## Abstract

Small-radius microring resonators with large free spectral range (FSR) are of great interest for optical communication and optical interconnect applications. The resonator loss of a waveguide-coupled ring resonator, if the gap width between the microring and the bus waveguide is extremely small, can be significantly influenced by the coupling loss which corresponds to the microring operated in a strong coupling regime. This effect is particularly prominent for small radius microrings. We have studied the coupling loss with respect to the gap width on a waveguide-coupled microring both experimentally and theoretically, using two-dimensional (2D) finite difference time domain (FDTD) and effective index method (EIM).

The coupling loss was confirmed by measuring transmission spectra of Si microring filters fabricated on silicon-on-insulator (SOI) wafers. Our experimental data show that the ring loss increases rapidly as the coupling gap decreases to less than 200 nm. The measured results show that the ring loss of a silicon microring with a radius of 2.75 *μ*m is around 0.01382 dB/circumference as the gap width is greater than 325 nm, referred to as the intrinsic ring loss. However, for a smaller gap of 150 nm, the loss of the microring increases to 0.07084dB/circumference. The added ring loss is attributed to the coupling loss at small coupling gap for small radius ring.

© 2013 OSA

## 1. Introduction

SOI waveguides exhibit a large index contrast between the core (Si) and claddings (SiOx), yielding strong optical confinement of silicon waveguide modes that can adapt to a small bending radius ring resonator with negligible bending loss. Thus there are considerable interests in producing ultra-compact ring resonators for integrated photonic devices by using the SOI platform. Various functional devices based on SOI waveguides and microrings have been demonstrated, such as arrayed waveguide gratings (AWGs) [1], microring resonators (MRRs) [2], MachZehnder interferometers (MZIs) [3], optical filter/add-drop multiplexers [4], modulators [5], bio-sensors [6], and so on. Among these devices, sharp bended Si waveguide structures are ubiquitous. Therefore, accurate analysis and characterization of SOI waveguide bends are necessary and important for design and fabrication of photonic integrated devices [7–9]. In the case of microring resonators, in order to achieve proper coupling condition, the gap width between the ring resonator and signal bus(es) has to be well designed. Most analyses ignore the presence of coupling loss in the bus-ring coupling region and assume the loss of a microring comes from the bending loss, scattering loss and absorption loss of the microring waveguide itself (termed intrinsic ring loss).

As the gap width between two waveguides becomes very small, the mode overlap increases such that the assumption of weak coupling is no longer valid. Therefore, the conventional coupled mode equations based on two-mode coupling should be modified. Not only the guided modes but also the leaky modes could be excited. This is particularly likely to occur for the bend waveguides, e.g. microrings since the evanscent wave extends farther outside the waveguide. Although it has been reported the small coupling gap introduces extra ring loss [10], there is still little discussion on this issue. Previously, we showed that the coupling induced radiation loss actually dominates for a 2.75 micron radius SOI ring resonator, as the gap width between a ring and a circular bus waveguide is less than 200 nm [11]. In this paper, we present a comprehensive, both theoretical and experimental, study on the coupling loss of a waveguide-coupled microring resonator, as a function of the coupling gap.

Theoretical analysis of the coupling loss on small-radius ring resonators is not trivial. Near the strong coupling region between the ring and the bus waveguide, in addition to the primary bus and ring waveguide modes, many other radiation or leaky modes are also involved in mode coupling. Thus it is difficult to directly calculate the entire mode-to-mode coupling processes through matrix analysis. A quasi-three-dimensional (3D) numerical analysis in combination with time-domain coupled-mode theory is utilized in this paper to extract the power coupling ratio as well as the power loss per circumference for a ring, including both the intrinsic ring loss (bending loss, scattering loss and absorption loss) and coupling loss. We fabricated a set of waveguide-coupled silicon microring resonators of 2.75 *μ*m in radius but varying the coupling gap width from 150 nm to 400 nm and measured their transmittance spectra. By fitting the data with the ring resonator model, we confirmed that the narrow coupling gap indeed introduces extra loss resulting from the ineffectual coupling to the radiation or leaky modes, in particular for small radius micro-ring, there is a large increase of the ring propsagation loss– we referred that as the coupling induced radiation loss. This introduces a significant challenge to the design and fabrication of ultra-compact microring-based planar lightwave circuits that usually require a large ring-bus coupling coefficient.

## 2. Theory and analysis method

Finite-difference-time-domain (FDTD) method [12] is widely applied as one of the most accurate numerical methods for solving waveguide problems, since it is directly derived from Maxwell’s equations with very few approximations. A 3D-FDTD method usually requires a large amount of computer memories and is very time-consuming. In this paper we use a quasi-3D-FDTD method, a 2D-FDTD plus an effective index method (EIM), to analyze the transmittance spectrum of a waveguide-coupled microring.

The time-domain coupled-mode theory [13] is used to model the transmittance spectrum of a waveguide-coupled microring, as shown in Fig. 1. The ring resonator is treated as a lumped oscillator with energy amplitude a(t). This oscillator has a resonant frequency of *ω*_{0} and an amplitude decay time-constant of
$\frac{1}{\tau}$. The decay rate is related to the power leaving the ring, which includes the power coupled externally to the transmitted wave
$\frac{1}{{\tau}_{c}}$, and all kinds of resonator propagation losses
$\frac{1}{{\tau}_{0}}$. Thus,
$\frac{1}{\tau}=\frac{1}{{\tau}_{c}}+\frac{1}{{\tau}_{0}}$. The dynamics of ring energy amplitude can be expressed by

*μ*is the coupling coefficient between the waveguide mode and the resonator mode at

*ω*

_{0}, and

*S*is the field of the incoming wave from the bus waveguide. The transmittance of waveguide power can be obtained as

_{i}In Eq. (2), for a given transmittance spectrum, the only fitting parameters are *τ _{c}* and

*τ*

_{0}, which are correspond to the power coupling ratio between the ring and the waveguide and the power loss ratio per circumference of the ring, respectively. The relationships are given by

*κ*

^{2}is defined as the fraction of power coupled to the ring from the bus waveguide (power coupling ratio), and

*α*̄

^{2}is the power loss ratio of ring waveguide (resulting from the bending, absorption, scattering and others, e.g. coupling loss in this study) per circumference, and R is the effective bending radius of the ring waveguide,and

*v*is the group velocity of lightwave in the micro-resonator. The total power loss coefficient of a ring is calculated by

_{g}*α*=−10×log10 (1−

*α*̄

^{2}) dB/circumference called ring loss coefficient. An equivalent expression of the transmittance spectrum [8] in wavelength domain near the resonance wavelength can be written by

*λ*

^{2}/(2

*πRn*) is the free spectral range of the resonator, and

_{g}*n*is the group index of the wave propagating inside the ring.

_{g}## 3. Device design and fabrication

A silicon waveguide is featured with a cross section of 500 nm in width and 250 nm in height to obtain single mode condition for TE polarization [14]. The radius of microring is 2.75 *μ*m and the device structure is configured as an all-pass filter fabricated by standard nanofabrication techniques on a silicon-on-insulator wafer with a 3 *μ*m buried oxide (BOX) and a 250 nm device layer. To achieve phase matching condition, the bus waveguide is a straight waveguide and the waveguide width is generally narrower than the ring waveguide width [15]. However, exactly controlling a bus waveguide width is not easy to achieve by considering the applicable fabrication tolerance in our lab. To avoid the phase mismatching effect, the bus waveguide is configured as a half ring structure with a cross section identical to the microring in our devices.

The coupling gap width between the bus waveguide and microring varies from 150 nm to 400 nm, with an increase step of 25 nm. The smart-cut SOI wafer (*Soitec ^{TM}*) was used to fabricate these microring filters with a spec of thickness variation less than 20 nm for the top device layer. To achieve stringent dimensional control on the small gap width between the bus and microring waveguide, a high-resolution e-beam lithography tool (

*ELIONIX*-

*ELS*750) was used to define the structure.

The SOI wafer was coated with 380 nm negative *NEB*-*A*_{4} photo-resist (PR). Then the SOI wafer was etched by a *Lam TCP*9400 poly-silicon etcher system. During the main etching step, the reactant gas is a mixture of *Cl*_{2}-*HBr* with 7:25 flow-ratio. Finally, the low temperature oxide (TEOS-oxide) is covered. Figure 2 shows the calibration data for gap width fabrication. We can find the gap width was defined well from 125 nm to 400 nm. As the gap width is designed to be less than 125 nm, the actual fabricated gap width is much smaller than designed value. This is caused by the electron scattering inducing the proximity effect during e-beam lithography.

## 4. Experimental results and discussions

A pulsed excitation method is adapted to calculate the wavelength spectrum of a ring resonator through the 2D-FDTD plus EIM. Based on this spectrum, we can extract the power coupling ratio and the per circumference power loss ratio via curve fitting technique based on the model derived from the couple mode theory in time domain. The calculated transmission spectrum for a single ring resonator with radius of 2.75 *μ*m and waveguide width 0.5 *μ*m is shown in Fig. 3. By using Eq. (4), the fitted amplitude coupling ratio *κ* = 0.11786 and the ring loss coefficient *α* =0.015 dB/circumference were obtained. Figure 3 also shows a slight asymmetry on the simulated transmission spectrum. This could be due to multiple modes interacting with a single eigen-mode ring resonator for different relative phases. For example, slot mode and higher order mode brought from the neighbor bus waveguide and ring resonator. The phenomenon is called Fano resonance and there are many other applications based on the effect [16–18].

Transmittance spectra of fabricated microring resonators were measured with a tunable laser (*Tunics T*100*S*-*CL*-*WB*) as the light source. The light was coupled to the input polished facet of the Si bus waveguide through a lensed fiber, with a 2.5 *μ*m spot size and 14 *μ*m working distance. The output optical signal was also collected by an identical lensed fiber connected to a power meter. The wavelength scanning range was from 1500 nm to 1580 nm with a resolution of 1 pm for TE-like modes. A normalized transmittance spectrum of a ring resonator with radius of 2.75 *μ*m is shown in Fig. 4. The doted spectrum is from the experimental data and the red line is the fitting curve from CMT. According to the results, the resonance wavelength is 1514.539 nm, the full width at half maximum (FWHM) is 60.43 pm, and the quality factor is about 25,000. The ring loss coefficient is 0.02747 dB/circumference and the amplitude coupling ratio is 0.069.

Figure 5 plots the round-trip power loss ratio as a function of the gap width. The red curve is from the experimental data and the black one is from the FDTD simulation. Both of the results show a trend of the resonator loss increasing as the gap width decreases. A slightly large loss measured from the experimental data could be contributed to the scattering loss induced by surface roughness, which is not modelled in the simulation. The larger error bar of the experimental data below 200 nm is contributed to the fact that at small gap width there is a larger variation on the definition of gap width. In principle, the intrinsic loss of a microring is independent of the gap width. Thus the excess loss could be resulted from coupling-induced power leakage [19]. Figure 5 indicates that the coupling loss increases rapidly when the gap width is below 200 nm. In addition, the data shows that at the gap width wider than 325 nm, the microring loss tends saturated, corresponding to an intrinsic ring loss coefficient about 0.01382 dB/circumference. As the gap width becomes narrower, the overlap integral of mode fields of bus waveguide and ring resonator raises rapidly, leading to the coupling coefficient increasing, and the result is shown in Fig. 6.

As noted in earlier discussion (section II), the resonator loss coefficient can be expressed by an intrinsic loss term and a coupling loss term (that is coupling gap dependent), $\frac{1}{{\tau}_{0}}=\frac{1}{{\tau}_{\mathit{intr}}}+\frac{1}{{\tau}_{\mathit{coup}}}$. The coupling loss, extracted from the total loss minuses an intrinsic ring loss, and adjusted for the excess intrinsic loss observed in our measurement, is shown in Fig. 7. It emerges as the coupling gap width is less than 300 nm. Our 2D FDTD calculations exhibit the same trend as our measured results. For more accurate comparison, 3D FDTD should be used. Engineering the power coupling ratio and the power loss coefficient of waveguide-coupled microrings are essential for designing high performance ring-based planar lightwave circuits. However, for strongly-coupled, ultra-compact microrings, the small coupling gap introduces extra loss, which is a challenge in achieving large power coupling ratio but keep the ring loss small. It would limit some applications such as using the microring for broadband tunable filter.

## 5. Conculsion

In conclusion, we have utilized the 2D FDTD method and EIM to numerically investigate the intrinsic ring loss and the gap-dependent coupling loss for a waveguide-coupled microring resonator. For a ring with a radius of 2.75 *μ*m, the coupling-induced loss could dominate the total loss as the gap width is less than 200 nm. The experimental results also confirmed the gap-dependent coupling loss effect. The intrinsic ring power loss coefficient of the 2.75 *μ*m SOI microring is about 0.01382 dB/circumference. As the gap width is below 200 nm, the total ring loss is over 0.03455 dB/circumference and the corresponding coupling loss is over 0.02 dB/circumference.

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