## Abstract

We propose a new waveguide resonator device with a mirror cavity and a multimode interference (MMI) coupler. We present simulation results for the silicon wire MMI coupler with suppressed reflections and its use as a coupling element in the resonator cavity, built on the silicon-on-insulator waveguide platform. Tapering structures used in the reflection suppression were optimized, and the wavelength dependency of a conventional MMI was compared to that of the MMI with reflection suppression. Equations relating the power transfer of the two-mirror MMI-coupled resonator and quality factor were derived. The device was also studied using finite difference time domain simulation by both pulse and continuous wave excitation. The resonator does not require bend waveguides, it has the advantages of having no bend loss and a compact layout. The resonator device has a very small footprint of 3 μm × 30 μm, and a quality factor of 516.

©2007 Optical Society of America

## 1. Introduction

Multimode interference (MMI) devices are widely used in integrated optics due to their broadband operation, polarization independent splitting ratio, and improved fabrication tolerances. MMI couplers and splitters on the silicon-on-insulator (SOI) platform have been studied and fabricated [1–3]. Ring resonators have been realized in silicon waveguides [4], including racetrack designs [5], and polarization insensitive structures [6]. The use of an MMI as the coupling device in a ring resonator on SOI was reported in [7] and a polarization insensitive MMI resonator in [8, 9]. MMI resonators have the added advantage of not requiring extended racetrack configurations or curved input/output channels. A significant problem in high index contrast waveguides, such as silicon wires, is the impendence mismatch of the access waveguides with the MMI slab region, resulting in reflections at the end walls of the MMI. Reflection suppression for an MMI fabricated on InP/InGaAsP was reported in [10], wherein the MMI had tilted end walls, and a tapered feature at the center between the input and output channels. Mirror cavity resonators have also been fabricated with photonic crystal waveguides [11].

In this paper we propose an original mirror cavity resonator with an MMI as the coupling element that has been designed for thin SOI waveguides. We present the Eqs. relating loss, splitting ratio, and mirror reflectivity to power transfer and quality factor of the device. Since our device does not require bend waveguides, it has the added advantage of having no bend loss and a compact layout. To reduce computational load, mirrors are modeled as perfect conductors, rather than a Bragg grating which can be used in the fabricated device. Optiwave’s OptiFDTD software is used to simulate the device by the finite difference time domain (FDTD) method. FDTD results are compared to analytical solutions. The FDTD simulation includes optimizing taper features for reflection suppression, namely the influence of the taper angle. Also, the wavelength dependency of the reflection corrected MMI is compared to a conventional MMI. The resonator spectral response is studied, including the quality factor and the free spectral range.

## 2. Theoretical analysis

The two-mirror MMI resonator can be analyzed as a multiple beam interference device [12]. We use the following notation for the field amplitudes (Fig. 1): *a _{1}* the input field,

*a*the transmitted output field,

_{2}*a*the reflected output field and

_{3}*a*the field in the resonator cavity access waveguide. The MMI coupling coefficients are

_{4}*t*and

*k*with

*γ*

^{2}=

*t*

^{2}+

*κ*

^{2}, mirror amplitude reflectivity is

*r*, and loss factor due to one round trip in the access waveguides of the cavity

*α*.

The phase change of the field corresponding to one cavity round trip is *φ*. Then the amplitude for field *a _{2}* is:

Equation (1) is a geometric series, which can be expressed as:

Similarly, the field amplitudes *a _{3}* and

*a*can be obtained:

_{4}Equations (2)–(4) are analogous to the ring resonator Eqs. derived in [12], with the inner circulation factor *α* = *αt*′*r*
^{2} for the two-mirror resonator. At resonance, *φ* = 2*mπ* (*m* is an integer), *a _{2}* → 0 due to the destructive interference between the transmitted field through the coupler and the cross coupled field. From Eq. (2) at resonance, and using the small angle approximation, the quality factor

*Q*can be expressed as:

Equation (5) incorporates the splitting ratio *t _{s}* and the power transfer factor

*γ*

^{2}, where

*t*′

^{2}=

*γ*

^{2}

*t*

^{2}

_{s}.

## 3. Method

#### 3.1. Simulation parameters

The 2D-FDTD simulation was based on a layout size of 15.7 μm × 4 μm with material refractive indices *n*
_{Si} = 3.476 and *n*
_{SiO2} = 1.44. Simulations were performed with the electric field in the plane of the layout shown in Fig. 2 (TM). The input and output channels were photonic wires 0.5 μm wide. The mesh size used in the simulations was 0.02 μm × 0.02 μm, while the time step was set according to the Courant Limit of $\Delta t\le \frac{1}{\left(c\sqrt{\frac{1}{{\left(\Delta y\right)}^{2}}+\frac{1}{{\left(\Delta z\right)}^{2}}}\right)}.$. The MMI width was 3 μm, and imaging plane position was found (without tapers) through simulations at 12.7 μm from the input plane for a 50:50 splitting ratio.

#### 3.2. Taper angle optimization

The input field *a _{1}* for the MMI coupler taper angle optimization was a continuous wave (CW) fundamental mode. Simulations were executed until the input wave propagated 20 MMI lengths. The power at the output channels was calculated as an overlap integral with the fundamental mode. Wavelength dependent loss was evaluated in the wavelength range

*λ*= 1.5 μm - 1.6 μm with a step size of 0.02 μm for both a conventional MMI and a reflection suppressed MMI. The influence of the taper angle on reflection suppression was evaluated for an angular range

*α*= 30° - 80° with a step size of 5°.

#### 3.3. Mirror resonator

Using the optimized MMI (*α* = 30°), a resonator device was designed with a cavity formed by two 1 μm wide mirrors terminating the upper access waveguides. Figure 2 shows the geometry of the mirror cavity MMI-coupled resonator. A femtosecond pulse with a half width of 1.4 × 10^{-14} s was injected from the input channel *a _{1}*. The time domain field at the output channels

*a*and

_{2}*a*was recorded while the pulse propagated through the 20 cavity lengths (10 round trips). The spectral response of the resonator was found by the Fourier transform of the time domain field. Full CW FDTD simulation was also performed for different wavelengths in the range of

_{3}*λ*= 1.5 μm - 1.6 μm.

## 4. Results

#### 4.1. MMI reflection suppression

We calculated the MMI loss without mirrors as a function of the taper angle *α* for different wavelengths in the range of *λ* = 1.5 μm - 1.6 μm (Fig. 3). It is observed that within this spectral range, for *α* = 30°, the MMI loss is reduced at all wavelengths, and the wavelength dependent ripple is also significantly reduced from ~1 dB down to ~0.1 dB. Due to low nearly wavelength independent loss, the optimal taper angle was set at 30° with a total MMI loss of 0.4 dB, at *λ* = 1.56 μm. Since the light diffracts from the photonic wire channel of a width 0.5 μm at a half angle of 21.8° (measured at *l*/*e ^{2}* far field irradiance asymptote), the taper is non-adiabatic and does not influence the imaging properties of the MMI.

Figure 4 shows the *H _{y}* field amplitude recorded as a function of time at the upper access waveguide. In the MMI without tapers (Fig. 3), the field propagates along one cavity length in 140 fs. Fluctuations in the field envelop are due to reflections at the MMI input and output boundaries. The fluctuations are separated in time by of approximately 280 fs, corresponding to the arrival of the reflected field after traversing two cavity MMI lengths. For the MMI with tapers at

*α*= 30° (Fig. 3), the field envelope fluctuations are substantially reduced. We found this type of time domain analysis is particularly useful to visualize and quantify the efficiency of reflection suppression when optimizing the taper angle.

#### 4.2 Resonator simulation

The spectral response of the resonator was determined by the Fourier transformation of the calculated time response. Results from FDTD simulations were then compared to Eq. (2) at resonance and the quality factor was compared to Eq. (5). The parameters used in Eqs. (2)–(5) were obtained from the FDTD simulations: *α*= 1 (lossless access waveguides), *γ*
^{2} = 0.912, and *r* = 0.994 (reflectivity of the perfect conductor mirror used). With the cavity length of 31.4 μm, the simulated free spectral range (FSR) was 22 nm. The quality factor *Q* = *λ*/Δ*λ*
_{FWHM} (where *λ* is the resonance wavelength and Δ*λ*
_{FWHM} is the resonance width measured at the full-width-at-half-maximum) was calculated as 310 (Δ*λ*
_{FWHM} = 5 nm), whereas *Q* = 300 was predicted by Eq. (5). Figure 5 shows the power in the fundamental mode in the output channels *P*
_{2} = *a*
^{2}
_{2} and *P*
_{3} = *a*
^{2}
_{3} as a function of wavelength.

Figure 6 shows the spectral response of the resonator with a cavity length of 59.4 μm, which was designed by extending the access waveguides from 1.5 μm to 8.5 μm, while maintaining the same MMI width. The FSR is 11 nm and Δ*λ*
_{FWHM} = 3 nm, yielding *Q* = 516, whereas *Q* = 518 was obtained from Eq. (5). Mirror width was also increased to 2 μm to test if 1 μm width was sufficient to fully reflect the evanescent field tail; since the 1 μm mirror reflectivity was 0.994, no significant improvement was expected. Identical values for *Q* were measured for both the 1 μm and 2 μm wide mirrors, indicating that 1μm mirror width is sufficient. Bragg grating structures were also simulated (2D FDTD) to determine their reflectivity and spectral selectivity. For a 10 μm long grating in a 0.5 μm wide SOI waveguide, FDTD simulations predict *r* > 0.96 for *λ* = 1.5 μm - 1.6 μm. Such gratings can be advantageously used in our resonator design, yielding a comparatively broad spectral range.

Our resonator device has similar properties to the device described in reference [13]. Since the FWHM is directly related to the resonator length [12], and the quality factor is directly related to the FWHM, the quality factor can be increased by increasing the resonator length. Also, MMI splitting ratio can be tailored by adjusting the MMI width [14], commonly referred to as MMI tapering. Splitting ratios of 85:15 have been reported in [15]. This could be used to increase the coupling coefficient to the resonator cavity, thereby decreasing the FWHM, hence further increasing the quality factor.

## 5. Conclusion

We have proposed and demonstrated by FDTD simulations a novel two-mirror resonator device with a reflection suppressed MMI coupler in the resonator cavity. We derived an analytical model for this device, which agree with results obtained through FDTD simulation. The reflection suppression was achieved by tapering the MMI access waveguides at α = 30°. The device has a reduced loss of 0.42 dB and a negligible wavelength dependent ripple of 0.1 dB. Also, the resonator has a very small footprint of 3 μm × 30 μm, which would be difficult to achieve with a conventional racetrack design. Only straight waveguides are used in the design, which minimizes device size and eliminates the waveguide bend loss.

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