## Abstract

A method for directly extracting the coupling coefficients, the resonant frequency detunings and the loss of an *N*
^{th}-order serially-coupled microring resonator filter from the measured power spectral responses is presented. The device parameters are obtained from the through-port complex transfer function, which is constructed from the experimentally extracted poles and zeros of the filter. We applied the method to determine the parameters of a symmetric microring doublet fabricated in the Siliconon-Insulator material platform. Simulated spectral responses using the extracted parameters showed good agreement with the measured data. The extracted parameters along with the poles and zeros of the device provide important information about the fabrication process and can be used to guide the post-fabrication trimming of high-order microring filters.

© 2008 Optical Society of America

## 1. Introduction

Microring resonators have been shown to have promising applications in optical spectral engineering, such as high-order filtering [1–3], dispersion compensation [4], and group delay engineering [5]. By arranging multiple resonators in a suitable coupling configuration, the spectral response of the device can be tailored to meet the requirements of a specific application. Due to its simple topology and relative ease of fabrication, the serial coupling configuration has been the most widely used, and devices based on this configuration have been demonstrated in various material systems [6–9]. Serially-coupled microring devices are often designed to have spectral responses with a flat-top passband and sharp skirt roll-off, which can be achieved by proper placement of the poles and zeros of the drop-port and through-port transfer functions of the device [10]. However, due to fabrication errors, the coupling parameters of the fabricated device are in general deviated from the designed values and the resonant frequencies of the microring resonators are always detuned from each other, causing the poles and zeros of the filter to be displaced from the desired locations. As a result, the fabricated device usually does not exhibit the desired spectral characteristics, and a post-fabrication index trimming method such as thermo-optic tuning is usually required to correct for the resonant frequency mismatch to optimize the spectral response. In order to expedite the process of spectral optimization, it is important to know the locations of the poles and zeros of the fabricated device, as well as the values of its parameters such as the coupling coefficients, the resonant frequency detunings and the loss in the microring resonators. Also, knowledge of the fabricated device parameters provides important information about the fabrication process, such as the process bias, uniformity, yield and repeatability.

In spite of their significance, to date there is no systematic method available for extracting the microring device parameters from the measured spectral responses. The difficulty of the problem lies in the fact that a relatively large number of unknown parameters, namely the coupling coefficients, the resonant frequencies and the loss in the microrings, need to be determined from just a few available measurements, which typically are the drop-port and through-port transmitted power responses. The conventional approach for determining the device parameters is to perform simulations of the microring device with the parameters varied by trial and error until some level of agreement is obtained between the simulated and measured transmission responses. For high order filters consisting of many microrings, the coupling parameters, the resonant frequency detunings and the microring loss affect the filter responses in a complex way, making the above trial-and-error approach a very time-consuming if not impractical process. Also, the values of the parameters obtained may not correspond to the physical device structure.

In this paper we show that it is possible to directly determine all the parameters of a general *N*
^{th}-order serially-coupled microring filter simply by measuring its transmission responses at the drop port and through port. In our approach, the poles and zeros of the filter are first experimentally determined from the measured power spectral responses, from which the complex transfer function at the through port of the filter can be constructed. Knowledge of the through-port transfer function then enables the coupling coefficients, the resonant frequency detunings and the loss in the microrings to be extracted. To demonstrate the proposed parameter extraction technique, we applied the procedure to extract the parameters of a symmetric microring doublet fabricated in the Silicon-on-Insulator (SOI) material system. Symmetric microring doublet structures are of special interest because they can serve as simple building blocks in a parallel-cascaded array for constructing more complicated filter responses, such as those containing transmission zeros [11]. In the realization of such a filter, knowledge of the parameters of the microring doublet in each stage is important for post-fabrication spectral optimization of the device response. We show that using the extracted device parameters, the spectral characteristics of the microring doublet can be reconstructed which are in good agreement with the measured responses.

## 2. Device model and parameter extraction procedure

We describe in this section the theoretical approach of the proposed parameter extraction technique, which is based on the energy coupling formalism of serially-coupled microring filters. Subsection A shows how the device parameters can be extracted from the complex transfer function at the through port of the device. Subsection B shows how the poles and zeros of the filter can be experimentally determined from the measured power spectral responses, which can then be used to construct the complex transfer functions of the filter.

#### 2.1 Parameter extraction from the through-port complex transfer function

Figure 1 shows the schematic diagram of an *N*
^{th}-order serially-coupled microring filter. Each microring resonator *k* is assumed to have non-identical resonant frequency denoted by ω* _{k}*=ω

_{ο}+Δω

*, where ω*

_{k}_{ο}is a reference frequency chosen near the passband center and Δω

*is the amount of detuning of ω*

_{k}*from ω*

_{k}_{0}. For simplicity we assume that the microring resonators have identical intrinsic loss caused by waveguide sidewall scattering, material absorption, bending and coupling losses, which result in an effective decay rate γ

*in the amplitude of the energy stored in each resonator. The rates of energy coupling between adjacent microrings are denoted by the coupling coefficients µ*

_{L}*,*

_{k}*k*=1 to

*N*- 1. Microrings 1 and

*N*are also coupled to the input and output waveguides via coupling coefficients µ

_{ο}and µ

*, respectively, which lead to additional energy loss at a rate of γ*

_{N}_{ο}=µ

^{2}

_{ο}/2 in microring 1 and γ

*=µ*

_{N}^{2}

*/2 in microring*

_{N}*N*. Denoting

*a*as the energy amplitude in microring

_{k}*k*, the rates of change in the energy amplitudes in the microrings due to loss and mutual couplings are given by [1]

$$\frac{d{a}_{k}}{dt}=\left(j{\omega}_{k}-{\gamma}_{L}\right){a}_{k}-j{\mu}_{k-1}{a}_{k-1}-j{\mu}_{k}{a}_{k+1},\phantom{\rule{.2em}{0ex}}k=2,3,\dots N-1,$$

$$\frac{d{a}_{N}}{dt}=\left(j{\omega}_{N}-{\gamma}_{L}-{\gamma}_{N}\right){a}_{N}-j{\mu}_{N-1}{a}_{N-1}.$$

Assuming harmonic time dependence *a _{k}* ∝

*e*, the above equations can be solved to give the following continued-fraction expression for the energy amplitude

^{jωt}*a*

_{1}in microring 1:

where *s*=*j*(ω-ω_{ο}) and *r _{k}*=γ

*-*

_{L}*j*Δω

*. At the through port of the filter, we have*

_{k}where *s _{i}* and

*s*represent the input and through-put energy signals, respectively. Denoting

_{t}*T*(

_{t}*s*)=

*s*/

_{t}*s*as the complex transfer function at the through port, we rearrange (3) to get

_{i}Expressing *T _{t}* (

*s*)=

*R*(

*s*)/

*Q*(

*s*), where

*R*(

*s*) and

*Q*(

*s*) are polynomials of degree

*N*with leading coefficient of 1, we can write

where *M*(*s*) is a polynomial of degree *N* - 1 whose leading coefficient is also assumed to be 1. By expanding the polynomial ratio *M*(*s*)/*Q*(*s*) as a continued fraction in the form of (4), the parameters µ* _{k}* and

*r*can be extracted to give the energy coupling coefficients µ

_{k}*, the resonant frequency detunings, Δω*

_{k}*=-Im{*

_{k}*r*}, and the intrinsic energy decay rate, γ

_{k}*=Re{*

_{L}*r*}. In the parameter extraction procedure, given the through-port complex transfer function

_{k}*T*(

_{t}*s*), we first determine

*M*(

*s*),

*Q*(

*s*) and µ

_{ο}from (5). Next

*Q*(

*s*) is divided by

*M*(

*s*) to get

which upon comparison with (4) gives Re{*a*}=γ* _{L}*+γ

_{ο}, Im{

*a*}=-Δω

_{1}, and

*b*=µ

^{2}

_{1}. Next,

*M*(

*s*) is divided by

*N*(

*s*) to obtain

*r*

_{2}and µ

_{2}, and the procedure is repeated until all the parameters

*r*and µ

_{k}*,*

_{k}*k*≤

*N*, are extracted. Finally the energy coupling parameters µ

*and the intrinsic energy decay rate γ*

_{k}*are converted to the field coupling parameters κ*

_{L}*and the power attenuation constant α (m*

_{k}^{-1}) in the ring waveguides via the relations [1]

and

$${\kappa}_{k}={\mu}_{k}{T}_{rt},1\le k<N-1,$$

$$\mathrm{and}\phantom{\rule{.9em}{0ex}}$$

$$\phantom{\rule{.2em}{0ex}}\frac{\alpha ={2}_{\gamma L}}{vg},$$

where *T _{rt}*=1/FSR is the round trip time of the microring resonators and

*v*is the group velocity in the microring waveguides.

_{g}#### 2.2. Pole-zero extraction from the power spectral responses

The above procedure provides a systematic way to directly extract the device parameters if the complex transfer function *T _{t}*(

*s*) at the through port of the microring filter is known. The through-port transfer function

*T*(

_{t}*s*) along with the drop-port transfer function

*T*(

_{d}*s*) can be constructed by extracting the poles and zeros of the filter from the measured spectral responses of the transmitted powers at the drop port and through port. For a serially-coupled microring filter, the transfer functions can be expressed as

where *p _{k}* are the poles of the filter,

*z*are the zeros of the through-port transfer function, and

_{k}**is a constant. The measured power spectral responses at the drop port and through port are**

*K*To extract the poles and zeros, we define two new power ratios *P*
_{1} and *P*
_{2} as

which are polynomials of degree 2*N* with real coefficients. In the parameter extraction procedure, *P*
_{1}(ω) and *P*
_{2}(ω) are obtained by performing least-square polynomial fit on the curves 1/|*T _{d}*(ω)|

^{2}and |

*T*(ω)|

_{t}^{2}/|

*T*(ω)|

_{d}^{2}computed from the power spectral measurements. From (12) and (13) it is seen that the roots of

*P*

_{1}(ω) and

*P*

_{2}(ω) appear in conjugate pairs as {-

*jp*,

_{k}*jp**

*} and {-*

_{k}*jz*,

_{k}*jz**

*}, respectively, for*

_{k}*k*=1 to

*N*. Thus the poles and zeros of the filter are simply the roots of

*P*

_{1}and

*P*

_{2}, respectively, which have been rotated by 90° in the

*s*-plane. Let {

*u*,

_{k}*u**

*} and {*

_{k}*v*,

_{k}*v**

*} be the roots of*

_{k}*P*

_{1}and

*P*

_{2}, respectively, for

*k*=1 to

*N*. Since all the poles of a microring filter must lie in the left-half of the s-plane, we choose those roots in the set {

*ju*,

_{k}*ju**

*} which have negative real parts as the poles of the filter. For most microring filters of interest such as the Butterworth and Chebyshev types, the zeros lie on the imaginary axis or in the left-half s-plane, so they can be chosen from the set {*

_{k}*jv*,

_{k}*jv**

*} which have zero or negative real parts. In the more general case, some zeros may also lie in the right-half s-plane, in which case one can refer to the original filter design to choose the appropriate zeros from the set {*

_{k}*jv*,

_{k}*jv**

*}*

_{k}*jv*

_{k}*jv*. Finally the complex transfer function

_{k}*T*(

_{t}*s*) at the through port can be constructed from the extracted poles and zeros according to (9).

## 3. Parameter extraction of an SOI microring doublet

We applied the above parameter extraction procedure to experimentally determine the parameters of a double microring filter fabricated in the Silicon-on-Insulator material system. Note that even for this simple device, there are six independent parameters to be extracted: Δω_{1}, Δω_{2}, µο, µ_{1}, µ_{2} and γ* _{L}*. The device was fabricated on an SOI wafer consisting of a 340nm silicon layer on a 1µm buried oxide layer. To obtain strong field confinement we designed the waveguides to have a width of 450nm. Figure 2(a) shows a schematic of the filter structure, which consists of two serially-coupled microracetracks with bending radii

*R*=5µm and straight waveguide coupling lengths of 2.5µm. The designed coupling gaps were

*g*

_{ο}=

*g*

_{2}=200nm and

*g*

_{1}=350nm, corresponding to calculated field coupling coefficients of ρ

_{ο}=ρ

_{2}=0.27 and ρ

_{1}=0.05 for the TE polarization. The theoretical coupling values were determined using the strongly coupled-mode theory [12] with the waveguide modes numerically computed from the two-dimensional index profile of the waveguides.

The device was patterned using a Raith 150 E-Beam Lithography System with polymethyl- methacrylate (PMMA) as the electron-beam resist. After developing the chip in an MIBK:IPA (1:3) developer, the resist pattern was transferred to the silicon layer by inductively-coupled plasma reactive ion etching (ICP RIE). The PMMA was then removed and the device was cleaved to expose the waveguide facets for fiber butt-coupling. Figures 2(b) and 2(c) show SEM images of the fabricated device and the coupling gap between the two microracetracks. From the SEM image we measured the bus-to-ring coupling gaps to be *g*
_{ο}=204 ± 5nm and *g*
_{2}=206 ± 5nm, and ring-to-ring coupling gap *g*
_{1}=337 ± 5nm. These values correspond to average calculated field coupling coefficients of ρ_{ο}=ρ_{2}=0.2744 and ρ_{1}=0.0609.

Using a tunable laser source with the polarization adjusted to TE, we butt-coupled light to the waveguide facets via tapered fibers and measured the transmitted powers as a function of the swept wavelength. The measured power spectral responses at the drop and through ports of the device are shown by the red and black dotted lines in Fig. 3. The drop-port response shows a passband with a 3dB-bandwidth of 50GHz and a ripple of about 1.1dB. The free spectral range of the device is 1.63THz, which yields a round trip time *T _{rt}* of 0.6135ps. Note that the through port response of the device is asymmetric, indicating that the resonant frequencies of the two microracetracks were detuned from each other. To extract the poles and zeros of the filter, we computed the power ratios

*P*

_{1}and

*P*

_{2}according to (12) and (13) and plotted the results in Fig. 4(a) as functions of the normalized frequency ω/ω

*, where ω*

_{c}*=2π×25GHz is the 3dB cut-off frequency. Performing least square fit on the data using a fourth-degree polynomial yielded the best-fit curves shown by the solid lines in Fig. 4(a). Slight deviations of the best-fit curves from the measured data can be seen which are due to Fabry-Perot oscillations in the measurements caused by reflections at the cleaved waveguide facets. The roots of the best-fit polynomials were next computed and their values are shown in Table 1. From the roots of*

_{c}*P*

_{1}and

*P*

_{2}we obtained the two poles and two zeros of the filter which are also given in Table 1. In Fig. 4(b) we show on the pole-zero diagram the roots of

*P*

_{1}and

*P*

_{2}(

*u*and

_{k}*v*), the extracted poles and zeros (

_{k}*ρ*and

_{k}*z*), and the designed poles and zeros (

_{k}*p′*and

_{k}*z′*) of the filter. The extracted poles are the roots

_{k}*u*

_{1}and

*u*

_{3}of

*P*

_{1}which have been rotated counter-clockwise by 90°. Similarly, the extracted zeros are obtained by rotating the roots

*v*

_{1}and

*v*

_{3}of

*P*

_{2}. Compared to the designed poles and zeros, the extracted poles and zeros have been shifted to the left due to loss in the microracetracks. Also the extracted poles and zeros no longer appear in conjugate pairs due to resonant frequency detunings of the two resonators.

From the extracted poles and zeros of the filter we constructed the through-port transfer function *T _{t}*(

*s*) and applied the parameter extraction procedure in Section 2A to determine the device parameters. The resulting values for the coupling coefficients, the resonant frequency detunings and the loss in the microracetracks are summarized in Table 2. The theoretical coupling values as computed from the measured coupling gaps are also shown in the table for comparison. The extracted bus-to-ring coupling values κ

_{ο}and κ

_{2}show that there is a 32% mismatch between the two values, as well as an 8% discrepancy between the average value of κ

_{ο}and κ

_{2}and the theoretical value. The extracted ring-to-ring coupling coefficient κ

_{1}is also about 12% larger than the theoretical value. The discrepancies between the extracted and theoretical coupling values can be attributed to random fabrication variations as well as fluctuations in the power spectral measurements due to Fabry-Perot oscillations. These oscillations introduced errors in the polynomial fits of

*P*

_{1}and

*P*

_{2}in Fig. 4(a), resulting in uncertainties in the extracted poles and zeros. From Table 2 it is seen that fabrication errors also caused a relative detuning of 8.6GHz (or 0.07nm) in the resonant frequencies of the two microracetracks. The extracted propagation loss in the microracetracks is 38dB/cm. This figure is higher than a loss figure of 29dB/cm which we obtained from an independent measurement using a single add/drop circular microring with a 5µm radius fabricated on the same SOI platform. The excess loss in the microracetracks compared to the circular ring is due to scattering from the junctions between the bent and straight waveguide sections in the microracetracks. With further optimization of the fabrication process, we expect the propagation loss in the microring waveguides to be further reduced.

From the extracted device parameters we computed the spectral responses of the device using power coupling analysis and plotted the results on top of the measured data in Fig. 3 (green and purple lines). It is seen that apart from the Fabry-Perot oscillations in the measured data, the computed responses are in good agreement with the measured responses. In particular, the computed drop-port response replicates the ripples in the passband and exhibits the same skirt roll-off as the measured response. The bandpass spectral shape is also repeated at the correct FSR intervals, which closely match with the measured data.

The extracted coupling coefficients and microring loss in Table 2 can yield valuable information about the fabrication process variations. By performing characterization of the same device in different dies or on different wafers, statistical correlation between the process parameters and the device parameters can be established. This information can in turn be used to predict the process yield as well as the expected variations in the device performance. Also, the extracted resonant frequency detunings of the microring resonators in Table 2 and the pole-zero diagram in Fig. 4(b) can be used to guide the post-fabrication optimization of the device response. For example, in the thermo-optic trimming of the microrings using micro heater elements, from the values of the frequency detunings one can determine the exact voltages to be applied to the heating elements to correct for the frequency mismatch. The extracted poles and zeros of the optimized device can then be compared to those of the filter design to provide a feedback measure of the trimming process.

## 4. Conclusion

We showed that the parameters of an *N*
^{th}-order serially-coupled microring filter, including the resonant frequency detunings, can be experimentally extracted simply from the spectral measurements of the transmitted powers at the drop port and through port of the device. We also showed that the poles and zeros of the device can be determined from these measurements. The dominant source of errors in the proposed method was found to be the Fabry-Perot oscillations in the measured data, which introduced uncertainties in the polynomial least-square fits of the power ratios. With a good design of the fiber-waveguide couplers to minimize reflections at the waveguide facets, the accuracy of the extracted parameters and the fit between the simulated and measured spectral responses can be further improved. We expect the parameter extraction method to be useful in providing important information about the fabrication process, as well as in assisting in the post-fabrication trimming of high-order microring filters.

## Acknowledgment

The authors would like to acknowledge funding for this work by the Natural Sciences and Engineering Research Council of Canada.

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