## Abstract

Ring resonators are waveguide realizations of Fabry-Perot resonators which can be readily integrated in array geometries to implement many useful functions. Its nonlinear phase response can be readily incorporated into a Mach-Zehnder interferometer to produce specific intensity output function. We present two generalized array configurations of ring-coupled MZI and discuss their characteristics in terms of the amplitude and phase response of the ring arrays as well as the transmission output of the MZIs. The two types of array have distinct transfer functions and effective phase shifts, and can be tailored to phase-engineer a wide-range of MZI transmission functions.

©2005 Optical Society of America

## 1. Introduction

Mach-Zehnder interferometer (MZI) provides an efficient means for converting phase modulation to intensity modulation. In this device, a single input wave is split between two arms, and a phase shift is induced either in a single arm or in both arms in a push-pull manner. There could be a single output or two complementary outputs depending on whether the two arms are combined via a Y branch or a 2×2 3dB coupler. The phase difference could be induced by an electro-optic or all-optical effect, a change in temperature or one of many other possible control parameters. As such, the MZI is a versatile device with many applications, for examples, as a space switch [1–2], intensity modulator [3], optical filter or sensor [4,5].

To engineer the performance of MZI, one could tailor the phase shift by introducing different passive or active building blocks into the MZI. One effective way is by coupling a ring resonator (RR) to one of the MZI arms. Since the ring resonator is a resonant structure, the phase accumulated inside the ring will enhance the phase difference between the MZI arms. We shall generally call this MZI a ring-enhanced MZI (REMZI). In this paper, we study the effect of introducing more complicated ring resonator configurations, in order to understand how far we can tailor the phase shift and the resulting MZI transfer function. The higher-order ring resonator structures are also filters [7] and have unique transmission and phase responses depending on the configurations. When coupled to MZI they greatly enhance the sensitivity of the MZI to frequency and effective index changes. In the most general case, we will consider a 2-D array of RRs, coupled to the MZI in two possible configurations to be discussed below. The resonators will at first be assumed lossless so as to focus our study solely on the phase shift introduced by the ring structures. We will then discuss the effect of loss.

## 2. Ring-enhanced MZI

In a REMZI, the ring resonator structure is coupled to one arm of an MZI. The outputs of the MZI can be expressed in general by the following matrix relation:

where the first and the third matrices on the right-hand side represent the output and input couplers of the MZI [7,9], respectively, and *T* exp(Δ*φ _{eff}*)≡

*t*in the second matrix summarizes the complex transfer function (amplitude and phase) of the ring structure which is coupled to the upper arm of the MZI. For generality we have included an additional phase bias Δ

_{eff}*φ*(active or passive) at the lower MZI arm. However, we will assume that Δ

_{b}*φ*=0 so as to focus on the phase shift introduced by the ring structures. To simplify the analysis, all rings and ring couplers are assumed to be identical, and for the input and output couplers 3-dB coupling ratios are assumed (i.e.,

_{b}*r*=

*κ*=1/√2). From (1), the bar and cross output powers are given by

$${P}_{c}={\mid {E}_{\mathit{cross}}\mid}^{2}=\left(1+{T}^{2}+2T\mathrm{cos}\Delta {\phi}_{\mathit{eff}}\right)\u20444$$

When *T*=1, (2) reduces to the usual equation *P _{b}*=sin

^{2}(Δ

*φ*/2) and

*P*=cos

_{b}^{2}(Δ

*φ*/2) for a symmetric MZI. The two outputs are complementary.

We first consider the simplest case of a single ring resonator (RR) coupled to a bus waveguide, as shown in inset A of Fig. 1, which forms the basic building block for all higher-order structures. The coupler acts as a partly transmitting mirror with a reflectivity (*r*), which is related to the coupling coefficient by $k=\sqrt{1-{r}^{2}}$. It can be shown that the transfer function *t _{eff}* is given by the expression ${t}_{\mathit{eff}}=\frac{r-a\mathrm{exp}\left(i\delta \right)}{1-ra\mathrm{exp}\left(i\delta \right)}$, where

*a*≡(-

*αL*/2) is the round-trip amplitude transmission factor, and $\delta =\frac{\omega}{c}{n}_{\mathit{eff}}{L}_{c}$ is the round trip phase,

_{c}*L*is the round trip length, α is the power loss coefficient, and

_{c}*n*is the waveguide effective index. The effective phase shift, Δ

_{eff}*φ*, therefore, can be expressed $\Delta {\phi}_{\mathit{eff}}=-{\mathrm{tan}}^{-1}\left(\frac{a\mathrm{sin}\delta}{r-a\mathrm{cos}\delta}\right)+{\mathrm{tan}}^{-1}\left(\frac{ra\mathrm{sin}\delta}{1-ra\phantom{\rule{.2em}{0ex}}\mathrm{cos}\delta}\right)$. Note that for a given

_{eff}*r*,

*t*is a function of only

_{eff}*a*and

*δ*. If the ring is lossless (

*a*=1), this configuration is an all-pass filter (i.e.,

*T*=1 for all frequencies) with a pure resonant phase response [7]. Resonance occurs when

*δ*=2

*mπ*, where

*m*is an integer. As a filter,

*δ*contains the frequency dependence, and can be written in the form $\delta =2\pi \frac{f}{\mathit{FSR}}=2\pi \left(m+\frac{\Delta f}{\mathit{FSR}}\right)$, where Δ

*f*≡

*f*-

*f*is the frequency detuning,

_{o}*f*is the resonance frequency, and $\mathit{FSR}=\frac{c}{{n}_{\mathit{eff}}\phantom{\rule{.2em}{0ex}}{L}_{c}}$ is the free spectral range. As shown in Fig. 1, we find that the

_{o}*effective*phase shift is 2π across a resonance; varying the reflectivity

*r*only changes the slope of the phase. In contrast, for the Fabry-Perot filter configuration shown as inset B, the effective transmission phase shift across a resonance is π. In this case there are two outputs as the ring is coupled to two waveguides; the output in the backward direction is defined as

*T*and that in the forward direction is denoted

*R*.

Using the single rings as building blocks, we next consider two general forms of higher-order 2-D periodic ring structures that may be coupled with the MZI. In each form we will discuss first the special 1-D geometry and then the more general 2-D configuration. For simplicity we assume all the rings and the couplers to be identical and lossless. Because the structure is periodic, the transfer matrix method can be used to evaluate *t _{eff}*. This method has been applied to 1-D structures [8–10] but, so far, not to a 2-D structure. Details of the 2-D transfer matrix formalism will be presented elsewhere [11].

## 3. Side-Coupled Ring Enhanced Mach-Zehnder Interferometer (SC-REMZI)

Figure 2 shows the generalized 2D array SC-REMZI configuration, which consists of columns of rings side-coupled to the upper MZI arm. The rings are mutually coupled in the same column, but not between columns.

The mutually coupled ring structure is known as a coupled-resonator optical waveguide (CROW) [6], through which light can propagate when the rings are on resonance. Being an extension of the single-ring all-pass filter, the effective phase shift in each ring is 2π, and the total phase shift for *M* rings is *M*×2π. However, because of mutual coupling, the resonance frequency shows *M*-fold splitting, and the accumulated phase shift increases with *M* in a staircase manner. When coupled to the MZI, this ripple-like phase shift is converted into an intensity pattern with *M* sharp oscillations in the transfer function, as shown in Fig. 3(a). Note that the split peaks (in red) are sharper relative to the original peak of the one-ring case (blue curve), and the spacing between them is determined only by the coupling coefficient between the resonators.

In the case of a single row of *N* identical rings which are coupled to one arm of the MZI, but not mutually coupled to each other, we merely have a series of all-pass filters each accumulating a phase shift of 2*π* across a resonance, giving a total phase shift of *N*×2*π*. Hence, the MZI transfer function shows *N* oscillations, with the most rapid change occurring at the resonance, as shown in Fig. 3(b).

In the 2D case, each column can be reduced to an equivalent resonator with an effective transfer matrix. The 2-D array is then reduced to a 1-D cascade of these equivalent resonators. Therefore, the 2-D array will contain the combined features of the row and column resonators, as clearly illustrated in Fig. 3(c). The general trends with the number of ring resonators in the row and column directions can be summarized as follows: The total accumulated phase shift is Δ*φ _{eff}*=2

*π*(

*N*×

*M*), where

*N*×

*M*is the total number of ring resonators. Part of this total phase is from the

*M*column resonators and contributes to the

*M*resonances in the MZI transfer function. The other part is from the

*N*row resonators and causes the (

*N*-1) dips within each resonance. In total, there are

*N*×

*M*peaks spread over a bandwidth determined by

*r*, where regions of rapid oscillation are interspersed with regions of more gradual change.

## 4. Coupled Ring Enhanced Mach-Zehnder Interferometer (C-REMZI)

In this configuration, the 2-D ring array is inserted into one arm of the MZI, sandwiched between two parallel bus waveguides. Along the arm propagation direction the rings are mutually coupled, similar to the CROW in the previous case except that the CROW is now coupled to a waveguide on each end.

Let us first consider the case of a single column of *M* resonators, where *M* is an odd integer. When the resonators are on resonance, input light will propagate through the array and be directed back into the MZI. Otherwise, the incident light will be reflected by the array and is assumed lost at the end of the input bus waveguide. This case has been demonstrated in [12]. Figure 5(a) shows the sharp transmission bands when the resonators are on resonance. The *M* ripples in the passband are due to the mutual coupling between the *M* resonators. Similarly, the phase response shows a ripple pattern and the total phase shift is *Mπ*, since the effective phase shift is π per ring in this case. The MZI output shows corresponding oscillatory behavior within the resonance bands. Outside the resonance band, the phase shift is zero since *T*=0. Here the output at the bar port is a constant 0.25 which is the fraction of light that has passed through the two 3-dB couplers. The total power is conserved when both the bar and cross output powers and the reflected power are accounted for.

In the case of the 1-D side-coupled array with *N* resonators (refer to Fig. 5(b)), light cannot propagate through the array when the resonators are on resonance, but is instead reflected in the opposite direction and re-directed into the MZI via the other bus waveguide. Furthermore, because the double channels provide many paths for the waves to feedback to earlier resonators, the array is similar to a distributed feedback grating, giving rise to Bragg resonances when the Bragg condition, *λ _{j}*=2

*n*/

_{eff}L_{b}*j*(

*j*=1,2,3…), is satisfied [10]. The transfer function, therefore, shows two types of resonances in general, one that depends on the resonator cavity length (

*L*

_{c}) and another that depends on the resonator spacing (

*L*

_{b}). These two resonances, however, overlap if we set 2

*L*

_{b}=

*L*

_{c}. This is the case shown in Fig. 5(b) and (c). In this case, the reflection bands may be considered as the photonic bandgaps (PBG) of the periodic structure. For an array of

*N*rings, there are

*N*zero crossings (or

*N*-2 sidelobes) between two adjacent bands. These sidelobes are due to interference between the reflected waves from the resonators.

As shown in Fig. 5(b), the shape of the transmission band rapidly changes from Lorentzian to more box-like as *N* increases up to a point, after which the shape changes only slowly, because the first few rings are more effective than the subsequent ones. Correspondingly, the phase shift within the band becomes more *linear* compared to the single-ring case, but the magnitude does not scale up linearly with *N*. On the other hand, near the band edges the phase becomes more nonlinear with increasing *N*, giving rise to more group delay (“slow wave”) at the band edges [6,10]. This behavior is similar to the FBG filter operating in reflection mode, and is related to the nature of a type of filter called *minimum phase* filter [13]. At points outside the band where *T*=0 there is a π-phase discontinuity which has no physical relevance. The resulting MZI output is sinusoidal near resonance where the phase is linear and more oscillatory near the band edges where the phase is nonlinear.

The 2-D array can be analyzed column-wise first, then row-wise. Column-wise, each column can be reduced to an equivalent resonator represented by an effective transfer matrix. The 2-D array is then reduced to a 1-D row array of equivalent resonators, hence the phase response is similar to that discussed above. However, because embedded in each equivalent resonator is a CROW with its PBG structure, the sidelobes in the array transmission spectrum is significantly suppressed compared to the 1×*N* case [11]. In response, the ripples outside the resonance band in the MZI output are much more subdued.

In conclusion, the general trends with the number of resonators in the C-REMZI configuration can be summarized as follows: Increasing the number of resonators in a column has the effect of increasing the number of resonances (oscillations), whereas increasing the number of columns has the effect of smoothening the resonance curve to become more sinusoidal due to the higher phase linearity near the resonance band.

## 5. The effects of coupling and loss

It is convenient to study the impact of the ring resonator on the MZI by varying the coupling coefficient $\kappa =\sqrt{1-{r}^{2}}$, where *r* is the reflectivity coefficient at the ring coupling region. For example, when *r*=1, the ring resonator structure is completely decoupled from the MZI. On the other hand, if *r*=0, then the rings become a physical extension of the MZI arm, and the phase shift generated in the rings is proportional to the path length. Consider for example a 3x3 array: the light will propagate through all 9 rings sequentially in the SC-REMZI case, but will propagate only through 3 rings in the C-REMZI case, hence the effective phase shifts are linear and total 18π and 3π, respectively, across a resonance as shown in Fig. 6(a).

The corresponding situation when *r*=0.8 is shown in Fig. 6(b). It is clear that increasing the reflectivity *r* has a band-limiting effect in that the MZI response is compressed to a band with a bandwidth determined by *r*. The higher the reflectivity (the lower the coupling), the narrower will be the transmission band and the more nonlinear the phase shift, and hence the more resonant will be the device output. The number of resonances in the output depends on how many rings are effectively coupled to the MZI arm. In this regard the side-coupled configuration is more effective, whereas in the C-REMZI case some of the rings in the array are quite redundant.

However, one must also consider another important effect that increases with the number of effectively coupled rings, that is, the response time of the device, which is the time it takes for an optical pulse to propagate through the large number of rings. The latency in a ring is proportional to the resonator’s finesse which increases with increasing r. Hence, the more resonant the device output, the higher will be the latency through all the rings. Clearly if the latency exceeds the optical pulse width then there will be no interference effect at the MZI output. Hence, the long response time of a large array of micro-rings will limit the device bandwidth and the signal bit rate that can be passed through it.

The array size is not only limited by the response time consideration but also by the propagation loss in the micro-rings. The ring loss is defined using the round-trip attenuation factor *a* (for a lossless ring, *a*=1). The effects of loss on the two REMZI configurations are shown in Fig. 7. It can be seen that even a 5% round trip loss has a detrimental effect on device performance, especially in the SC-REMZI configuration where the light propagates through all the rings. Since the signal must pass through all the rings, all the rings must have precisely overlapping resonant frequencies, identical coupling coefficients, and very low loss. Therefore, fabrication requirement is stringent, and these REMZI designs are practical only if very low loss and uniform rings can be realized. Recently, very high order multi-ring filters with very high Q resonators have been realized using low-loss Hydex material [14], showing the technological possibility of realizing large and uniform resonator arrays.

## 6. Discussion

We have presented in this paper two generalized array configurations of ring-coupled MZI, namely SC-REMZI and C-REMZI, and discussed their characteristics in terms of the amplitude and phase response of the ring arrays as well as the transmission output of the MZIs. Because the rings are resonant devices, they greatly enhance the sensitivity of the MZI to frequency and effective index changes. In the side-coupled (SC) structure, the array is coupled to waveguide only on one side and hence the wave can only propagate in the forward direction. In the coupled-REMZI case, the array is coupled on both sides to waveguides so there is optical feedback between the resonators. Because of this fundamental difference the two types of array have distinct transfer functions and effective phase shifts, which can be tailored to phase-engineer a wide-range of MZI transmission functions.

From the application point of view, the side-coupling configuration is probably better as it can be designed to give any number of sharp resonances within a small band in the MZI output, which may be a desirable feature for some nonlinear switching and sensing applications [4,5]. The 2-D array coupled between two waveguides, on the other hand, has been shown to form a near-ideal bandpass filter characterized by a flat-top, square and ripple-free amplitude response and a largely linear phase response, if the array is sufficiently large (e.g., *M*=3 and *N*=10). This characteristic is attributed to the 2-D nature of the photonic bandgap exhibited by the 2-D periodic structure [11]. Despite the large number of rings, the phase response is not greatly enhanced and is approximately linear except near the bandedges. For this reason this 2-D ring structure does not significantly modify the MZI output. Of course, MZI is not the only way to utilize the phase response of these ring resonator structures, but it is probably the simplest and the most effective device. In this study, we have shown the amplitude and phase characteristics of various ring configurations and how they can be used to physically engineer the MZI transmission behavior. We have also discussed some of the practical issues involved, including the effects of loss, delay time, and fabrication requirement.

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