## Abstract

The existence of localized or defect modes in a periodic array of symmetric and lossless micro-ring resonators is demonstrated for both finite and infinite structures using the transfer matrix method, for two types of array: one consisting of a cascade of coupled resonators coupled to an input and an output waveguide, and another consisting of uncoupled resonators periodically coupled between two bus waveguides. The defect can be introduced either by removing one ring, or by making one ring bigger or smaller. The 1-D periodic dielectric waveguide structures consisting of micro-ring resonators can exhibit photonic bandgaps, and when point defects are introduced defect states can form within the bandgaps, giving rise to donor and acceptor modes similar to other photonic crystals. The results based on the transfer matrix model agree with the finite-difference time-domain method, and are compared with those of a quarter-wave mirror stack.

©2005 Optical Society of America

## 1. Introduction

Micro-ring resonators have been used as the building blocks for the synthesis of high-order optical filters [2–9]. Two model configurations have been used for these filters, as shown in Fig. 1. In one configuration which we refer to as Type I, the resonators are periodically coupled to two side waveguides but not mutually coupled, with equal spacing between adjacent resonators [3–6]. If the periodic spacing is chosen so that the responses of all the rings add in phase, then the configuration is analogous to a hybrid combination of distributed-feedback (DFB) grating and ring resonators, with each ring acting as a “tooth” in the grating that has high reflectivity at resonance [4]. Hence high reflection at the drop port with flat bandpass response and out-of-band sidelobes can be obtained with a relatively small number of rings. In another configuration which we call Type II, the rings are mutually coupled in a linear cascade, each ring forming a Fabry-Perot cavity and the array is equivalent to a chain of coupled Fabry-Perot or mirror stack [7–11]. The array is coupled to an input and output bus waveguides. In this case the light is transmitted when the resonators are on resonance, and reflected when off resonance.

Like other photonic crystal structures [13], these linear periodic structures of micro-ring resonators can form photonic bandgaps, frequencies where light cannot propagate through the structure. These bandgaps correspond to the reflection resonances of the filters discussed above. The introduction of irregularities (or defects) in such a photonic crystal can generate a localized state in the bandgap, where light is localized near the defect forming a so-called defect mode. Defect mode is useful because it can suppress spontaneous emission [14], and may be used in the design of microcavity lasers and filters. The existence of this mode was first demonstrated by McCall [15] in the microwave region. Guided and defect modes in the optical region have been predicted for a periodic dielectric waveguide consisting of dielectric rods or air holes [16–17], and experimentally demonstrated in a chain of coupled vertical-cavity microresonators [18]. It is the purpose of this paper to investigate the nature of guided and defect modes in the two types of ring resonator arrays mentioned above which, to our knowledge, have not been studied so far. We use a transfer matrix analysis which has the advantage that it can be coupled easily with the Bloch theorem to derive the photonic bandgap (PBG) properties of these periodic structures. Moreover, it is convenient to represent defect in the form of matrices. In the limit of weak coupling it can approximate the tight binding approach [7], and in the time domain analysis it agrees well with temporal coupled mode theory [19]. In the following we will outline the basic model of calculations in Section 2, and derive the PBG structure and the defect modes for Type I model system in Section 3I, and for Type II model system in Section 4. In Section 5 we compare the results with the Finite Difference Time Domain (FDTD) simulations. In the concluding section we discuss some possible applications and analogies with other periodic systems.

## 2. Basic model of calculations

A possible way to introduce a point defect into a Type I model structure is to remove one ring from the arrays (as shown in Fig. 2(a)), and for the Type II model system is to make one of the rings have a different radius than the others (Fig. 2(b)). In general, the system can be divided into three sections comprising the two periodic sub-arrays and the one defect in between. The periodic sub-arrays can be regarded as frequency-dependent mirrors, and the defect forms a Fabry-Perot cavity as shown in Fig. 3. The *R*
_{1,2} and *T*
_{1,2} in Fig. 3 are the reflection and transmission coefficients of the respective periodic sub-arrays. The *L*
_{defect} represents the defect dimension, which could be the defect inter-ring spacing in the case of Type I, or half of the ring circumference in the case of Type II.

For convenience, we can express the phase change accumulated by light in a defect as δ_{D} = γ*L*
_{defect}, where γ is the propagation constant. The system transmission coefficient (*T*) is found by totaling the transmitted waves.

$$=\frac{{T}_{1}{T}_{2}\mathrm{exp}\left(-j{\delta}_{D}\right)}{1-{R}_{1}{R}_{2}\mathrm{exp}\left(-j2{\delta}_{D}\right)}$$

In general, for the lossless case where |*R*|^{2} + |*T*|^{2}=1, the transmission in (1) is maximized when the condition *ϕR*
_{1} + *ϕR*
_{2} - 2*δ*_{D}
= *m*(2*π*)is satisfied, where *ϕ*_{R}
are the phases of the two mirrors. This resonance condition determines the defect mode frequencies. Normally the defect mode exists for *m* = 0. For a symmetric structure the defect modes are then determined by the condition

For an *infinite* structure in which the defect is sandwiched between two semi-infinite, symmetric sub-arrays, *ϕ*_{R}
can be obtained as follows. In each sub-array, let the transfer matrix relation for the unit cell be represented by

where *a* is the incident field amplitude and *c* the reflected field amplitude. The matrix [*m*] depends on the specific model structure and will be derived in the next two sections for Type I and Type II structures, respectively. Here we note only that if the unit cell is *reciprocal*, i.e., the amplitude and phase response from left to right is the same as from right to left, then a general property of [*m*] is that it is *unimodular*, i.e., det[*m*] = *m*
_{11}
*m*
_{22} - *m*
_{12}
*m*
_{21} = 1. Furthermore, to apply Bloch theorem to the periodic structure, we assume the rings to be lossless, for which we will have *m*
_{22} = ${m}_{11}^{*}$ and *m*
_{12} = ${m}_{21}^{*}$. We can then write the Bloch theorem in the form

where *k* is the Bloch wavevector, and Λ is the period of the unit cell. Combining (3) and (4) gives the eigenvalue equation

which leads to the characteristic equation (or dispersion relation):

The eigenvector may be written as

from which we obtain the complex reflection coefficient:

It is shown in Appendix A that |*R*| = 1 within the bandgap where *k* is imaginary. An infinite periodic structure, therefore, exhibits true photonic bandgaps in which |*R*| = 1. The resonance for Eq. (1) occurs when the denominator is zero, that is when. |*R*|^{2} exp[-*j*2(*ϕ*_{R}
- *δ*_{D}
)] = 1, and thus can occur for an infinite lossless structure. The defect mode existing in this cavity will have an infinite Q.

For *finite* structures, even if the resonators are lossless, |*R*| is always less than 1 although it can be nearly equal to 1 near the middle of the reflection spectrum. This implies that resonance can not occur unless we introduce gain in the defect. In order to excite the defect modes in simulations, we allow the input sub-array to have a finite transmission (*T*
_{1}) by choosing the appropriate coupling parameters and array size. The existence of the defect mode then selectively enhances the transmitted light at the defect frequencies by the Fabry-Perot effect. The defect mode will have a finite Q (or linewidth) that depends on its location within the bandgap. To find the defect frequencies for a finite structure we must first determine the reflection spectra of the periodic sub-arrays which will depend on the number of resonators. This will be discussed below.

## 3. Type-I microresonator structure

In this periodic structure, although the individual resonators are not directly coupled to each other, they are still coupled indirectly through the input and output waveguides. Because the double channels provide many paths for the waves to feedback to earlier resonators, they allow Bragg resonances to develop in addition to the ring resonances. These occur when the *Bragg* condition, *γL*_{b}
= *jπ* (*j* = 1,2,3…), is satisfied [12], where *L*
_{b} is the resonator spacing.
Hence, the device behaves as a hybrid between ring resonator and distributed-feedback grating [6]. For simplicity, we will consider only the case where 2*L*
_{b} = *L*
_{c}, since in this case the two resonances overlap and the photonic band structure is simpler [22].

The unit cell is a Fabry-Perot cavity consisting of a single ring coupled to two parallel bus waveguides, as shown in Fig. 4. Assuming a single input *a*, the output in the backward direction is defined as *T* and that in the forward direction is *R*, where

$$T=\frac{c}{a}=-\frac{{t}^{2}{\tau}^{1/2}{e}^{-\mathit{j\delta}/2}}{1-{r}^{2}\tau {e}^{-\mathit{j\delta}}}$$

In these expressions, *r* and *t* are the reflectivity and transmittivity at the coupler between the waveguide and the resonator, which are assumed to be phase-matched and lossless so that *r*
^{2} +*t*
^{2} = 1; τ = exp(-α*L*_{c}
/2) is the round-trip amplitude transmission factor, and *δ* = *γL*_{c}
is
the round trip phase shift; *γ* = (*ω*/ *c*)*n*_{eff}
is the propagation constant, *L*_{c}
is the round trip length, *α* is the power loss coefficient, and *n*_{eff}
is the waveguide effective index. δ contains the frequency dependence and can be written as

where *FSR* = *c*/*n*
_{eff}
*L*
_{C} is the *free spectral range*, ∆*f* = *f*-*f*_{o}
is the frequency detuning, *f*_{o}
is a resonance frequency, and *m* = *f*_{0}
/*FSR* is an integer denoting the resonance order. Hence the frequency response is periodic in the normalized frequency (Note that we ignore the small frequency dependences in the coupling coefficient and in the waveguide effective index).

The fields in the left and the right planes of the Fabry-Perot cavity is given by the transfer matrix relation

Note that the transfer matrix is unimodular (i.e., the determinant is 1), even in the presence of waveguide loss. This is because the Fabry-Perot filter is symmetric and reciprocal. For the *n*th ring cavity in the Type I array, the input and output field amplitudes are represented by (*a*
_{n}, *d*
_{n}), and (*b*
_{n}, *c*
_{n}), respectively, where, by continuity, we have *b*_{n}*e*
^{-iγLb} = *a*
_{n+1} and *d*_{n}
= *C*
_{n+1}
*e*
^{-iγLb} . Together with Eq. (11) this gives the recursive matrix relation

Note that the matrix [*m*] is again unimodular. Furthermore, in the *lossless* case (*τ* =1), it can be shown that (*R*
^{2}-*T*
^{2})/*R* = 1/*R*
^{*}, so that *m*
_{22} = ${m}_{11}^{*}$ and *m*
_{12} = ${m}_{21}^{*}$. For an infinite structure, the dispersion relation using Eq. (6) and *m*
_{11} = (1 /*R*)exp(*jpδ*), where *p* = *L*_{b}
/*L*_{c}
, is given by

where *R* =|*R*| exp(*jφ*_{R}
) is given in Eq. (9). The real value of *k* as a function of δ (for *p* = 0.5) is plotted on the left half of Fig. 5. It is apparent that real *k* is possible only when the right-hand side is between -1 and +1; those frequencies for which the right-hand side becomes greater than +1 or less than -1 are not allowed and are considered to form the PBG (in which *k* is imaginary).

For a finite structure of *N* identical and lossless resonators, the transmission can be obtained by iterating the transfer matrix *N* times:

Sylvester’s theorem [20] can be applied to give

Note that cosθ = ½(*m*
_{11}+*m*
_{22}) = ½(*m*
_{11}+${{m}_{11}}^{*}$) is real, and that the matrix *M* is again unimodular, and *M*
_{22} = ${{m}_{12}}^{*}$ and *M*
_{12} = ${{m}_{21}}^{*}$. Equation (15) can also be expressed in terms of the Chebyshev polynomials of the second kind. Using the condition *d*
_{N} = *c*
_{N+1} = 0, one obtains the array *reflectivity* as

and the array *transmittivity* as

where |*R*_{I}
|^{2} + |*T*_{I}
|^{2}=1 as expected by conservation of energy, so the transmission and reflection spectra are complementary. The reflection is maximum when the resonators are on resonance (where δ/2π is an integer). In the limit of large *N*, |*R*| approaches 1 and the reflection bands become the photonic bandgaps.

On the right half of Fig. 5 we plot the calculated transmission amplitude response |*T*_{I}
|^{2} for N = 20 and r^{2}=0.9. Note that for this value of N, the photonic bandgaps are fully formed around the resonance wavelengths *λ*_{m}
= *n*_{eff}*L*_{c}
/*m* (or δ/2π = *m*). The width of the bandgap is dependent on the coupling coefficient between the waveguide and the resonator. The stronger the coupling (i.e., the smaller the r^{2}), the stronger will be the feedback between the resonators and the wider the bandgap. The ripples in the transmission band are due to the Fabry-Perot interference within the periodic structure. This can be seen from the linear dispersion of the periodic structure, which indicates that it behaves as a uniform slab in the transmission band.

The dependence of the reflection amplitude and phase responses on the number of resonators is more clearly shown in Fig. 6. Note that with increasing *N*, the amplitude response changes from Lorentzian for a single ring to become more box-like, while the phase response *ϕ*_{R}
changes from being highly nonlinear at the center for the single-ring case to become more linear at the center and nonlinear at the band edges. This behavior is similar to the Bragg grating filter, and is related to the nature of a type of filter called *minimum phase* filter (MPF) [20], which are filters in which the amplitude responses are related to the phase responses by the Hilbert transform. One consequence of the Hilbert transform is that the phase is linear when the amplitude is flat, but acquires higher order terms when the amplitude response changes radically as occurs near the passband edges.

The defect modes for an infinite Type I structure, calculated using Eqs. (2), (8) and (12) are shown in Fig. 7(a). The defect frequencies in terms of δ are plotted as a function of the defect lengths (normalized to the ring circumference). Similarly, Fig. 7(b) shows the defect modes for a finite structure consisting of a defect cavity sandwiched between two sub-arrays with *N* = 10, calculated using the transfer matrix method. The agreement with the infinite case is excellent. The inset (c) shows the cross section view for the case *L*
_{defect} = 0.75 (i.e., a defect with *L*
_{c} = 3/4 of the regular ring circumference), which has two narrow transmission resonances in the bandgap. It is easier to excite the defect mode when the bandgap is small. Hence, the *r*
^{2} is set to 0.9 to ensure that *R*
_{1,2} < 1 so that the defect modes can be easily excited. However, note that there is a region near the center of the bandgap (δ/2*π*~1) where we could not observe the transmission resonances because *R*
_{1,2} ~ 1 here and it is difficult to excite the defect modes. It can be shown that the full-width half-maximum linewidth of the resonances is approximately given by ∆*δ*_{FWHM}
=(1/${R}_{1}^{2}$)(1 - ${R}_{1}^{4}$)^{1/2}, which is expected from the Fabry-
Perot analogy, where higher mirror reflectivity will give rise to narrower oscillating modes. Hence, as the defect mode approaches the band edges, the linewidth broadens because the reflectivity here is smaller, and it is easier for the defect mode to couple to the guided modes so that the defect mode has a smaller Q.

The calculated Bloch wave amplitude distribution based on the eigenvector of Eq. (7) is given in Fig. 8. It shows that the light is localized in the defect and decays exponentially away from the defect. The field in the defect cavity forms a standing wave with a field enhancement factor analogous to that of a Fabry Perot cavity.

## 4. Type II microresonator structure

We next perform the same transfer matrix analysis for a cascade of resonators in which the resonators are directly coupled to each other in a lattice architecture. The cascade, in turn, is coupled to an input waveguide and an output waveguide. Propagation through this structure is allowed only when the resonators are on resonance. When the resonators are off resonance, the wave is reflected. Hence, we expect the photonic bandgap structure to be complementary to that of Type I. The transfer matrix relation from one unit cell to the next is given by

Note that the matrix [*p*] is unimodular, and that *p*
_{22} = ${p}_{11}^{*}$ , and *p*
_{12} = ${p}_{21}^{*}$ in the lossless case. Applying Eq. (4) to this case gives the dispersion relation

where Λ is the period of the unit cell, and *t* is the coupling factor of the coupler. Equation (19) is similar to Eq. (13) (when *p* = 0.5), as they both originate from Eq. (4) which implies that the characteristic equation is determined by the “transmittivity” from one cell to the other multiplied by an oscillatory function. In the Type II configuration, the transmittivity is *t*, but for Type I, it is given by *R* [see Fig. 4]. The oscillatory function is sine in one case and cosine in the other, reflecting the fact that the photonic bandgaps are complementary, as shown below. The real and imaginary parts of *k* as a function of δ are shown on the left half of Fig. 9. The broad bandgaps or reflection bands, where *k* is imaginary, are centered at δ/2π = *m* + ½, complementary to those of Type I structure where the (narrow) bandgaps are centered at θ/2π = *m*. The width of the bandgap is again determined by *r*
^{2}. This can be seen from the condition cos(*k*Λ)= 1, which gives the transmission bandwidth as

The larger the *t* (i.e., the smaller the *r*
^{2}), the greater the interaction between the rings, hence the broader the transmission bandwidth or the *narrower* the bandgap. This trend is opposite to that for the Type I structure.

For a *finite* structure containing *M* identical, lossless resonators, the transmission can be obtained by iterating the transfer matrix *M* times:

where the matrix [*P*] is again unimodular, and *P*
_{22} = ${P}_{11}^{*}$ , and *P*
_{12} = ${P}_{21}^{*}$ can be found with
the use of Eq. (15). Making use of the fact that *d*
_{M+1} = 0, and *b*
_{M+1} = -*ra*
_{M+1}, the coupled array reflectivity *R*_{c}
= *b*
_{1}/*a*
_{1} is obtained as:

Similarly, making use of the identity *c*
_{M+1} = *jta*
_{M+1}, the array transmissivity *T*
_{c} = *c*
_{M+1}/*a*
_{1} is given by:

It can be shown that |*R*_{c}
|^{2} + |*T*_{c}
|^{2} = 1 as expected. The transmission amplitude and phase responses calculated for a structure of 5 rings are shown on the right half of Fig. 9. For a finite CROW of *M* rings, there are *M*-fold splitting in the transmission resonance. Likewise, in the phase response, there are *M*-fold ripples. The effective phase shift across each ring is π, so the total phase shift is *M*π.

Figure 10(a) shows the defect modes in the bandgap centered at δ/2π = 1.5 for the infinite Type II structure, calculated based on Eqs. (2), (12) and (18). The defect frequencies in terms of δ are plotted as a function of the defect radius (normalized to the regular ring radius). It is in excellent agreement with Fig. 10(b) for the finite case of a defect cavity sandwiched between two sub-arrays with *M* = 5. The *r*
^{2} is set to 0.3 so as to make it easier to excite the defect modes. In contrast with Fig. 7(b), all defect modes can be observed due to the fact that *R*
_{1,2} < 1 for low *r*
^{2} in a Type II structure. The inset in (c) shows a particular transmission profile when the defect ring has a radius of 2/3*R*.

In analogy with the electronic band structure, the continuous band below the photonic bandgap corresponds to the photonic valence band, and that above the bandgap corresponds to the photonic conduction band. This can be seen from the photonic band structure of Fig. 9, where the negative curvature at the top of the bottom band (centered at δ/2π =1) implies a negative photon “effective mass” and hence represents a valence band edge, while the positive curvature at the bottom of the upper band (centered at δ/2π =2) corresponds to a conduction band edge. With respect to the valence and conduction bands, the defect modes can be classified as either donor or acceptor modes, just as in other photonic crystals. We note that when the defect ring has a smaller radius compared with the other rings, the defect mode shifts from the bottom edge of the bandgap into the forbidden band as its radius is reduced. Hence, the defect mode is called the acceptor mode in a photonic crystal structure. Likewise, if the defect ring has a larger radius, then the defect mode moves from the top edge of the bandgap down into the gap, and corresponds to the donor mode in photonic crystal structure. As the radius increase further a second donor mode appears.

## 5. Comparison with FDTD simulations

To test the reliability of the transfer matrix model, we compare the Type II results with the finite difference time domain method. For the Type I side-coupled structure FDTD is more difficult as more rings are required to form defect modes. In the FDTD simulations, a pulse of 15 *fs* is launched into one end of a finite Type II structure containing a single defect. The Type II structure consists of two 3-ring sub-arrays with the ring radius *R* = 1.7 μm, sandwiching a defect ring with a radius of 2/3*R*. This defect size is known to generate a defect mode at the center of the forbidden band, i.e, at δ/2π =1.5, as shown in Fig. 10(c). Taking into account the wavelength dependence of the waveguide effective index (*n*
_{eff}), the possible input center frequencies (wavelengths) are estimated to be 172 THz (1.744 μm) and 202 THz (1.4850 μm). To excite the defect mode, the number of rings is kept small, and the waveguide-ring and ring-ring separation is set to be small so that the structure exhibits a large *t* (or small *r*).

The localization of light in the defect ring is clearly shown by the snapshot of the steady-state power distribution given in Fig. 11(a). The transmission or drop port output, normalized to the input, is shown in Fig. 11(b), along with the transfer matrix results. The agreement between the two methods is reasonably satisfactory, except that the transmission output is relatively low for the FDTD case because of the high radiation loss associated with the small rings. The transmission is lower at higher frequencies because of the weaker coupling at shorter wavelengths which reduces *t*. The circled peaks in the center of the forbidden bands correspond to the defect modes and are close to the estimated frequencies. Because the free spectral range of the defect ring is 1.5 times that of the regular rings, the adjacent defect mode overlaps with the continuous band, and so the next visible defect mode occurs in the third bandgap from the first one. The frequency shift of the defect mode from that given by the transfer matrix method is probably due to the mesh discretizaton involved in FDTD, as pointed out before in [19], and also because in the transfer matrix method the wavelength dependence of *n*
_{eff} is ignored, and a frequency shift term due to the self-coupling of the resonators is omitted.

## 6. Discussion and summary

In summary, we have shown that periodic dielectric waveguides consisting of micro-ring resonators can exhibit a photonic bandgap structure, and when point defects are introduced defect states can form within the bandgaps, giving rise to donor and acceptor modes similar to other photonic crystals. So far the rings are assumed to be lossless. The filter response and photonic bandgap structure of these ring arrays are very sensitive to and are quickly degraded by loss [22]. However, finite structures with moderate number of rings can be made very low loss using special waveguide material, as demonstrated recently [9]. In very low loss structures the defect modes will have very high Q, giving rise to some interesting potential applications. First, transmission through such a defect may be very sensitive to the environment making it a potentially very compact sensor. Secondly, if there is some mechanism to tune the defect frequencies then this system may be used as a high-Q tunable filter. Finally, if the defect could be made of an active medium, then a single-mode microcavity laser with extremely small linewidth may be achieved in which spontaneous radiation can only emit into the defect mode and very large spontaneous coupling factor may be obtained [13].

A waveguide-coupled micro-ring resonator is an integrated-optic form of the Fabry-Perot (FP) interferometer with the distinct advantages that no cleaved mirrors are needed and the reflected light is spatially isolated from the input light. Type I and Type II structures are just different arrangements of multiple FP resonators. Type II, in particular, is equivalent to a cascaded FP cavity (or mirror stack) as shown in Fig. 1(b). Hence, one should be able to show that the transmission properties are similar to those of a mirror stack. Not surprisingly, Fig. 12, which shows the eigen-frequency (or resonance wavelength λ) solutions for all propagating and localized modes in a mirror stack consisting of a defect sandwiched between 6 layers (or 3 unit cells) of quarter-wave mirrors on each side, is remarkably similar to Fig. 10(b) for the equivalent Type II structure. The layer refractive indices are *n*
_{1} = 2.5 and *n*
_{2} = 1.5, and the thicknesses are made quarter wave to maximize the reflection around 1.55 μm (i.e. λ_{0}), i.e., *d*
_{1} = λ_{o}/4*n*
_{1} = 0.155 μm and *d*
_{2} = λ_{o}/4*n*
_{2} = 0.2583 μm, respectively. Adding the two layers yields Λ = 0.4133, as denoted in Fig. 12. The defect length *L*
_{defect} is varied across this value. We see that in the region where *L*
_{defect} < Λ, there is an acceptor mode moving upward from valence band to conduction band. For *L*
_{defect} > Λ, there are donor modes moving downward from conduction band to valence bands, in exact analogy with the CROW structure. The parallelism between the quarter-wave mirror stack and the coupled resonator array provides yet another basis for understanding the photonic band structure of micro-ring resonator arrays, and reflects a common principle underlying wave propagation through all periodic structures.

## Appendix A: Proof that |*R*| =1 in the forbidden region for Eq. (8)

In the forbidden region, *k* = -*jγ*, and so cos(*kΛ*) =cosh(*γ*Λ) = 1/2(*m*
_{11} + *m*
_{22}). Using exp(-*γ*Λ) = cosh(*γ*Λ) - sinh(*γ*Λ), and the fact that *m*
_{11} = ${{m}_{22}}^{*}$, *m*
_{12} = ${{m}_{21}}^{*}$ and cosh^{2}(x)-sinh^{2}(x) = 1, we can write Eq. (8) in the form

Note that because (*m*
_{22}-*m*
_{11}) is a pure imaginary number, in taking the magnitude we can simply multiply (*m*
_{22}-*m*
_{11}) by (-*i*).

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