1. Introduction

Micro-ring resonators have found wide applications in optical filtering/add-drop multiplexing [1–4], signal processing [5–9], bio-sensing [10–12], and so on. In most cases, high quality factor (Q) and deep transmission notches are desired. There have been many reports of high Q rings in silicon-on-insulator (SOI) structure [13–16]. However, there are still two main issues that are not fully addressed. Firstly, it is challenging to achieve deep notches (>30dB) in the transmission spectra for a single-waveguide-single-ring structure. To reach the “conventional critical” coupling, the ring resonator intrinsic quality factor (Q_i) and coupling quality factor (Q_e) with the waveguide must be equal and the resonant channel is then completely dropped. The ring/waveguide distance and their individual widths have to be carefully tuned for that purpose. Secondly, for small-radius rings, the free-spectral range is usually large, which limits the number of channels that can be adopted for operation. In this paper, we show that by introducing mutual mode coupling the above mentioned two issues can be improved. Depending on the strength of this mutual coupling, the transmission spectra can greatly enhance the notch depths and further split the resonance. The split resonances allow more wavelengths for signal processing and thus increase the system capacity [9].

We present two ways of generating the mutual mode coupling. In Section 2, we show the effects of nanometer-scaled gratings on the ring sidewalls. In Section 3, the directional mutual coupling takes place between two concentric rings. Both sections start with the time-domain coupled mode analysis [17–18], followed by experimental demonstration.

2. Grating-assisted single-ring counter-directional mutual mode coupling

2.1 Coupled mode analysis

The system schematic is shown in Fig. 1. To simplify the analysis, we assume that wave coming from the waveguide generates only one of the propagating modes in the ring, which in turn generates the other counter-propagating mode due to the grating that is present along the ring sidewalls. For example, the incident wave S₊₁ only generates the resonant mode a, and there is no direct coupling between S₊₁ and the mode b. The mode a and the mode b are related by the mutual coupling coefficient u. For power conservation, u is a real number. The change rate of the energy stored in the ring resonator for the counter-clockwise traveling mode a is thus given by

where ω_a is the resonant frequency of the counter-clockwise mode in the ring, τ is the photon life time, κ is the coupling coefficient between the ring and the waveguide. The reciprocal of photon life time is the decay rate 1/τ, which is related to the power coupling to the waveguide (1/τ_e) and power dissipating due to intrinsic losses (1/τ_i). Thus, 1/τ=1/τ_e+1/τ_i. The quality factor (Q) is decided by the photon life time, i.e., Q=ω₀τ/2. The intrinsic Q and the ring/waveguide coupling Q are denoted as Q_i and Q_e, respectively. The waveguide/ring coupling coefficient and the power decay rate are related by |κ|²=2/τ_e.

Similarly,

Assume the propagation constant in the waveguide is β. The outgoing waves are related by incoming waves by

and

The appendix in Ref. [18] has gives the strict derivation of the relations between incoming and outgoing waves. From Eqs. (3) and (4), the output waves are related to the incident waves by subtracting the part that is coupled to the resonator and multiplying a phase shift due to the distance between the two reference planes.

 

Fig. 1. Schematic of a ring resonator side coupled to a waveguide. Wave propagating from left to right (S ₊₁) in the waveguide only generates the counter-clockwise travelling mode a(t) in the ring. Clockwise travelling mode b(t) are coupled with a(t) by coefficient u due to the grating along the ring. The grating is indicated as the red dashed circle.

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We study the steady-state solution and assume e^jωt time dependence for the resonator and waveguide modes. With only input from port 1 and S ₊₂=0, the power of the outgoing wave can be derived from Eq. (1)–(4).

where $A=j\left(\omega -{\omega }_a\right)+\frac{1}{{\tau }_a}$ and $B=j\left(\omega -{\omega }_b\right)+\frac{1}{{\tau }_b}$ .

The transfer function can be written as

where C=AB+u ² and D=C-|κ ² _a|B.

From Eq. (6), the transfer function in general is not of Lorentzian shape. The Q value of the system cannot be estimated by the division of resonance frequency by 3dB bandwidth. The transmission spectrum has to be fitted to obtain the Q factors numerically.

Let’s consider the conventional case, when mutual coupling coefficient u=0.

When the intrinsic Q and coupling Q of mode a(t) become the same, i.e., when Q_ae=Q_ai, T becomes zero at resonant frequency and the channel is completely dropped. This is the conventional critical coupling.

Further on, let’s analyse the case when u≠0. To simplify the analysis, we assume the two travelling modes a(t) and b(t) are degenerate in the sense that their resonant frequencies and decay rates are the same, thus A=B. For complete channel drop, both the real part and imaginary part of D must equal zero.

Since u is a real number, Eq. (8) can only be satisfied when Q_ai>Q_ae. In Eq. (8), u_m is defined as the optimal mutual coupling coefficient.

 

Fig. 2. Illustration of the dependence of transmission spectra on the mutual coupling coefficient u. With the presence of mutual coupling, the notch becomes deeper. At optimal coupling um, complete channel drop can be reached. When u further increases, mode splitting occurs.

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It is convenient to define a mutual coupling Q factor Q_u, and a mutual coupling life time τ_u=1/|u|. Assume ω_a=ω_b=ω ₀, $Q_u=\frac{{\omega }_0{\tau }_u}{2}=\frac{{\omega }_0}{2\mid u\mid }.$ .

When u=u_m, $\frac{1}{Q_{\mathrm{um}}^2}=\frac{1}{Q_b}\left(\frac{1}{Q_{\mathrm{ae}}}-\frac{1}{Q_{\mathrm{ai}}}\right)$ .

For Q_a=Q_b, $\frac{1}{Q_{\mathrm{um}}^2}=\frac{1}{Q_{\mathrm{ae}}^2}-\frac{1}{Q_{\mathrm{ai}}^2}$ .

Note that Q_u is not related to any power loss in the system. It is merely a value that manifests the mutual coupling rate for the two traveling modes in the ring.

Figure 5–3 illustrates the transmission properties when Q_u→∞ (u=0), Q_u>Q_m (u²<u_m ²), Q_u≈Q_m (u²≈u_m ²) and Q_u<Q_m (u²>u_m ²) for the case when Q_ai>Q_ae and A=B.

In practice it is difficult to guarantee the conventional critical coupling for complete channel drop. In our experiments, we fix the ring/waveguide widths and search for the optimal ring/waveguide airgap to reach critical coupling. The variation step size is as small as 20 nm and still we can easily miss the critical coupling point. The mutual coupling, however, offers another freedom when designing high notch filters. From Fig. 2 and the associated equations, the presence of the mutual mode coupling improves the notch depth in all cases despite the fact that it is difficult to pin-point the exact value for Q_u. When the coupling further increases, the resonance notch will split into two. This can also be a useful feature as it fills the vacant frequency band and more wavelengths can be adopted for operation.

2.2 Device fabrication

We use a commercial single-crystalline SOI wafer (SOI Tech) with top silicon layer thickness 250nm and silica buffer thickness 3µm. The waveguide and ring pattern is first defined in the electron beam lithography (Raith 150, 25kV) with the negative resist ma-N 2405. Pseudocircular scan mode is chosen for the exposure of the rings. In this scan mode, the ring area is broken down into concentric polygons. The corner of these polygons will give rise to the “nodes” along the ring after resist developing. These “nodes” naturally form a grating on the sidewalls. By setting up the number of the vertices (corners) of the polygons, the E-beam scan step size, and the exposure dose, the period and amplitude of the grating can be tuned. For the optimal case, the grating only helps mutual mode coupling and does not significantly deteriorate the intrinsic Q of the ring. The scanning electron microscopy (SEM) photo of the grating on the ring sidewall is shown in Fig. 3(a). The width of the grating ridge is ~20nm. The period extends from ~50 nm to ~100 nm. The grating provides a weak perturbation along the ring. This perturbation creates a small reflection, which accumulates and generates the counter-travelling modes.

 

Fig. 3. The SEM photo of (a) the grating on the side-walls of a 20-µm-radius ring resonator and (b) gold grating couplers at the tapered waveguide end to assist vertical light injection/detection with a single mode fiber.

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Reactive ion plasma etching (ICP DRIE, STS) is then performed to transfer the pattern to the silicon layer. To couple light efficient from single mode fibre to silicon waveguide, gold gratings are added to both ends of the waveguides [19] during the single-resist-layer lift-off process. For period 590nm, filling factor 34% and thickness 25nm, the grating couples only TE light with an optimal fibre-to-fibre loss below 20 dB. The SEM photo of the gold grating coupler is shown in Fig. 3(b). The waveguide width starts with 10µm and gradually tapers down to 480nm. The ring cross-section is 480nm (wide) by 250nm (thick). The width of the air gap between the ring and waveguide is 100±10nm to ensure good coupling with the waveguide and thus Q_ae<Q_ai.

2.3 Experimental results

Figure 4 shows the SEM photo and measurement results of a 5-µm-radius ring, where obvious mode splitting occurs. The notches around 1550 nm are fitted using Eq. (6). The intrinsic Q value obtained is 6.0×10⁴ and the coupling Q is 2.0×10⁴. The mutual coupling Q of 2.27×10³ is much smaller than Q_um of 2.12×10⁴, indicating a strong mutual coupling and thus mode splitting occurs. The split notches are still deeper than the case without mutual coupling (12dB compared to 6dB).

 

Fig. 4. (a) SEM photo of a 5-µm-radius ring. (b) broad spectrum transmission. (c) Curve fitting using Eq. (6) for one of the notch groups. The intrinsic Q value obtained is 6×10⁴ and the coupling Q is 2×10⁴. The split notches are deeper than the case without mutual coupling (12dB compared to 6dB).

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Figure 5 shows the SEM photo and measurement results of a 10-µm-radius ring. The intrinsic Q value obtained is 1.5×10⁵ and the coupling Q is 1.2×10⁴. The Q_u value obtained from curve fitting is 1.218×10⁴, close to Q_um 1.204×10⁴, resulting in a notch depth of ~ 40 dB. Also note that the precision of the short-spectrum scan, as shown in Fig. 5(c), is much enhanced compared to the fast broad spectrum scan from 1530nm to 1580nm, where the notch bottom is missed due to insufficient scan points.

 

Fig. 5. (a) SEM photo of a 10-µm-radius ring. (b) broad spectrum transmission. (c) Curve fitting using Eq. (6) for one of the notch groups. The notch depth is much improved, 40dB compared to the case without mutual coupling, where the notch is barely visible.

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Fig. 6. (a) SEM photo of a 20-µm-radius ring. (b) broad spectrum transmission. (c) Curve fitting using Eq. (6) for one of the notch groups. The intrinsic Q value obtained is 2.6×10⁵ and the coupling Q is 1.9×10⁴.

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Figure 6 shows the SEM photo and measurement results of a 20-µm-radius ring. The intrinsic Q value goes up to 2.6×10⁵. The obtained Qu value 2.15×10⁴ is larger than Q_um 1.91×10⁴, and the notch depth is ~20dB.

3. Directional mutual mode coupling between two concentric rings

3.1 Coupled mode analysis

Since the deeper notch and mode splitting are the result of the secondary coupling between the ring resonator modes, it is possible to find another solution without generating the grating along the ring. The schematic is shown in Fig. 7. A second ring is placed, concentrically, inside the outer-ring. The waveguide/ring, ring/ring separation is set so that the waveguide mode only couples to one of the traveling mode in the outer-ring. The outer-ring mode then couples, along the same direction, with the inner-ring mode.

We redefine u as the mutual inter-ring coupling coefficient. From the same time-domain coupled mode analysis, Eqs. (1)–(4) can be re-written as

 

Fig. 7. Schematic of two concentric ring resonators side coupled to a waveguide. Wave propagating from left to right (S ₊₁) in the waveguide only generates the counter-clockwise travelling mode a(t) in the outer ring. Counter-clockwise travelling mode b(t) in the inner ring can then be generated by evanescent directional coupling from a(t).

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Again for steady-state solution we assume e^jωt time dependence for the resonator and waveguide modes. With only input from port 1, and S ₊₂=0. The solution of the transfer function is exactly the same as the previous case.

where $A=j\left(\omega -{\omega }_a\right)+\frac{1}{{\tau }_a}$ , $B=j\left(\omega -{\omega }_b\right)+\frac{1}{{\tau }_b}$ .

and C=AB+u ², D=C-|κ ² _a|B.

In the previous case, it is easy to assume the two counter-propagating modes share the same resonance frequency and decay rates because they exist in the same ring. For the concentric double ring structure, the two modes are intrinsically different. Nevertheless, we consider the situation when the ring radius is much larger than their separation (R_in≫d₁), and we can still assume A≈B to simplify the case. Then the dependence of the transfer function T on the mutual inter ring coupling coefficient u is similar to that illustrated in Fig. 2.

 

Fig. 8. (a) SEM photo of a single 40-µm-radius ring. (b) SEM photo of a dual 40-39-µm-radius ring structure. (c) broad spectrum transmission for both cases. The double ring structure exhibits much enhanced notch depth. (d)-(e) Curve fitting for two of the deep notches using Eq. (6).

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3.2 Experimental results

The magnitude of the mutual inter-ring coupling coefficient mainly depends on the perimeters of the rings and the separation width between them. Here we experimentally demonstrate the enhancement of notch depth by use of a second ring. The results and SEM photos are shown in Fig. 8. The outer ring has a radius of 40 µm. The inner ring radius is 39 µm. The waveguide/ring gap width is 150 nm and the ring/ring gap width is 480 nm. In Fig. 8 (d)–(e) the curves were fitted using Eq. (6) assuming A=B.

The scan-mode in electron beam lithography has been switched to real circular scan and the periodic nodes have disappeared. The coupling between the propagating and counter-propagating modes in the individual ring is thus eliminated. Therefore, the notches appear very shallow, around 1-3dB, in the transmission spectrum for the single ring structure, seen as the black solid curve in Fig. 8(c). However, by adopting a second ring and allowing inter-ring mutual coupling, the notch depths are much improved to 25-30dB, seen as the red dashed curve in Fig. 8(c) and Fig. 8(d)–(e). For the notch around 1549.6nm, the intrinsic Q and coupling Q are estimated to be 1.6×10⁵ and 1.3×10⁴, respectively.

Apart from improved notch depth, concentric double-ring structure holds another advantage for bio-sensing applications. The modal area has increased, which allows a wider region for detecting bio-molecules. It is possible to further increase the number of coupled rings and research work is still under way. This paper only studies the single-waveguide-single-ring structure as a drop-filter. For add-drop applications in telecommunications, a second waveguide must be placed. For the two-waveguide-single-ring structure, the values of Q_e, Q_i and Q_u, hence the structure parameters, have to be re-adjusted for the optimal channel transfer.

4. Conclusion

To summarize, we have analyzed the effects of mutual mode coupling in ring resonators by time-domain coupled mode theory. Resonance-splitting and improved notch depth can occur depending on the strength of this mutual coupling. Two ways of generating mutual mode coupling have been presented and demonstrated in SOI micro-ring structure. The grating on the ring sidewalls created from the pseudo-circular scan in electron beam lithography causes the mutual coupling between the two counter-propagating modes in the same ring. By careful tuning its period and amplitude, the grating only affects the mutual coupling and does not deteriorate the intrinsic Q factor of the ring itself. Split-wavelength separation of 0.68 nm, notch depth of 40 dB, and intrinsic Q factor of 2.6×10⁵ are demonstrated for the 5, 10, 20-µm-radius rings, respectively. Without the help of the sidewall gratings, mutual coupling can take place between concentric rings. Notch depth improvement larger than 25dB has been demonstrated on the 40-39-µm-radius double-ring structure. The increased modal area and improved notch depths are promising advantages for bio-sensing applications. We hope our study on the mutual mode coupling will shed some insights in the research and applications of the micro-ring resonators.