## Abstract

A kind of Mach-Zehnder optical switch with a dual-bus coupled ring resonator as a two-beam interferometer is proposed and investigated. The analysis based on the transfer matrix method shows that a sharp asymmetric Fano line shape can be generated in the transmission spectra of such a configuration, which can be used to significantly reduce the phase change required for switching. Meanwhile, it can also be found that complete extinctions can be achieved in both switching states if the structural parameters are carefully chosen and the phase bias is properly set. Through tuning the phase difference between the arms of the Mach-Zehnder interferometer, complete extinction can be easily kept within a large range of the ring-bus coupling ratios in the OFF state. By properly modulating the phase change in the ring waveguide, the shift of the resonant frequency and the asymmetry of the transmission spectra can be controlled to finally enable optical switching with a high extinction ratio, even complete extinction, in the ON state. The switching functionality is verified by the finite-difference time-domain simulation.

© 2009 Optical Society of America

## 1. Introduction

Ring resonators are versatile elements widely used for various applications, including filters [1], modulators [2], switches [3, 4], and sensors [5]. A ring-resonator-based all-pass filter has proved capable of providing phase correction and equalization [6]. A ring resonator coupled to one of the two arms of a Mach-Zehnder interferometer (MZI) has been used to improve the linearity of modulation [7, 8]. Moreover, an interleaver based on the cross-ring resonator [9], and a group of interferometric optical switches with the double-ring coupler have been proposed for optical communication [10].

The transmission spectra of the ring cavities usually exhibit symmetric line shapes. However, it has been proposed that if the sharp asymmetric Fano-resonance line shapes can be generated in the transmission spectra, the phase change required for modulating or switching the light can be dramatically reduced [11]. The recent studies have shown that a ring enhanced MZI (REMZI) can produce the Fano-resonance lines in the transmission spectra [12–14]. In this paper, we report that the MZI with a dual-bus-coupled cross-ring resonator as a two-beam interferometer is able to generate the Fano-resonance line shapes in the spectra. The ring resonator is coupled to both arms of the MZI. Based on this configuration, a 1×2 optical switch is proposed. The OFF/ON switching functionality is realized by introducing a phase shift to the ring waveguide. With the employment of the transfer matrix approach, our analysis shows that the complete extinction can be achieved in the OFF state through adjusting the phase difference between the MZI arms. Meanwhile, the complete extinction can also be realized in the ON state by utilizing the influence of the phase change in the ring waveguide on the asymmetry of the output spectra.

In section 2, the schematic diagram of the optical switch, and the expressions of the output electric fields are illustrated. In section 3, the various line shapes of the transmission spectra under different phase biases are discussed. In section 4, the variations of the transmission spectra resulted from introducing a phase shift to the ring waveguide are examined. As a result, the proper pairs of the ring-bus coupling ratios for complete switching are obtained. In section 5, the switching functionality is verified by the finite-difference time-domain (FDTD) simulation.

## 2. The structure, and the expressions of the output electric fields

The schematic diagram of the MZI with a cross-ring resonator coupled to the both arms is shown in Fig. 1(a). In the configuration, the input beam will be first divided into two by a 1×2 splitter. Then the two split beams will be simultaneously coupled into the ring resonator through the two coupling zones. The cross-ring structure termed a twist ring [15], as shown in Fig. 1(a), is chosen to realize two-beam interference in the ring resonator. In addition, a phase shifter is set on the ring waveguide for switching the light. A phase bias controller is set on one of the interference arms of the MZI to provide with a static phase difference between the arms.

Since the high-performance cross grid can be always obtained through appropriate design [16], we assume that the cross grid of the cross ring used here is an ideal one with no crosstalk, no loss, and no reflection. Thus the cross-ring resonator framed by the dash line in Fig. 1(a) can be equivalently represented by the dual-bus-coupled ring resonator shown in Fig. 1(b). The dual-bus-coupled ring resonator that is often discussed under a single input condition with the electric fields *E*
_{i1}=0 or *E*
_{i2}=0, has been widely used as an add-drop filter [17], but it will be employed as a two-beam interferometer in this paper and named as the ring-resonator two-beam interferometer (RRTBI). The transfer matrix approach is a convenient technique for analyzing the ring resonators and shows excellent agreement with experimental results. This approach is used to obtain the expressions of the output electric fields in this paper. *E*
_{i1} and *E*
_{i2} are the electric fields of the input bus waveguides, while *E*
_{o1} and *E*
_{o2} represent that of the output bus waveguides. Since the bandwidth of the resonance is far smaller than the resonant frequency, the power coupling coefficients *κ*
_{1} and *κ*
_{2} can be considered as constants, where the superscripts represent the coupling regions. Assuming that the evanescent coupling is lossless, *E*
_{o1} and *E*
_{o2} can be determined through the transfer matrix approach and expressed as:

$${E}_{o2}=\frac{-{s}_{1}{s}_{2}{\gamma}^{m}\mathrm{exp}\left(-j{\phi}_{1}\right)}{1-{c}_{1}{c}_{2}\gamma \mathrm{exp}\left(-\mathrm{j\phi}\right)}{E}_{i1}+\frac{{c}_{2}-{c}_{1}\gamma \mathrm{exp}\left(-\mathrm{j\phi}\right)}{1-{c}_{1}{c}_{2}\gamma \mathrm{exp}\left(-\mathrm{j\phi}\right)}{E}_{i2},$$

where *j* = (-1)^{1/2}. *s _{i}*=

*κ*

_{i}^{1/2}is the ring-bus amplitude coupling coefficient, and

*c*= (1-

_{i}*κ*)

_{i}^{1/2}is the amplitude transmission coefficient, where

*i*= 1, 2.

*γ*= exp(-

*αL*/ 2) is the round-trip amplitude attenuation factor for a ring resonator with length

_{u}*L*and linear loss coefficient

_{u}*α*.

*φ*=

*βL*is the round-trip phase where

_{u}*β*is the propagation constant in the ring waveguide.

*φ*is equal to the sum of the phases

*φ*

_{1}in the upper semicircular waveguide with length

*L*

_{1}and the phase

*φ*

_{2}in the lower semicircle with length

*L*

_{2}, as shown in Fig. 1(b).

*m*is the quotient of

*L*

_{1}and

*L*, and

_{u}*n*is the quotient of

*L*

_{2}and

*L*. We have

_{u}*m*+

*n*= 1.

## 3. Fano-resonance transmission spectra in the OFF state

If the input light is split into the two beams with the same power, and the loss induced by the phase bias controller is neglected, the magnitude of *E*
_{i1} and *E*
_{i2} are equal. Assuming *m*=*n*=0.5, this device configuration is expected to exhibit a periodic spectral response. The period is 4*π* according to Eq. (1). The transmission spectra *I*
_{o1} = |*E*
_{o1}|^{2} and *I*
_{o2} = |*E*
_{o2}|^{2} are shown in Fig. 2(a), when *c*
_{1} = 0.9, *c*
_{2} varies from 0.8 to 0.9 and *γ* = 0.99. As it is known, the cavity is on resonance when *φ* = 2*qπ* (*q* is an integer). It has been noted that the two-beam interference in the ring waveguide affects the resonant intensity. If the phase difference *δ* = arg(*E*
_{i1} / *E*
_{i2}) is equal to zero and *q* is even, the RRTBI can produce constructive interference and support resonance in the cavity (the even mode with respect to the mirror plane in Fig. 1(b)). When *δ*=0 and *q* is odd, the RRTBI can produce destructive interference and weaken resonance (the odd mode), so that the energy stored in the cavity drops. As a result, the power dissipation decreases.

The Fano-resonance line shapes can be obtained through tuning *δ*, as shown in Fig. 2(c) and (d). When *δ* = 0 or *δ* = *π*, the output profiles exhibit the symmetrical resonance dips, but the values of *φ* for supporting the even and odd resonant modes are exchanged, as shown in Fig. 2(b). To illustrate the relationships in the transmission spectra around the two types of the resonant points, we define Δ*φ* = −2*πn _{eff}L_{u}*Δ

*λ*/

*λ*

_{0}

^{2}to denote the deviation of the phase, where Δ

*λ*is the wavelength detuning,

*λ*

_{0}is the resonant wavelength and

*n*is the effective refractive index of the ring waveguide. We define

_{eff}*I*and

_{oo}*I*to represent the transmission spectra in the range of 2

_{oe}*π*around the odd and even mode resonant points respectively. Considering

*I*and

_{oo}*I*are the functions of Δ

_{oe}*φ*and

*δ*, according to Eq. (1),

*I*(Δ

_{oo}*φ*,

*δ*) =

*I*(Δ

_{oe}*φ*,

*δ*+

*pπ*) is obtained, where

*p*is odd. Thus we just need to pay our attentions to the transmission spectra in the range −

*π*<Δ

*φ*<

*π*. The terms

*φ*

_{1}and

*φ*

_{2}in Eq. (1) can be replaced by 0.5Δ

*φ*.

To find the condition to obtain a complete extinction, *I*
_{o1} = 0 is assumed. Substituting *E*
_{o1}=0 in Eq. (1), *E*
_{i2} can be expressed as a function of *E*
_{i1}:

Since |*E*
_{i1}| = |*E*
_{i2}|, according to Eq. (2), the value of Δ*φ* is determined by

For a given pair of the ring-bus coupling ratios, through tuning *δ*, there are two values of Δ*φ* symmetrical about the resonant frequency which satisfy Eq. (3). The positive value of Δ*φ* is shown in Fig. 3(a). In the lossless case (*γ* = 1), the value ranges of *c*
_{1} and *c*
_{2} derived from Eq.(3) are surrounded by the following curves, as shown in Fig. 3(a):

The output responses of the points labeled A–F in Fig. 3(a) are shown in Fig. 3(b). As *c*
_{1} and *c*
_{2} are set closer to the curves described by Eq. (4), Δ*φ* drops to 0. The spectral profile finally turns to a symmetric resonance dip given by line F in Fig. 3(b). As *c*
_{1} and *c*
_{2} are chosen towards the curve given by Eq. (5), Δ*φ* approaches to *π*. While one of *c*
_{1} and *c*
_{2} wanes to zero, the other drops to 2^{-1/2}, which means the corresponding coupling zone turns to a 3-dB coupler. In this case, the device is degraded to a conventional simple MZI.

A sharp asymmetric line shape can be used to greatly reduce the required phase change. Thus, the control of the asymmetry of the Fano spectra is significant for the switching operation. Considering the lossless case (*γ* = 1), the slopes of *I*
_{o1} and *I*
_{o2} at the resonant frequency (Δ*φ*=0), can be derived from Eq. (1), respectively:

Eq. (6) reveals the effect of the tuning of *δ* on the asymmetry. *δ* = 0.5*π* maximizes the gradients for a given pair of *c*
_{1} and *c*
_{2}. Moreover, for more oblique resonance spectra, the values of *c*
_{1} and *c*
_{2} have to be chosen to get closer to each other, and tend towards unit.

As it is known, the loss in the ring resonator is unavoidable, such as the bend loss, the scattering loss, and the material loss. Assuming *γ* = 0.85, the value ranges of *c*
_{1} and *c*
_{2} are shown in Fig. 4. Similarly, the complete extinction can be realized by tuning the values of Δ*φ*, *c*
_{1} and *c*
_{2}.

Based on the above analysis, it can be concluded that the OFF state with *I*
_{o1}=0 can be easily realized through biasing the phase difference *δ*. This characteristic enables some high-extinction applications. Optical switch is one of such applications. In the next section, it will be shown that the ON state with *I*
_{o2}=0 can be achieved by introducing a proper phase shift to the ring waveguide.

## 4. Fano-resonance transmission spectra in the ON state

Before introducing a phase change *φ _{d}* to the ring waveguide, let’s tune the phase bias to have

*I*

_{o1}=0. For one thing, it is easy to understand that the phase shift

*φ*gives rise to a variation of the resonant frequency and the transmission spectra shift as shown in Fig. 5. To represent the deviation, we define Δ

_{d}*θ*= −2

*πn*Δ

_{eff}L_{u}*λ*/

*λ*

^{2}

_{0off}, where

*λ*

_{0off}is the resonant wavelength in the OFF state. For another thing, the changes in the maximums, the minimums, even the shapes, happen to the spectra. This is because the phase change in the ring waveguide alters the phase difference between the two interference paths in the RRTBI. For comparison, the phase change in the ring waveguide in a conventional REMZI just produces a shift of the transmission spectrum without the change in asymmetry. If these two functionalities, varying the asymmetry of the Fano-resonance spectra and shifting the resonant frequency, of the phase change in the ring waveguide can be matched, a complete switching can be achieved.

To obtain *I*
_{o2} = 0 in the ON state, we assume that the phase shift is applied to the lower semi-circle waveguide. Applying *E*
_{o2} = 0 to Eq. (1), *E*
_{i1} can be expressed as:

With Eq. (7) and Eq. (2), we can obtain the following equation to determine the required phase change *φ _{d}*:

Combining Eq. (8) with Eq. (3), for a given value of *γ*, the value ranges of *c*
_{1} and *c*
_{2} turn to a pair of curves. Figure 6 shows the relationships between the structural parameters and the phase change *φ _{d}*. Based on the analysis in section 3, since the values of

*c*

_{1}and

*c*

_{2}increase and get closer to each other, the sharpness of the asymmetric Fano-resonance spectra increases. As a result,

*φ*and Δ

_{d}*φ*decrease. Accordingly, the energy stored in the cavity increases so that the output power decreases as shown in Fig. 7. Certainly, there is a tradeoff between the insertion loss and the required phase change.

The modulation depth is another important performance parameter of an optical switch. Because the transmitted power from the dark output port is zero, the modulation depth depends on the bright one (in our case, Output 2 in the OFF state, Output 1 in the ON state). A larger modulation depth can be obtained at the expense of a larger phase change, as shown in Fig. 7. If *φ _{d}*>0.1

*π*, more than 93% of the modulation depth can be achieved. The small difference between the output powers

*I*

_{o1}and

*I*

_{o2}shown in Fig. 7, is because of the unequal ring-bus coupling ratios. The imbalance between the two coupling ratios results in different requirements of the energy stored in the cavity for the complete extinctions.

## 5. FDTD simulation

To verify the switch functionality, 2-D FDTD simulation is used. If the cross grid of the cross-ring resonator is an ideal one with no crosstalk, no loss, and no reflection, the transmission spectra of the two output ports of the cross-ring resonator in Fig. 1(a) are the same as the transmission spectra of the conventional ring resonator in Fig. 8. A pair of coherent beams with the same electric field magnitude is launched to the conventional ring resonator to imitate the 1×2 beam splitter. The phase shifter is set on the lower semicircle, as shown in Fig.8. As an example, a change Δ*n* in refractive index is introduced through the phase shifter. The parameters set for the simulation are described below. The radius of the ring resonator is 2μm and the waveguide width is 0.2μm. The refractive indices of the core and cladding are 3.477 and 1.444 respectively. The imaginary part of the refractive index of the ring waveguide is 2.025×10^{-4}, which leads to *γ* =0.99. The left and right gaps between the ring and the bus waveguides are 0.16μm and 0.147μm, which are used to obtain *c*
_{1} = 0.896 and *c*
_{2} = 0.869. The length of the phase shifter is a quarter of the ring waveguide. Figure 8(a) shows *I*
_{o1} becomes zero after adjusting the phase difference between *E*
_{i1} and *E _{i2}*, which leads to

*δ*=0.41

*π*, when

*λ*=1.591μm. Tuned with Δ

*n*= 0.052 for the modulation length of about 3μm, the light is routed to the other output port as shown in Fig. 8(b). If the radius is increased to obtain a phase shifter with the length of more than 170μm, the required change in refractive index would be reduced to less than 0.001.

## 6. Conclusions

In this paper, an improved Mach-Zehnder optical switch with high extinction ratios is proposed. A cross-ring resonator is utilized as a two-beam interferometer to generate the Fano-resonance line shapes in the transmission spectra. The analysis shows that complete extinction can be achieved through biasing the phase difference between the MZI arms in the OFF state. This characteristic gives rise to the potential in some high-extinction applications. To realize switching, a phase change is introduced to the ring waveguide. The sharp asymmetric response profile results in a large decrease in the required phase change. We present the two functionalities of the phase change in the ring waveguide: altering the asymmetry of the transmission spectra and shifting the resonant frequency. These two functionalities provide with an optional way to control the Fano line shapes. If the structural parameters are carefully chosen, the complete extinction can also be obtained in the ON state.

## Acknowledgments

This work was supported by the Major State Basic Research Development Program under Grants 2007CB307003, the Natural Science Foundation of China under Grants No. 60676028, and the Science & Technology Program of Zhejiang Province under Grant 2007C21022.

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