## Abstract

We present a filter design formalism for the synthesis of coupled-resonator optical waveguide (CROW) filters. This formalism leads to expressions and a methodology for deriving the coupling coefficients of CROWs for the desired filter responses and is based on coupled-mode theory as well as the recursive properties of the coupling matrix. The coupling coefficients are universal and can be applied to various types of resonators. We describe a method for the conversion of the coupling coefficients to the parameters based on ring resonators and grating defect resonators. The designs of Butterworth and Bessel CROW filters are demonstrated as examples.

©2011 Optical Society of America

## 1. Introduction

A coupled-resonator optical waveguide (CROW) consists of a chain of weakly coupled resonators in which light propagates by virtue of the coupling between adjacent resonators [1]. Such a waveguiding mechanism is unique and can be realized in various types of resonators, such as ring resonators [2,3], grating resonators [4], and 2D photonic crystal resonators [5]. One of the key features of CROWs is that optical pulses can propagate at a significantly reduced group velocity, dictated by the inter-resonator coupling. This ability to “slow” down light may find applications such as optical delay lines, optical buffers, interferometers, and nonlinear optics [3,6,7].

The first proposal and analysis of CROWs was based on infinite-length chains whose dispersion properties can be derived and are controlled by essentially a single parameter, the coupling coefficient *κ* [1]. In practice, an infinite-length CROW has to be terminated and coupled to the outside world. The resulting finite-length CROW requires a proper design because the reflection at the two boundaries leads to Fabry-Perot-type oscillations and therefore ripples in the passband of the transmission spectra, resulting in signal distortion. To minimize the reflection, the boundary coupling coefficients should be properly chosen [8]. The coupling coefficients near the boundary can also be apodized to adiabatically transform between the CROW mode and the waveguide modes [9,10].

A further optimization of CROW delay lines consists of a judicious choice of all the coupling coefficients. Each resonator in a CROW can be considered as a feed-back loop which contributes a pole to the transfer function of the CROW. Therefore, the transfer function of an *N*-resonator CROW is an *N*-pole optical filter. The coupling coefficients of CROWs can be chosen to achieve desired properties such as maximally flat transmission (Butterworth filters) or maximally flat group delay (Bessel filters) over a prescribed bandwidth. Optical bandpass filters are important elements in optical signal processing, especially for wavelength-division-multiplexed (WDM) systems [11]. High-order bandpass filters based on coupled ring resonators have been extensively studied and experimentally demonstrated [12–16]. Filters based on coupled-resonator systems can also be realized on grating resonators [4,17] and photonic crystal defect resonators [5,18].

A prerequisite for the synthesis of high-order optical filters is a robust and systematic approach to directly relate the desired filter transfer function to the parameters of the CROWs. Several methods have been proposed. For ring or Fabry-Perot resonators, the transfer matrix method (TMM) can be applied to analyze the forward and backward fields inside the resonators. If the cavity lengths are nearly identical, each delay is an integer multiple of a unit delay, and the CROW can be considered as a digital optical filter and analyzed by the Z-transform formalism [11]. and [19] derived extraction procedures to convert digital filter responses to the field coupling coefficients between ring resonators. A simpler method for the analysis of CROWs is the time-domain coupled-mode theory (CMT), which considers the whole field as a superposition of individual resonator modes and is independent of the type of resonators. The derived coupling coefficients are more general but have to be converted to the parameters of the type of resonators used. In [20], coupling coefficients were extracted by direct comparison of the transfer function derived from CMT and the desired filter response. The approach becomes impractical for high-order filters. Another filter design approach is based on equivalent circuits and the techniques of microwave filters. Circuit-based methods for the synthesis of high-order filters have been proposed in [21] and [22].

In this paper, we present a filter design formalism based on CMT and the recursive properties of the coupling matrix. Coupling coefficients are extracted using recursive relations. In contrast to [20], this formalism does not need a direct comparison and is robust for high-order filters. These universal coupling coefficients can then be converted to the parameters of the specific type of resonators comprising the CROWs. We propose a method for the conversion of the coupling coefficients. This method utilizes the resonance splitting of two coupled resonators for inter-resonator coupling and the transmission of 2-resonator CROWs for waveguide-resonator coupling. It is more accurate than the approaches proposed in [20]. Another interesting property of this formalism is that the time-domain coupling coefficients are proportional to the bandwidth of the filters. For the same kind of filters, the bandwidth can be changed easily without having to rederive the coupling coefficients. We will first describe the formalism for lossless resonators. In the presence of uniform loss or gain, a predistortion technique is applied. We demonstrate the designs of Butterworth and Bessel CROW filters on ring resonators and grating defect resonators.

## 2. Finite-length CROWs

We start with the theory of CROWs using coupled-mode theory. Figure 1(a)
illustrates a CROW with an infinite number of identical resonators. The resonant frequency of each resonator is *ω*
_{0}, and the inter-resonator coupling coefficient is *κ*. For an input frequency *ω*, the mode amplitude of the *k*-th resonator can be written as *a _{k}*(

*t*)exp[

*iωt*], where

*a*(

_{k}*t*) is the slowly-varying amplitude. ${\left|{a}_{k}(t)\right|}^{2}$ represents the energy stored in the resonator. Since the resonators are coupled only to their neighbors, the coupled-mode equation of each resonator, assumed lossless in this section, can be written as [20,23]

*ω*≡

*ω*−

*ω*

_{0}and

*κ*is a real number. At steady state,

*da*/

_{k}*dt*= 0, and Eq. (1) becomes a recursive formula for

*a*, $\kappa {a}_{k+1}+\Delta \omega {a}_{k}+\kappa {a}_{k-1}=0.$ The general solution of this recursive formula is ${a}_{k+1}=\gamma {a}_{k},$ where

_{k}*γ*is the solution of $\kappa {\gamma}^{2}+(\Delta \omega )\gamma +\kappa =0.$ For a propagating mode (CROW mode), $\gamma =\mathrm{exp}[-iK\Lambda ],$ where

*K*is a wave number and Λ is the distance between adjacent resonators. When $\left|\Delta \omega \right|<2\kappa ,$

*ω*and

*K*represents the dispersion curve of the CROW (shown in Fig. 1(b)), which defines the CROW band within which light can propagate. Frequencies outside the CROW band are forbidden since

*K*is complex and $\left|\gamma \right|\ne 1.$

In practice, an infinite-length CROW has to be terminated and coupled to the outside world. Shown in Fig. 1(c), the first and last resonators of a CROW are coupled to the input and output waveguides. The coupling to the waveguides can be modeled as external losses, 1/τ* _{e}*, of the end resonators. When a CROW mode propagates to the boundary, the discontinuity between the CROW and the waveguide causes reflection, leading to Fabry-Perot-type oscillations. The reflection can be minimized by choosing 1/τ

*properly. Figure 1(d) illustrates the difference between a finite-length and an infinite-length CROW at the boundary. In the case of a finite CROW, the*

_{e}*N*-th resonator is coupled to the output waveguide, while in the case of an infinite CROW, it is coupled to the next resonator. The differential equations for these two cases are respectively

To match the boundary, the right-hand sides of Eqs. (3)a) and (3b) should be equal so that the *N*-th resonator cannot tell the termination of the CROW. Since ${a}_{N+1}=\gamma {a}_{N}$ for a CROW mode, the equality of Eqs. (3)a) and (3b) requires

For CROW modes, $\left|\gamma \right|=1$. Equation (4) requires 1/*τ _{e}* =

*κ*and

*γ*= −

*i*, which corresponds to the center of the CROW band (Δ

*ω*= 0). Figure 1(e) compares the transmission spectra of 10-resonator CROWs with 1/

*τ*=

_{e}*κ*and 1/

*τ*= 0.1

_{e}*κ*respectively. For 1/

*τ*=

_{e}*κ*, the spectrum is flat at the band center. The ripple amplitudes increase at frequencies close to the band edge since the boundary is only matched for Δ

*ω*= 0. For 1/

*τ*= 0.1

_{e}*κ*, the ripples are large over the whole bandwidth. The optimal boundary condition 1/

*τ*=

_{e}*κ*leads to maximally-flat transmission spectrum for finite-size CROWs with uniform coupling coefficients. To further reduce the Fabry-Perot oscillations over the whole CROW band, one can taper the coupling coefficients to adiabatically transform between the CROW mode and the waveguide modes [9,10]. The spectra of transmission and group delay can be further improved by choosing all the coupling coefficients so that the transfer function of the CROW is equal to the transfer function of a desired filter, which will be described in the next section.

## 3. Synthesis of bandpass filters based on CROWs

Consider a CROW which consists of *N* identical resonators and is coupled to input and output waveguides (Fig. 2
). All the *N* + 1 coupling coefficients are allowed to take on different values. The coupled-mode equations obeyed by the complex amplitudes of the *N* resonators are

The right-hand side of each equation consists of a detuning term (−*i*Δ*ωa _{k}* for each

*k*) and two coupling terms to the neighboring resonators, except for the first and last resonators which have only one neighbor. 1/τ

_{e}_{1}and 1/τ

_{e}_{2}are external losses of the first and last resonators due to coupling into the waveguides. The input mode with power ${\left|{s}_{in}\right|}^{2}$ is coupled into the first resonator via a coupling coefficient

*μ*

_{1}. It can be shown from conservation of energy and time reversal symmetry that ${\mu}_{1}=\sqrt{2/{\tau}_{e1}}$ [20,24]. At steady state, the left-hand sides of Eq. (5) are all 0. By replacing

*i*Δ

*ω*with the Laplace variable

*s*, Eq. (5) can be rewritten as

The $N\times N$ coupling matrix A is a tridiagonal matrix. The vector ** a** which contains all the mode amplitudes can be solved by inverting A. The transmitted and reflected amplitude,

*s*and

_{out}*s*are given respectively by

_{r}*s*with a leading term

*s*. Therefore, $T(s)$is an all-pole function with

^{N}*N*poles.

#### 3.1 N-th order all-pole bandpass filters

The transfer function of an all-pole lowpass filter with *N* poles can be written as

*k*,

*b*

_{N}_{-1},…,

*b*

_{0}are constants. Typical examples of all-pole filters are Butterworth, Chebyshev, and Bessel filters. We substitute

*s*with

*i*(

*ω*−

*ω*

_{0})/

*B*, where

*B*is a bandwidth parameter,

*T*(

*s*) then describes a bandpass filter which is centered at

*ω*

_{0}and of bandwidth scaled by

*B*. Figure 3(a) shows the transmission and group delay spectra of a Butterworth filter and a Bessel filter which feature maximally flat transmission and maximally flat group delay, respectively.

Since the amplitude transmission of an *N*-resonator CROW (Eq. (8)) and the transfer function of an *N*-th-order all-pole lowpass filter (Eq. (9)) are both all-pole functions with *N* poles, we present in what follows a formalism for designing the coupling coefficients of CROWs so that the amplitude transmission of the CROW is equal to the desired *T*(*s*).

#### 3.2 Extraction of coupling coefficients for a desired filter response

The tridiagonal matrix A in Eq. (6) has the following recursive properties of the polynomials *p*
_{1} through *p _{N}*:

*p*is the determinant of the buttom-right

_{k}*k*×

*k*submatrix of A (a principal minor of A). For example,

*p*is the determinant of A, and

_{N}*p*

_{1}= A

*. Each*

_{N,N}*p*is a polynomial in

_{k}*s*with a leading term

*s*. Once we know both

^{k}*p*and

_{N}*p*

_{N}_{-1}, all the coupling coefficients $(1/{\tau}_{e1},{\kappa}_{1},{\kappa}_{2},\cdots ,{\kappa}_{N-1},1/{\tau}_{e2})$ can be extracted step by step, using Eq. (10). For example, when dividing

*p*by

_{N}*p*

_{N}_{-1}, the quotient is $s+1/{\tau}_{e1}$, and the remainder is ${\kappa}_{1}{}^{2}{p}_{N-2}$. Then we can continue to divide

*p*

_{N}_{-1}by

*p*

_{N}_{-2}.

*p _{N}* and

*p*

_{N}_{-1}can be obtained from the transmission and reflection of the CROW. The amplitude transmission $T\equiv {s}_{out}/{s}_{in}$ and reflection $R\equiv {s}_{r}/{s}_{in}$ can be shown from Eqs. (7)a) and (7b) as

*k*is a constant. Given a desired $T(s)$,

*p*is already known. To find

_{N}*p*

_{N}_{-1},

*R*(

*s*) is also required and can be related to

*T*(

*s*) using conservation of energy, ${\left|T(i\omega )\right|}^{2}+{\left|R(i\omega )\right|}^{2}=1$, for a lossless system. Power spectral factorization is performed to obtain

*R*(

*s*) from ${\left|R(i\omega )\right|}^{2}$. In the power spectral factorization, there are at most 2

*choices of the numerator of*

^{N}*R*(

*s*), each of which will correspond to different coupling coefficients. We choose the

*R*(

*s*) whose coefficients are real and whose distribution of the zeros is the most symmetric around 0. If

*R*(

*s*) is complex, the resonant frequencies have to be detuned from

*ω*

_{0}, and the resonators are not identical. The details of finding

*R*(

*s*) are described in the appendix.

#### 3.3 Coupling coefficients of Butterworth and Bessel CROWs

Here we use an *N* = 4 Butterworth filter to demonstrate the extraction of coupling coefficients. The transfer function $T(s)=1/({s}^{4}+2.613{s}^{3}+3.414{s}^{2}+2.613s+1)$. The steps are listed in Table 1
. For Butterworth filters, the power spectral factorization for solving *R*(*s*) is unique and simple. The numerator of *R*(*s*) is *s ^{N}*. Table 2
lists the extracted coupling coefficients for Butterworth and Bessel filters of

*N*= 6 and 10. Note that the extracted coefficients are normalized by the bandwidth parameter

*B*, which can be selected to control the bandwidth of the CROW filter.

The coupling coefficients of Butterworth CROWs are symmetric. At the center of the CROW, the coupling coefficient is about 0.5, which corresponds to a CROW band from Δ*ω* = −1 to 1. The coupling coefficients gradually increase toward the two ends of the CROW. This adiabatic transition of the coupling coefficients reduces the reflection at the boundary, and Butterworth CROWs are one of the optimal designs which remove the oscillations in the transmission spectra. Bessel CROWs, which possess maximally flat delay, do not have symmetric coupling coefficients. With the proper choice of *R*(*s*) in the power spectral factorization (see Appendix), the coefficients are nearly symmetric.

Figure 4(a)
compares Butterworth CROWs comprising *N* = 6, 10 and 20. As the order increases, the transmission spectra become flatter in the passband and the roll-off at the band edges is steeper. To see how tolerant the Butterworth CROWs are under random change of the coupling coefficients, Fig. 4(b) shows the transmission spectra of *N* = 10 Butterworth CROWs whose coupling coefficients are multiplied by a random variable which is uniformly distributed between 0.9 and 1.1. In other words, the standard deviation of the coupling coefficient is 5.8% of its original value. From the transmission spectra of 10 different simulations, the transmission is above 94% over most of the bandwidth.

#### 3.4 CROWs with the presence of loss or gain

The filter design formalism described above assumes that the resonators are lossless. In practice, there are loss mechanisms, including intrinsic radiation loss of the resonator design, absorption loss of the material, and scattering loss due to imperfection of the fabrication. The modal loss can be modeled by the loss rate of the resonators, 1/*τ _{i}*, which is related to the quality factor

*Q*of the resonators by $Q=\omega {\tau}_{i}/2$. The differential equation of the

*k*-th resonator in Eq. (5) can be rewritten as

If the loss rates of all the resonators are identical, the coupling matrix in Eq. (6) is modified by replacing *s* with s + 1/*τ _{i}*. Therefore, for an all-pole filter response

*T*(

*s*) designed for lossless resonators, the transmission in the presence of loss is given by $T\text{'}(s)=T(s+1/{\tau}_{i})$, whose poles are shifted to the left by 1/

*τ*.

_{i}To obtain the desired filter response in the presence of loss, the poles can be shifted to the right by 1/*τ _{i}* in the design to pre-compensate for the left shift due to the loss. This technique is called predistortion of the filters [25]. First, the poles of the desired function

*T*(

*s*) in Eq. (9) are shifted to the right by 1/

*τ*. The constant in the numerator has to be decreased so that the magnitude of the new transfer function

_{i}*T*

_{0}(

*s*) is always smaller than or equal to 1. As a result, ${T}_{0}(s)=\alpha T(s-1/{\tau}_{i})$, where

*α*is a constant and is smaller than 1. In the presence of loss 1/

*τ*, the transfer function is $T\text{'}(s)={T}_{0}(s+1/{\tau}_{i})=\alpha T(s)$, which is recovered to the desired response except for the attenuation factor

_{i}*α*.

If the CROW is pumped with uniform gain, the amplifying CROW can be modeled with a negative 1/*τ _{i}*. The factor

*α*is greater than 1 and is an amplification factor. Table 2 lists the predistorted design for

*N*= 10 Butterworth CROWs with 1/

*τ*= 0.05

_{i}*B*(lossy) and −0.05

*B*(amplifying) respectively. Their transmission spectra with and without loss/gain are plotted in Figs. 5(a) and 5(b) respectively. Since the group delay is greater at the band edge, frequencies near the band edge experience larger loss and gain. Consequently, the amplitude responses are predistorted accordingly before the loss or gain, as can be seen in Fig. 5. For Bessel filters, since the group delay is almost constant over the bandwidth, the characteristics of the filters remain the same in the presence of small gain or loss.

## 4. CROW filters based on microring resonators

A microring CROW consists of a chain of coupled ring resonators (Fig. 6(a) ). Light is coupled in and out of the CROW via the input and output waveguides. Assuming the coupling region is sufficiently long compared to the wavelength, only light circulating in one direction in thering is phase-matched to the input and is excited. The coupling between two adjacent rings can be analyzed, using the notation in Fig. 6(b), by

*η*and

*t*are respectively the dimensionless coupling and transmission coefficients over the coupling region. Assuming the coupling is lossless, ${\eta}^{2}+{t}^{2}=1$. The transmission and reflection of microring CROWs can be analyzed using transfer matrix formalism [26].

The field coupling coefficient *η* at the coupling region is related to the time-domain coupling coefficient *κ* between two rings. In [20], these relations were derived: $\eta =\kappa /{f}_{FSR}$for inter-resonator coupling and ${\eta}_{in,out}=\sqrt{2/({\tau}_{e}{f}_{FSR})}$ for waveguide-resonator coupling, where ${f}_{FSR}={v}_{g}/2\pi R$ is the free spectral range of the ring resonators. However, these formulas are only valid when the coupling is sufficiently weak, because the field inside the ring is not uniform when *η* is not much smaller than 1 [15]. In what follows, we describe a method which is valid for any reasonable coupling.

*Inter-resonator coupling:* Consider two identical resonators coupled to each other with a coupling coefficient *κ* (Fig. 7(a)
). The resonant frequencies of the two eigenmodes are *ω*
_{0} ± *κ*. Figure 7(b) illustrates the corresponding case for ring resonators. The field coupling coefficient is *η*. The fields b_{1,2} and c_{1,2} are related by the coupling (Eq. (13)), and propagation along the rings leads to

*θ*(Δ

*ω*) is the round-trip phase of the rings at

*ω*=

*ω*

_{0}+ Δ

*ω*and is equal to −Δ

*ω*/

*f*. Combining Eqs. (13) and (14), exp[−

_{FSR}*iθ*(Δ

*ω*)] is the eigenvalue of the coupling matrix in Eq. (13), which leads to $\theta (\Delta \omega )=\pm {\mathrm{sin}}^{-1}\eta $. Therefore, the frequency splitting, which is equal to the coupling coefficient

*κ*, is $\Delta \omega =\pm {f}_{FSR}{\mathrm{sin}}^{-1}(\eta )$. As a result,

*Waveguide-resonator coupling:* In Fig. 7(c), the two resonators in Fig. 7(a) are both coupled to an output waveguide with an external loss, 1/*τ _{e}*. By writing down the coupled-mode equations of the two resonators and solving for the steady-state solution at

*ω*=

*ω*

_{0}, the amplitude transmission ${s}_{out}/{s}_{in}$ is given by $(2\kappa /{\tau}_{e})/(1/{\tau}_{e}{}^{2}+{\kappa}^{2})$, which equals 1 only when the boundary condition $1/{\tau}_{e}=\kappa $ is satisfied. We can use this structure to derive the conversion of 1/

*τ*. Figure 7(d) illustrates the corresponding structure for ring resonators. The condition that the transmission is unity at

_{e}*ω*=

*ω*

_{0}can be derived as

For a given 1/*τ _{e}*, we first use Eq. (15) to find an inter-resonator coupling

*η*which corresponds to a coupling coefficient $\kappa =1/{\tau}_{e}$, and then use Eq. (16) to obtain

*η*.

_{i}With Eqs. (15) and (16), we are ready to convert the coupling coefficients *κ* in Table 2 to the microring couplings *η*. The only constraint is that *κ* does not exceed (*π* /2)*f _{FSR}*, or

*ω*/4, the maximal coupling which ring resonators with a free spectral range

_{FSR}*f*can achieve (see Eq. (15)). We consider examples of silicon microring CROWs. The mode index and group index of the silicon waveguides are respectively 2.4 and 4. The ring radius is 30 μm so that one resonant wavelength is at 1570.8 nm, and the free spectral range

_{FSR}*f*is 398 GHz. The bandwidth of the filters can be chosen by setting the bandwidth parameter

_{FSR}*B*. For example, the bandwidth of Butterworth filters is 2

*B*(Fig. 3(a)). If we choose

*B*=

*ω*·0.005, where ${\omega}_{FSR}=2\pi {f}_{FSR}$, the bandwidth is 2·398GHz·0.005 = 3.98GHz. The converted

_{FSR}*η*for Butterworth and Bessel filters with

*B*=

*ω*·0.005 and

_{FSR}*B*=

*ω*·0.05 are listed in Table 3 .

_{FSR}The transmission spectra of the microring CROWs in Table 3 were calculated using the transfer matrix formalism and were compared with the original transmission spectra based on the *κ* in Table 2, calculated using CMT. Figure 8(a)
shows the transmission spectra of Butterworth filters with *B* = *ω _{FSR}*·0.005. Since

*η*are sufficiently weak (the largest

*η*is 0.338), the two spectra are nearly identical. Figure 8(b) shows the spectra for

*B*=

*ω*·0.05, where the coupling is stronger. Although there are small passband ripples whose amplitude is about 0.0002, the spectrum still closely agrees with the desired response. Therefore, the conversion is valid even when

_{FSR}*η*is as high as 0.852, whereas the same

*κ*would be converted to

*η*= 1.102 using the formula proposed in [20]. Note that for a bigger bandwidth or higher filter order,

*κ*at the boundary will increase and might exceed the upper limit of

*κ*, (

*π*/2)

*f*. Therefore, resonators with large

_{FSR}*f*are beneficial. However, for ring resonators with very small radii, say less than 5 μm, the assumption of the transfer matrix formalism that the coupling region is sufficiently long compared to the wavelength is no long valid, and the coupling of modes will be complicated since modes in the opposite direction will also be excited. Figure 8(c) shows the spectra of transmission and group delay for an

_{FSR}*N*= 6 Bessel CROW with

*B*=

*ω*·0.05. Figure 8(d) compares the transmission spectra of Butterworth CROWs with 6 and 20 rings.

_{FSR}## 5. CROW filters based on grating defect resonators

CROWs can be realized on a waveguide grating with multiple “defects”. A Bragg grating is a periodic perturbation to the waveguide. When an artificial defect is introduced in a grating, a defect mode is created with a resonant frequency inside the grating band gap. If the defect length corresponds to a quarter wavelength (a quarter-wave-shifted defect), the mode resonates at the Bragg frequency of the grating. This defect mode consists of a forward and a backward waveguide mode. The envelope of the field distribution is centered at the defect and decays exponentially in the grating, as shown in Fig. 9(a)
. If the grating consists of multiple defects (Fig. 9(b)), each defect mode interacts with its neighbors via their evanescent tails. The coupling between adjacent defects can be controlled by the spacing between the defects (*L*
_{1},*L*
_{2},…,*L _{N}*

_{+1}in Fig. 9(b)). These defect resonators form a grating CROW.

CROWs based on waveguide gratings are attractive because the coupling between CROWs and waveguides is natural and easy to implement. For strong gratings, the size of grating resonators can be as small as several microns, which is much smaller compared to ring resonators [4]. However, the defect resonators in a strong grating require a proper design to reduce the radiation losses due to spatial Fourier components which are coupled to the radiation modes of the waveguide [27].

CROWs based on weak gratings can be analyzed by coupled-mode equations [17,28]:

*a*and

*b*are the amplitudes of the forward and backward waveguide modes,

*δ*is the detuning of the propagation constant from the Bragg condition of the grating, and

*κ*(

_{g}*z*) is the coupling coefficient of the grating. In a grating CROW, the phase of

*κ*(

_{g}*z*) is shifted by

*π*at each quarter-wave-shifted defect, as shown in Fig. 9(b), while the amplitude remains the same. For an input mode

*a*(−∞) from the left, the field distribution and the transmission of a grating CROW with a given

*κ*(

_{g}*z*) can be solved using Eq. (17) with the boundary condition

*b*(∞) = 0.

The conversion from the coupling coefficients *κ* in Table 2 to the lengths *L*
_{1},*L*
_{2},…,*L _{N}*

_{+1}in grating CROWs applies methods similar to those employed in Section 4.

*Inter-resonator coupling:*
Fig. 9(c) shows the case of two defects which are separated by a distance *L* in an infinitely long grating. It was shown in [17] that under the assumption of $\mathrm{exp}(-{\kappa}_{g}L)\ll 1$, the resonance splitting due to the coupling is $\Delta \omega =\pm {\kappa}_{g}{v}_{g}\mathrm{exp}(-{\kappa}_{g}L)$, where *v _{g}* is the group velocity of the waveguide mode. Since the Δ

*ω*is equal to the coupling coefficient

*κ*of the two resonators,

*ω*≡

_{g}*κ*.

_{g}v_{g}*Waveguide-resonator coupling:*
Fig. 9(d) shows a finite grating with two defects. The external loss 1/*τ _{e}* of the defect modes into the waveguides is controlled by the length

*L*. The amplitude transmission of this grating at Bragg frequency can be solved, using Eq. (17), as $1/\mathrm{cosh}[{\kappa}_{g}(2{L}_{i}-L)]$. The transmission is unity if ${L}_{i}=L/2$, which corresponds to the boundary condition

_{i}*κ*= 1/

*τ*. Therefore,

_{e}Using Eqs. (18) and (19), the coupling coefficients in Table 2 are converted to the lengths of the grating sections, which are listed in Table 4
. We consider Si waveguides whose group index is 4 at the wavelength of 1570 nm. The grating strength *κ _{g}* is 0.1/μm, so

*ω*/(2

_{g}*π*) is 1.19 THz. The transmission and group delay spectra of N = 6 Butterworth and Bessel filters with

*B*=

*ω*·0.05 are plotted in Fig. 10 .

_{FSR}## 6. Conclusion

We have demonstrated a formalism for choosing the coupling coefficients of CROWs to achieve desired filter responses such as maximally flat transmission (Butterworth filters) and maximally flat group delay (Bessel filters). The formalism uses CMT and the recursive relations of the coupling matrix to extract the coupling coefficients. Compared to TMM, the design using CMT is simpler since the field in each resonator is represented by only one variable. The universal coupling coefficients can be applied to any type of resonators or even the coupling of different types of resonators. The bandwidth of the filters can be changed easily by selecting the bandwidth parameter *B*. Furthermore, predistortion techniques can be applied for the design of lossy or amplifying CROW filters.

The disadvantage of CMT is that it assumes weak coupling between resonators and that the field distribution in each resonator remains unchanged. It is less accurate than TMM, which directly analyzes the fields inside the resonators. The time-domain coupling coefficients of CMT have to be converted to the parameters of the type of resonators comprising the CROWs. We have demonstrated the conversion to the field coupling coefficients for microring resonators and the lengths of grating sections for grating defect resonators. The formulas for the conversion are valid for any reasonable coupling.

## Appendix

In what follows we describe the details of finding *p _{N}*

_{-1}for the purpose of using Eq. (10) to extract the coupling coefficients of CROWs. The formalism is similar to the z-domain digital filter design described in [11,19].

Assuming the CROW is lossless, *T*(*s*) and *R*(*s*) are related by ... For an all-pole filter of order *N*, *T*(*s*) can be written as

*q*

_{1}through

*q*are the poles and

_{N}*k*is a constant. All the filter responses

*T*(

*s*) we consider in this paper are real functions, so the poles come in complex conjugate pairs. Therefore,

The denominators of *T*(*s*) and *R*(*s*) are identical, as can be seen in Eqs. (11)a) and (11b). We assume that the numerator of *R*(*s*) is $p(s)=(s-{z}_{1})(s-{z}_{2})\cdots (s-{z}_{N})$, where *z*
_{1} through *z _{N}* are the zeros of

*R*(

*s*). The goal of power spectral factorization is to find

*z*

_{1}through

*z*so that ${\left|p(i\omega )\right|}^{2}=({\omega}^{2}+{q}_{1}{}^{2})({\omega}^{2}+{q}_{2}{}^{2})\cdots ({\omega}^{2}+{q}_{N}{}^{2})-{k}^{2}$. Each zero

_{N}*z*is selected from a pair (

_{i}*z*, −

*z**), where

*z*is a complex number, so there are at most 2

*combinations of zeros. In general,*

^{N}*p*(

*s*), and thus

*p*

_{N}_{-1}, are not real. For a filter with a complex

*p*

_{N}_{-1}, the resonant frequencies of the resonators have to be detuned, and Eq. (6) is modified as

*ω*

_{0}. Equation (10) is also modified by replacing

*s*with

*s*−

*δ*, so

_{i}*δ*can be extracted during the extracting process.

_{i}The zeros of *R*(*s*) can be chosen so that *p _{N}*

_{-1}is real. Figure 11 shows three different choices of zeros for an

*N*= 7 Bessel filter. They correspond to different coupling coefficients

*κ*and frequency detuning

*δ*, as listed in Table 5 . The first one is often referred to as “minimum phase”, where all zeros are located inside the left-half

*s*-plane (Fig. 11(a)). It corresponds to zero frequency detuning and monotonically increasing

*κ*. In the second one, the zeros are all located at the first and third quadrants (Fig. 11(b)). The resulting values of

*κ*are symmetric, but the frequency detuning is nonzero since

*p*

_{N}_{-1}is complex. In our CROW filter design, we prefer a nearly symmetric

*κ*without frequency detuning. Consequently, we choose the zeros that are the most uniformly distributed around the origin and are complex conjugate pairs, as shown in Fig. 11(c). Note that although the three CROW filters in Table 5 look very different, they have the same

*T*(

*s*) and ${\left|R(i\omega )\right|}^{2}$, except that the phase of

*R*(

*s*) is different.

## Acknowledgments

This work was supported by National Science Foundation and The Army Research Office.

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