Fig 3 we have varied

*t*

_{intra}^{2}while keeping

*t*

_{inter}^{2}fixed. Similar band diagrams are obtained if

*t*

_{inter}^{2}is instead varied with a fixed

*t*

_{intra}^{2}but with the sign of

*F*reversed. An intuitive insight on the dependence of the movement of the photonic bands on F and the formation and subsequent widening of PBG at resonance when ∣

*F*∣ changes are given in the next section in terms of the spectral properties of a single unit cell of (

**AB**).

#### 2.3 Band structures of finite CROW with periodic order

### 2.3.1 Matrix analysis of finite CROW

The dispersion diagrams in Fig 2–3 based on the dispersion relation of (7) strictly apply only for Bloch-wave propagation in an infinite periodic CROW. Any actual physical realization of CROW is of finite length. The exact dispersion relation for a finite CROW has been derived using the tight binding approximation in [25] and the transfer matrix method in [26]. However, it is not necessary to use the exact dispersion relation when modeling finite CROW as the general features of the transmission and phase responses of a finite CROW approximately satisfy the photonic band properties of the dispersion diagram for an infinite one. The transmissivity *T* of the finite CROW is defined as T = c_{N+1,1}/a_{0,2} and can be derived by employing (4) and the identity c_{N+1,1} = ita_{N,2}, where t = t_{inter}, and assuming that d_{N+1,1} = 0:

Making use of the Chebyshev polynomials in (4), the transmissivity expression of (8) for the general case of N unit cells, each consisting of dual asymmetrical rings, can be expressed as:

where *C* = exp(*iδ*
_{2}), *D* = exp(-*i*0.5[*δ*
_{1} - *δ*
_{2}], *E* = exp(*i*0.5[*δ*
_{1}+*δ*
_{2}]), *F* = exp(-*i*0.5[*δ*
_{1}-3*δ*
_{2}]), *G* = exp(*i*0.5[*δ*
_{1}+3<*δ*
_{2}]) and τ_{p} = 1. We use (8) and (9) to analyze the transmissivity and corresponding phase response of a finite CROW for different defect size tuning in section 2.3.3.

### 2.3.2 Formation of localized defect bands in finite CROW

We first give a physical picture of the formation of mini-passbands or defect bands due to the presence of multiple periodic defects by using a CROW with 7½ unit cells, with each cell having 2 rings, all with identical coupling strength t^{2} = 0.7. The defect ring, denoted as **D**, has twice the cavity size of regular rings (i.e., *γ* = 2) for this illustration. Initially, when the CROW contains only 1 defect, a single high-Q defect mode is localized in the PBG as shown in Fig. 4(a). As the number of periodic defects increases progressively, the localized states on separated defects interact with one another and their mutual coupling causes the splitting of the original degenerate eigenmodes. The initial splitting of the single cavity defect mode into symmetric and anti-symmetric modes is analogous to the splitting of degenerate atomic levels into bonding and anti-bonding orbitals when 2 atoms interact in a diatomic molecule. The extent of this splitting is proportional to the overlap of the localized eigenmodes of the periodic defects. The overlapping of the localized states will induce the formation of a defect mini-band in what was formerly a PBG in the original defect-free CROW. The final defect band in Fig 4(d) is thus the result of the N-fold splitting of the frequency *ω*
_{0} of the defect ring into a band of frequencies *ω*. The frequencies *ω* < *ω*
_{0} correspond to the symmetric defect modes while *ω* > *ω*
_{0} are the anti-symmetric defect modes. Note that all the defects in Fig 4 have *γ* = 2. If the defects have other integer sizes, such as *γ* = 3, 4 .., then there will be (*γ*-1) mini defect bands, each with N-fold splitting, as we shall see in the next section. This physical picture of the formation of defect bands via coupled defect states can generally also be applied to the case of quasi-periodic CROWs in subsequent section, as it will then be observed that there is *strong periodic correlation* in the quasi-periodic ring sequence. Besides this paper, recent works on quasi-periodic structures in [17–20] and references therein have also given much detailed treatment on the physical origins of the localized states in deterministic aperiodic ordered systems.

### 2.3.3 Results and discussions

### 2.3.3 (a) Effects of size tuning the periodic defects in finite CROW

We consider the effect of size tuned defects on the case of a periodic CROW structure with 7½ unit cells: {(AB)(AB) … . A} for t_{intra}
^{2} = t_{inter}
^{2} = 0.7. Fig. 5 shows the transmission and phase responses of this CROW configuration for each specific value of *γ* at 1, 2 and 3. It can clearly be seen that the general features of the amplitude responses of this finite CROW agree well with the dispersion diagrams for an infinite CROW as shown in Fig 2. Size tuning the defects *γ* such that *γ* is an integer introduces an equal number *γ* of mini-PBGs instead of a single wide PBG. In between the mini-PBGs are passbands with N-fold splitting because of mutual coupling between the N defects (where N = 7). Likewise, there are N-fold ripples, each of π phase shift, in the corresponding phase response at δ = π(2m+1). Strictly speaking, the stopbands in a finite CROW are only approximately PBG as they exhibit only near-zero amplitude. They occur near where the anti-resonance condition is satisfied by rings B, i.e, *mγ* = (*n*+1/2) (where *n* is an integer). For example, if *γ* = 3, then *m* = 7/6, 9/6 (1.5), and 11/6. However, only at *m* = 1.5 are rings A also at anti-resonance, hence this PBG is the widest and “deepest”.

Similarly, the passbands in between the PBGs occur where some or all of the rings resonate, i.e., at frequencies where *mγ* is an integer. For example, for *γ* = 3, they occur at *m* = 1, 4/3 (1.33), 5/3 (1.67) and 2. However, only at *m* = 1 and 2 are all rings resonant and hence these passbands are broadest and have 2N resonant peaks, while the other two passbands are associated mainly with rings B and thus narrower with only N peaks. They may be called the “defect” or secondary passbands and together with the passbands at δ = 2mπ, which we term as primary passbands, give a multi-passband transmission spectrum.

The transmission of a 7½ unit cells is also illustrated for non-integer *γ* at *γ* = 2.3 and 2.7, as shown in Fig. 6. The primary passbands and defect bands are no longer fixed at δ = 2mπ and δ = π(2m+1) but generally moves *down* the wide PBG when *γ* is tuned *up* from 2.3 to 2.7, in agreement with the characteristics of the dispersion graph for an infinite CROW shown below the transmission graph in Fig 6. Mini-PBGs are also formed within the wide PBG. The width of these mini-PBGs can generally be tuned by adjusting the non-integer *γ* that control the number and position of the defect bands. Non-integer *γ* thus provide a more dynamic tuning scheme in contrast to integer *γ*.

### 2.3.3 (b) Effects of tuning the coupling strengths on the transmission at resonance

The effect of tuning the coupling strength in a finite CROW is presented here for two cases: t_{intra}
^{2} = t_{inter}
^{2} and t_{intra}
^{2} ≠ t_{inter}
^{2}. To focus only on coupling tuning without the complication of size tuning, we first assume *γ* = 1 in Fig 7. If t_{intra}
^{2} = t_{inter}
^{2}, then all the rings are identical and also coupled identically and the whole array behaves as a single CROW. The passband at resonance order *m* = 1 is shown in Fig. 7(a) for a CROW with 7 unit cells. Increasing *t*
^{2} enhances the mutual interaction between the rings and results in broader 2N-fold splitting of the resonance.

The case of t_{intra}
^{2} ≠ t_{inter}
^{2} is shown in Fig. 7(b) for t_{intra}
^{2} > t_{inter}
^{2} . It can be observed that as ∣*t _{inter}*

^{2}-

*t*

_{intra}^{2}∣ = ∣F∣ increases, the primary passband is split into 2 smaller passbands separated by a mini-PBG. The mini-gap can be widened by simply increasing the intra-ring coupling relative to the inter-unit cell coupling. The origin of this mini-PBG can be understood as follows: By increasing the difference between

*t*

_{inter}^{2}and

*t*

_{intra}^{2}, the CROW is no longer a chain of identical rings but becomes a chain of unit cells each containing a two-coupled-ring system, and as the coupling between the rings increases, the characteristic feature of a strongly-coupled two-ring system, which is a split resonance with increasing splitting, becomes more and more dominant. This, and additional feature, will arise even for the case

*γ*= 2, 3, …(integer values). Similar effect can be achieved by decreasing t

_{inter}

^{2}while keeping t

_{intra}

^{2}fixed. In this case, decreasing t

_{inter}

^{2}reduces the linewidth, and thus lowers and broadens the transmission dip between the symmetric and anti-symmetric modes. With suitable choice of

*F*for a unit cell so that a stopband forms, a PBG can thus be formed

*at resonance*by directly coupling N unit cells together in a CROW. This is illustrated in Fig. 8. The transmission for a unit cell is shown in Fig. 8(a). For the transmission of a CROW shown in Fig. 8(b), N unit cells are coupled together resulting in two N-split Lorentzians within δ = 2πm that are separated from each other by a stopband. As the number of unit cells increases, the stopband deepens and broadens such that an approximate PBG results at resonance. Thus, in summary,

*coupling tuning*may be useful, along with the earlier discussed

*size tuning*, to introduce more flexibility in generating multi-passband transmission as well as PGB engineering at both on and off resonance.

### 2.3.3 (c) Unit cell with other configurations

For this work, we have focused on PBG engineering of CROW based on the unit cell of (**A B**) for simplicity. In principle, it is also possible to change the spectral and PBG properties by using any arbitrary combinations of rings of different size and number in each unit cell. To illustrate this, we have shown in Fig. 9 the dispersion diagram of an infinite CROW of a different unit cell design of (**ABB**), where *γ* = L_{B}/L_{A} = 2 in Fig 9(a) and *γ* = 3 in Fig 9(b). The corresponding transmission for a finite CROW with around 7½ unit cells ((**ABB**)^{7}
**A**)) with r^{2} = 0.3 are shown at the top of each dispersion diagram. In contrast to the perfect CROW of (AAA…A) a CROW with unit cell of (**ABB**) produces 3-fold splitting in the passband and *p*-fold splitting in the stopband where *p* depends on *γ*. For *γ* = 2, p = 3 and for *γ* = 3, p = 5. This splitting into *p* mini-PBG occurs for two reasons. First is the splitting similar to that in a CROW with unit cell (**AB**) as shown in Fig. 5. A further splitting occurs in the secondary passbands within the stopband because there are actually two coupled B rings in the unit cell (ABB). The transmission of the finite CROW matches well with the dispersion diagram at the defect region around non-integer *m*, except with just 7½ unit cells and the 3-fold splitting, the mini-PBGs in the primary passbands around integer *m* are not fully formed.

### 2.3.3 (d) Effects of addition of new defects

Up till now, we have only looked into the effects of periodic defects, all of similar size tuning. It is also possible to introduce a new defect, which we denote as G, into the periodic CROW to generate additional localized states. We have shown in Fig. 10 the PBG at δ = π(2m+1) for a periodic CROW of 6½ unit cells before (shown in blue graph) and after (shown in red graph) the defect G is added at the centre of the CROW. The new defect G is thus embedded between two equal arrays of unit cell of dual asymmetrical rings so that the CROW arrangement becomes: {0 **D** 0 **D** 0 **D G** D 0 **D** 0 **D** 0}, resulting in *disorder* in the periodic CROW. We set the cavity size D and G to be twice and thrice that of the size-untuned ring 0 respectively. To excite the defect mode of G, *r*
^{2} is set at 0.3, which results in a new localized state within each of the original mini-PBGs that are generated by the cascade of {0 **D**}. This shows the possibility of PBG engineering within these mini-PBGs by implementing *disorder* in the original periodic ordered system. As D is twice the size of ring 0, there is 1 defect band and 2 mini-PBGs within the wide PBG around δ = π(2m+1), in accordance with our discussion earlier. The defect band has 6-fold splitting corresponding to the total number of defect D. The new localized “defect mode” of G appears as a sharp resonance line in the middle of each mini-PBG in this case. More generally, the position of the localized state of G can be tuned by its size relative to the regular ring 0. As mentioned earlier, the passbands in between the PBGs occur where some or all of the rings resonate. The localized state within each mini-PBG can thus only be observed if the defect mode of G is coupled to the propagating mode of the ring array. This can only take place if the ring array is finite and when the reflectivity *r*
^{2} is small enough to excite this new defect mode.

## 3. CROW with quasi-periodic order

#### 3.1 Theory and matrix formulation

So far, we have only considered CROW with periodic size tuning where all unit cells are identical. In this section, we consider a deterministic aperiodic array where each “unit cell” is different, containing a different number of rings determined according to a certain design rule. Specifically, we consider the case where the sequence of rings in the array is given by the Fibonacci number sequence F_{n} = (1,1,2,3,5,8,13…), where the n^{th} member of the sequence is defined by the recursive relation F_{n} = F_{n-1}+F_{n-2}, and where we define F_{0} = 0 and F_{1} = 1. The CROW analog is a symbolic version of the Fibonacci sequence: the starting blocks are single rings C_{0} = {B} and C_{1} = {A}. Applying the recursive rule, the subsequent sequences are then C_{2} = {B, A}, C_{3} = {A,B,A}, C_{4} = {B,A,A,B,A}, C_{5} = {A,B,A,B,A,A,B,A}, etc. In this way, we will get ring CROW arrangement that is perfectly ordered but is not periodic. Thus, such CROWs fall in between periodic ordered and disordered systems: their sequences are aperiodic but they are generated by a known ordering rule, thus forming a *deterministic aperiodic order* in the ring arrangement. The deterministic aperiodic order generated by the rules of the Fibonacci sequence is generally known as *quasi-periodic* [21]. Fig. 11 shows a 5-member quasi-periodic CROW based on the Fibonacci sequence of order 4, or C_{4}.

Using the convention in Fig. 11 and following the previous matrix formulation in (4), the field amplitudes (a_{N+1}, b_{N+1}) in the N^{th} lattice of the quasi-periodic CROW can be defined as [a_{N+1} b_{N+1}]^{T} = [H_{N}] [a_{1} b_{1}]^{T}, where [H_{N}] is the transfer matrix for the Fibonacci sequence C_{N} of order N, which satisfies the same recursive relation: [H_{N}] = [H_{N-1}] [H_{N-2}]. Hence, for C_{4}: [H_{4}] = [H_{3}] [H_{2}] = [H_{3}] [A] [B] = [A] [B] [A] [A] [B], where [A] and [B] are the transfer matrices of unit cell (A) and (B), respectively. The transmissivity T = C_{N+1}/a_{1} of the finite quasi-periodic CROW of order *N* can be derived by substituting the matrix elements H_{Nij} for M_{ij} in Equation (8), and we will use it to analyze the transmission spectra of the Fibonacci class of CROWs, assuming all the rings have identical coupling scheme.

#### 3.2 Results and discussions

It can be seen in Fig. 12 that even for such quasi-periodic arrays, PBGs are still formed around δ = π(2m+1) and states localizations near these PBGs are still possible. The general trend is that with increasing order of the Fibonacci sequence, which leads to more rings in the CROW, the resonance spectrum at δ = π(2m+1) in between the mini-PBGs develops from a single high Q localized resonant peak to a mini-band. The spectral shape of the mini-band will evolve to take on more complicated and interesting appearance with higher order of the Fibonacci sequence. At the same time, the resonance bands at δ = 2πm become significantly broadened with more transmission dips and splittings. It is generally more difficult to excite the localized states centered at δ = π(2m+1) when the number of rings in the quasi-periodic CROW is large as more light is being reflected. Thus, r^{2} is set to 0.3 for the Fibonacci sequences C_{5} and C_{6} while for C_{3} and C_{4}, r^{2} can take higher value of 0.9 for the resonance centered at δ = π(2m+1) to be noticeable.

As a general rule, as the number of rings increases for higher order of the Fibonacci sequence, more transmission dips and ripples develop at both the primary and secondary passbands, which are also widened, as shown in Fig 12. This trend as well as the presence of mini-bands and mini-PBGs around δ = π(2m+1) bear close resemblances to the PBG structures of periodic CROW discussed earlier, thus relating quasi-periodic ordered ring system with periodic one. It can be suggested that the presence of mini-PBGs and mini-bands in the PBG structures of quasi-periodic Fibonacci ring CROW is a direct result of the *strong periodic correlation* in the Fibonacci sequence, which is also highlighted in [17]. For quasi-periodic CROW system like the Fibonacci CROW, there exists short-range periodic order among certain constituent sequences of the ring cascade while the earlier discussed periodic ring system has long-range periodic order. In their PBG structures, therefore, quasi-periodic systems will exhibit several spectral properties of their periodic counterparts while at the same time have distinctive features, which are the signatures that characterize their quasi-periodic nature.

The results of Fig. 12 are for the cavity size ratio of *γ* = 2. In particular, C_{3} and C_{4} each exhibit a single high-Q resonant peak, which makes them suitable for ultra-narrow passband applications. The number of such ultra-narrow defect modes around δ = π(2m+1) for the Fibonacci CROW of C_{4} is equal to (*γ -1*) for integer *γ*, as illustrated in Fig. 13. For order N > 4, the defect bands will generally have more complex shapes which depend also on *γ* if other values of *γ* are used.

## 4. Effect of losses

Thus far, we have assumed all the ring resonators to be lossless. In this section, we model the effect of losses for a periodic CROW with 3½ unit cells: {(AB)(AB)(AB)A}, and an quasi-periodic CROW of C_{4} as discussed in the previous section. The ring losses is defined using the round-trip attenuation factor *τ* (for a lossless ring, *τ* = 1), which is denoted as *τ _{A}* for intra-ring A and

*τ*for intra-ring B, and is mainly due to sidewall roughness induced scattering losses and bending losses. In the presence of losses, the PBG is not strictly formed or defined as the propagation vector will be complex within these entire bands.

_{B}As shown in Fig. 14, it is found that even a small loss has a detrimental effect on the amplitude response. For satisfactory amplitude response, the round-trip attenuation factor *τ _{A}* must be larger than 0.99 for the periodic CROW with 3½ unit cells. With increasing number of rings, the amplitude response also suffers greater attenuation.

In conclusion, all CROW are suitable for practical applications only if all the rings have very low losses, τ > 0.99. This requirement might severely restrict the practical implementation of our proposed CROW designs. However, it has already been experimentally demonstrated [9, 27] that very large arrays of high-Q (low losses) rings with small bend radii for filter applications are possible. This implies that it is possible to achieve a round-trip amplitude transmission factor of τ > 0.99. Similarly, the ultra-narrow high-Q resonance peak of the Fibonacci CROW as shown in Fig 14(b) is very sensitive to and quickly degraded by any slight loss. However, as long as the losses can be kept to a minimal, such deterministic aperiodic structure will be suitable as a compact sensor.

## 5. Comparison with FDTD simulations

The transfer matrix model used so far is only approximate as it neglects several factors which will become apparent when compared with the benchmark Finite Difference Time Domain (FDTD) method [28]. Below we present this comparison for the Fibonacci sequence C4. In the FDTD simulation, a Gaussian pulse is launched into the input bus of the quasi-periodic CROW. In order to reduce computational resources and time, the CROW are designed on high-index contrast silicon-on-insulator photonic wire, which allows each ring to have a relatively small footprint of bend radius 2μm and 1μm, respectively, for rings A and B. To obtain high-Q resonances, the waveguide-ring and ring-ring separations are set to be relatively large such that all the rings are weakly coupled (~ *r ^{2}* = 0.9). The transmission at the output port, normalized to the input power, is shown in Fig. 15.

In general, both the FDTD and transfer matrix give similar trends in the transmission spectrum, with localized states generated within a PBG sandwiched between two primary passbands with resonance splitting. However, this agreement stops here and there are significant differences, in that the output is comparatively lower, and the primary passbands are rather asymmetrical, in the FDTD case. These are due mainly to the effect of bend losses. The transmission spectrum is expected to improve if larger ring radius and smaller grid size are used, which will, however, increase the computational time. We note that the transmission of the localized mode is generally lower at lower wavelengths and this is due to the wavelength dependence of the coupling strengths and losses, which is not accounted for in our transfer matrix model.

Also, there is a slight shift of the localized mode away from the centre of the PBG. This is most likely due to mesh discretization in the FDTD [4] as well as the effect of self coupling that will cause a frequency shift but is not taken into consideration in the transfer matrix model. Finally, we note that FDTD can be used to analyze the effect of small deviations in *γ* from integer values. In general, the results are expected to be more complicated and irregular due to the presence of the Vernier effect.

## 6. Discussions and conclusions

In summary, we have studied for the first time the effects on the PBG structure of ring-based CROW in the presence of multiple periodic and quasi-periodic “defects”, all of similar sizes. Due to space limitation, we have presented only the most basic configurations with more intuitive results in this paper. In this section we discuss their potential applications and also possible variations and extension for our future work.

For the periodic case, the interaction of the multiple defects results in splitting of the eigenmodes corresponding to distinct localized states, leading to the formation of defects-induced mini-passbands within the broader PBG. The number and position of such narrow defect passbands can be tuned by adjusting the relative size of the intra-rings. If the intra and inter unit cell coupling strengths are asymmetrical, new PBG forms at resonance. These features make the deployment of such schemes for practical PBG engineering very attractive in the absence of active elements to achieve tuning. Other possible applications include multipassband filters after the ripples are flatten by aperiodization techniques highlighted in [29]. To borrow further from the electronic analogy, we can consider a random distribution of defects, which is the normal state of defects, instead of the periodic distribution, and also “cluster” defects such as {AAAAbbbAAAAAAbbbbAAAAA…}.

Somewhat more orderly than the random distribution is the quasi-periodic system exemplified by the Fibonacci sequence studied in Section 3, where we show that a sharp localized resonant state still exists within the PBG, that slowly transforms to a broad passband as the order of the Fibonacci sequence is increased. If the total losses are sufficiently low, the ultra-narrow passband of the quasi-periodic CROW based on a low-order Fibonacci sequence such as C_{4} may be highly useful as a compact sensor for bio-photonic and chemical applications. Deterministic aperiodic CROW based on other sequences, such as the Cantor and Thue–Morse sequences and also hybrid ordered system whereby there is a mix of both periodic and quasi-periodic order are also possible and are expected to show interesting spectral properties. As a general rule, systems with perfect order or complete disorder are simple, and it is in the middle where there is a mix of order and disorder that complexity arises that may produce novel properties and applications. Finally, we may also consider different types of arrays such as SCISSOR [11] for both the periodic and quasi-periodic designs considered in this work.

Note that the focus of this work is on how the *PBG structures* are affected by periodic and quasi-periodic orders, using *multiple defects*. As such, several other optical behaviors of our proposed devices are not covered in this work due to space constraint. One of these is the dispersion of the devices, which besides losses, is the primary restriction to the application of these ring arrays. With suitable choice of coupling strength and the use of external dispersion compensation schemes [30], the dispersion can generally be reduced. For other applications, the dispersion can be turned into advantage, and hence is an interesting subject for our further study. In addition, the presence of multiple defects in both the periodic and quasi-periodic ordered ring systems would influence the slow wave characteristics of the devices, that would in turn affect the non-linear phase sensitivity because of the relation: *dϕ*/*dI _{i}* =

*LS*

^{2}(2

*πn*

_{2}/

*λ*), where

*dϕ*/

*dI*is the non-linear phase sensitivity,

_{i}*L*is the length of the device,

*S*is the slowing ratio,

*n*

_{2}is the Kerr coefficient of the material and

*I*is the input intensity. Also, the presence of multiple coupled defects impacts the ring intensity build-up factor and this could enhance the bistable or multistable performance of our proposed devices. These mentioned issues will be analyzed in our future work.

_{i}## Acknowledgments

The first author would like to express his heartfelt gratitude and deepest respect to the late Professor Mee Koy Chin for his spirited guidance and inspiration. Special thanks are also given to Mr. Landobasa Tobing and the reviewers for valuable comments that improved the manuscript. This work is supported by the Ministry of Education, Singapore, under grant ARC 16/07.

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