1. Introduction

Silicon optical microring resonator is one of the basic building blocks for future integrated photonic circuits due to its compact size and versatile functionalities. It has attracted considerable research interests in recent years, and various kinds of functional devices have been implemented based on microring resonators including switches [1, 2], modulators [3–5], filters [6–12], optical delay lines [13–19], WDM multiplexers [20], and phase coherent devices [21]. For a single microring resonator, its Lorentzian resonance line-shape cannot satisfy the ideal “box-like” spectrum requirement for an optical filter [6]. Its large group delay dispersion (GDD) around resonance also causes distortion to the transmitted optical signal [14]. Hence, coupled resonators in series- or parallel-coupling configurations are always used to improve the filtering and delay performances [10–12, 19, 22, 23].

Due to fabrication errors, however, it is hard to accurately align the resonance frequencies for all resonators and their coupling coefficients can also easily deviate from designed values. These uncertainties will eventually deteriorate the device performances. In addition, coupling can also induce resonance frequency shifts (CIFS) [24, 25], which can further complicate the device design in compensation for this effect. Therefore, there is a great need for fewer resonators to accomplish the same function. One way to reduce the number of resonators is to couple different resonance modes in one physical resonator to get the desired high-order resonance features. A single microring resonator with clockwise (CW) and counter-clockwise (CCW) degenerate resonance modes is quite suitable for this purpose, as its resonances are “auto-aligned” with each other and any fluctuation in the resonator physical parameters can affect these modes equivalently. The coupling between CW and CCW modes can be enabled by small perturbations inside the resonator, which can be a notch or just fabrication-induced sidewall roughness [15, 26–28]. The perturbations cause backscattering, which leads to the resonance splitting and broadening. However, these perturbations are usually very small in the nanometer scale and almost impossible to be fabricated in a controlled way, resulting in reliability issues. Moreover, the coupling strength, which determines the resonance splitting level, is fixed and difficult to be changed once the device is fabricated, further limiting its application.

Based on the counter-propagation mode coupling, in this paper we propose a novel reconfigurable optical resonance structure to perform as a second-order resonance system. Compared to the microring resonator with backward coupling, it overcomes the shortcoming of relying on perturbation to induce mutual mode coupling. Instead, the mode coupling is enabled by waveguide evanescent coupling and thus can be easily controlled and made tunable. As the CW and CCW modes are possessed by one physical resonator, changing one resonator means simultaneously changing two modes, and thus tuning is more power-efficient. Our device features compact size, easy reconfiguration, and high tolerance to fabrication errors, suitable for optical filtering and delay applications.

2. Device structure

Figure 1 shows the schematic structure of our proposed device, which is essentially a single waveguide folded to self-couple with itself to form resonances. Light from the input port passes the first coupler (input coupler) and splits into two paths. At the second coupler (crossing coupler), part of the light crosses over and propagates forward and part is reflected back. The two branches of forward propagating light recombine at the third coupler (output coupler) and the recombined light either circulates back to the input coupler or directly couples out. Therefore, the forward propagating light passes three couplers to form a CW resonance loop, indicated by the large blue arrows in Fig. 1. Similarly, the reflected light travels backward to form a CCW resonance loop, indicated by the small red arrows in Fig. 1. The crossing coupler determines the mutual mode coupling strength between the CW and CCW resonance modes. Light can be reflected back via the CW and CCW resonance loops to the input end. As both the CW and CCW resonance modes are excited, a standing-wave pattern is formed in the resonator.

 

Fig. 1 Schematic of the waveguide self-coupling based reconfigurable resonance structure. Electric fields before and after each coupler are labeled. The large blue and small red arrows indicate clockwise (CW) and counter-clockwise (CCW) resonance loops, respectively.

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In order to conveniently control the mutual mode coupling strength, we use a push-pull Mach-Zehnder Interferometer (MZI) [29] to replace the crossing coupler, as shown by the inset in Fig. 1. We also add another push-pull phase shifter in the split paths so that the resonance input/output coupling can be varied (to be proved by theoretical analyses). With these two phase shifters incorporated, the device transmission and reflection spectra can be easily tailored, as will be seen in the following sections.

3. Modeling

We use the transfer matrix method to get the transmission and reflection transfer functions of the device [30]. The electric fields before and after each coupler are labeled in Fig. 1, with superscript ‘ + ’ indicating left to right propagation and ‘-’ for right to left propagation.

The electric fields before and after input/output couplers are related by the following transfer matrix:

Assuming the input is only from the upper waveguide, i.e., $E_{a1}^{+}=E_{in}$ and $E_{b3}^{+}=0$, then$E_{a1}^{-}$ is the reflected electric field and$E_{b3}^{-}$is the transmitted electric field. Solving the linear equation Eq. (5), we can get the electric field transfer functions for the transmission and reflection ports. To simplify our analysis, we only consider the case when the input and output couplers are ideal 3-dB couplers, i.e., t_a² = t_b² = 0.5, and the transmission and reflection functions are expressed as (in z-domain):

4. Analysis and discussion

The transmission and reflection responses of our structure are dependent on the two phase shifts Δϕ_io and Δϕ_x. The resonance structure can be reconfigured for different applications by tuning these two parameters. In the following subsections, we first consider several limiting cases and then analyze its optical performances as a tunable filter and a delay line. The device tolerance to structural parameters and temperature variations is discussed in the last subsection.

4.1 limiting cases

In certain special configurations, the resonance structure can regress to simpler structures, which can be regarded as the limiting cases for our device.

When Δϕ_io = 0 (or t_io = 1), the amplitude responses for the transmission and reflection ports are |H_t(z)| = |t_x| and |H_r(z)| = |κ_x|, which implies light is partially reflected (without forming resonances) and its reflectivity is determined by the cross coupling coefficient. When Δϕ_x varied from 0 to π/2, the reflection increases from zero to full reflection.

When Δϕ _x = 0 (or t_x = 1), the crossing coupling is at its maximum and the light just crosses over without inducing backward coupling. The transmission function can then be simplified to

 

Fig. 2 (a) and (b) Transmission intensity and group delay responses for the device configured as a notch filter. (c) and (d) Transmission and reflection intensity and group delay responses for the device configured as an add-drop filter.

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When Δϕ_io = π/2 (or t_io = 0), the device transfer functions then become

When Δϕ_x = π/2 (or t_x = 0), all the light is reflected back at the crossing coupler and the device is essentially composed by two separate Sagnac loop mirrors. If the input and output couplers are ideal 3-dB couplers, then all the light can be reflected back to the input-port.

4.2 Tunable filter

When both Δϕ_x and Δϕ_io are small, the mutual mode coupling between the CW and CCW resonance modes and the coupling to the external access waveguide are both weak, and then we can configure the device as a second-order filter. The complex poles of the transfer equations Eqs. (6) and (7) are essentially the complex eigenmodes of the device. The complex phase of the pole can be written as ${\tilde{\varphi }}_{p0}=\pm {\varphi }_{p0}+i\gamma /2$, where ϕ_p0 and γ are associated with the real-valued resonance frequency and linewidth:

The resonance structure has two eigenmodes with split frequencies and identical linewidths. When their frequency separation is less than the linewidth, i.e., 2ϕ_p0 < γ, these two modes are regarded as undistinguishable. Under weak coupling (t_io ≈ 1 and t_x ≈ 1), the condition for the undistinguishable eigenmodes can be approximated to

In order to make use of mode mutual-coupling to expand the passband for a filter application, Eq. (13) should be satisfied.

To get a flat-top passband, the coupling coefficients k_io and k_x should satisfy a more stringent requirement. Stating from the pole and zero positions and their function on reflection spectrum, we deduce the relation between the poles and zeros required to generate a flat-top passband. The reflection transfer function has two conjugate poles (p₀ and p₀*) and two conjugate zeros (z₀ and z₀*):

The conjugate poles correspond to two split peaks. When the poles are closer to each other or farther away from the unit circle, the linewidth of the peaks becomes larger than their separation and these two peaks will merge into one and we can no longer distinguish them (see Eq. (13)). Hence, the spectral profile between the poles shown as a peak or a valley is quite dependent on the pole positions in the unit circle.

The existence of the conjugate zeros further tailors the spectrum. The poles and zeros have an identical real part at_xt_io, but the zeros have a larger imaginary part than the poles have, meaning that the zeros are located outside the poles in the pole-zero diagram, i.e., ϕ_z0 > ϕ_p0. Thus, the addition of zeros makes the reflection passband smoother and also drop faster. To get an exact flat passband, the second-order derivative of the intensity transfer function should be zero at the uncoupled resonance frequency, i.e., d ²[|H_r(z)|²]/dϕ ² = 0 at ϕ = 0. After mathematical treatment, we get the following condition for the flat-top passband response:

Figure 3(a) shows the relation curves between Δϕ_io and Δϕ_x for various loss factors to satisfy the flat-top passband response. The addition of the waveguide propagation loss up-shifts the Δϕ_x versus Δϕ_io curve, which means strong mutual mode coupling is needed to compensate for the loss-induced resonance broadening. The solid curves in Fig. 3(b) show the reflection spectra of our device for three different Δϕ_io’s under the flat-top condition. The intensity remains almost flat between the pole frequencies. Beyond the pole frequencies, the intensity drops rapidly and reaches the minimum around the zero frequencies. Beyond the zeros, the intensity increases slightly. To compare with a conventional microring ADF, we also plot the drop spectra of a double-ring ADF (dotted curves) with its waveguide-ring and ring-ring coupling coefficients being κ_io and κ_x, respectively. From the comparison, we see that our device has a faster roll-off but a lower out-of-band rejection ratio because of the excess zeros in the transfer function.

 

Fig. 3 (a) Relation between Δϕ_x and Δϕ_io for flap-top filters. (b) Solid curves: reflection intensity spectra of our device; dotted curves: drop-port intensity spectra of an add-drop filter. (c) FSR-normalized transition and reflection bandwidths as functions of Δϕ_io. (d) Insertion loss and out-of-band rejection ratio as functions of Δϕ_io. In (b)-(d), the flap-top passband condition is satisfied.

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The reflection bandwidth is defined as the frequency separation between the poles. i.e., BW_ref = tan⁻¹(κ_x/t_iot_x)/π (normalized to the free spectral range (FSR)). The roll-off rate is related to the transition bandwidth, defined as the frequency separation between the pole and the zero, i.e., BW_tran = [cos⁻¹(t_iot_x)-tan⁻¹(κ_x/t_iot_x)]/2π (normalized to the FSR). Figure 3(c) shows BW_ref and BW_tran change with Δϕ_io for various loss factors. Larger Δϕ_io leads to a wider reflection passband but a slower roll-off. Phase shift Δϕ_io not only affects the filter bandwidth but also the insertion loss and out-of-band rejection ratio (or the cross-talk level). The normalized reflection intensity at the resonance frequency is I_r0 = |H_r(1)|², and thus the insertion loss (at the reflection end) is IL = −20lg(|H_r(1)|). The out-of-band rejection ratio is defined as OBR = 20lg(|H_r(1)|/|H_r(−1)|), indicating the contrast between the on- and off-resonance reflection intensities. Figure 3(d) shows the IL and OBR of the flat-top filter change with Δϕ_io for various loss factors. Larger Δϕ_io has lower insertion loss but higher cross-talk level.

The waveguide loss also plays an important role in the filter performances. Under the flat-top passband condition and with a certain Δϕ_io, larger waveguide loss slightly increases the reflection bandwidth but also deteriorates the filtering performance in terms of insertion loss and cross-talk level as seen from Figs. 3(c) and 3(d). For narrow-bandwidth filters (Δϕ_io is small), loss can more significantly degrade its performances, because a narrow bandwidth means a high-Q resonance and hence it is more sensitive to loss.

4.3 Tunable delay line

As like microring resonators, group delay is greatly enhanced around resonance, making it suitable for optical signal delay applications. Neglecting the linear transmission factor t_wg in Eq. (7), the reflection group delay at the uncoupled resonance frequency (the operating frequency) can be expressed as

As phase changes nonlinearly with frequency around resonance, GDD can be strong enough to significantly distort the optical signal. At the operating frequency, even dispersive orders of GDDs are all zero, but odd dispersive orders of GDDs still exist. To flatten the group delay, the third order GDD should be zero at the operation frequency, i.e., d ³Φ_r(ϕ)/dϕ ³ = 0 at ϕ = 0, where Φ_r(ϕ) is the phase of H_r(z). From Eq. (7) and after some mathematical manipulation, we get the condition for the flat-top group delay response:

Therefore, the pole and zero should satisfy Eq. (18) or Eq. (19) to achieve a flat-top group delay response. Substituting p₀ with Eq. (14), Eq. (19) becomes

Figure 4(a) shows the dependence of Δϕ_x on Δϕ_io for different loss factors based on Eq. (18). Δϕ_x is almost linearly proportional to Δϕ_io in a small range of interest, which implies that Δϕ_x and Δϕ_io can be simultaneously tuned by one external voltage as long as their corresponding active region lengths are in scale. This can greatly simply the drive circuit for group delay tuning. Figure 4(b) shows the insertion loss changes as a function of Δϕ_io. Higher waveguide loss increases the insertion loss, similar to the flat-top filter case. Yet distinct from the previous case, larger Δϕ_io can also lead to a higher insertion loss, more evident for the lossless case (a = 1), as not all light is reflected back at the operating frequency. Figures 4(c) and 4(d) show the group delay bandwidth (approximately the filter bandwidth) and group delay (at the operating frequency) change as functions of Δϕ_io. A large group delay can be obtained by using smaller Δϕ_io and Δϕ_x yet with a sacrifice of smaller bandwidth, limited by the so-called delay-bandwidth product in resonance systems [32]. Large group delay corresponds to a high-Q resonance, which is more sensitive to loss. A small increase in loss can significantly deteriorate its group delay response. Hence, to get a high group delay, not only should Δϕ_io and Δϕ_x be small, the loss should also be kept low enough.

 

Fig. 4 (a) Relation between Δϕ_x and Δϕ_io for flap-top group delay responses. (b)-(d) Insertion loss, normalized group delay bandwidth, and group delay change as a function of Δϕ_io. (e) and (f) Typical reflection intensity and group delay spectra. In (b)-(f), the flat-top group delay condition is satisfied.

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Figures 4(e) and 4(f) show the normalized reflection intensity and group delay spectra for Δϕ_io = 0.06π, 0.08π and 0.1π. As the third-order GDD is eliminated at the operating frequency, the group delay is almost flat within its bandwidth, and it can be continuously tuned for a relatively large range. For instance, if the device size is L = L₁+L₂+L₃ = 60 μm, the waveguide group refractive index n_g = 4.0, and the loss factor a = 0.997 (corresponding to a 5 dB/cm average waveguide propagation loss), then FSR ≈ 10 nm and T_R ≈ 0.8 ps at 1.55 μm wavelength, and the group delay thus can be continuously tuned for 28 ps with its bandwidth kept >10 GHz.

We also compare our device to a conventional double-ring ADF for signal delay application. In the ADF configuration, the resonances of the two rings should be well-aligned and the coupling strength of the three couplers (input, output and inter-ring couplers) should be well-controlled in order to obtain flat-top group delay responses. Yet in our structure, the resonances are always aligned because they are from degenerate CW and CCW modes of one physical resonator, and hence it alleviates the stringent fabrication requirement. Moreover, our structure has only two tuning elements compared to three in the double-ring ADF, so it is more compact and has less parameters to control, resulting in easy tuning and less power consumption.

Figure 5 shows the total phase shift as a function of group delay for our device and a double-ring ADF. The ADF-based tunable delay line structure is schematically illustrated in the inset. As the mutual mode coupling is much weaker than the input/output coupling (evident from Eq. (21)), the total phase shift for our device is nearly half of that for the ADF structure. The phase shift can be implemented by free carrier injection in silicon [31], and thus the power consumption is also almost one half less for our structure. Likewise, for resonance frequency tuning, because both CW and CCW modes can be simultaneously tuned by one waveguide section (e.g., waveguide L₁) in our structure, the power consumption spent on the frequency tuning is also one half smaller for our structure.

 

Fig. 5 Comparison between our device and a double ring ADF delay line in terms of total phase shift versus group delay.

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4.4 Tolerance to structural parameters

In the previous analysis, we assume the input and output couplers are both ideal 3-dB couplers. However, in reality, due to fabrication errors, the waveguide width and gap separation between waveguides are hard to control very precisely, and as a result, the coupling strength always deviates from what is designed. To investigate how the performances of the high-order filter and delay line deteriorate from the optimum ones, we resort to Eqs. (1)-(5) to numerically calculate the reflection intensity and group delay spectra.

For the filter configuration, we fix the phase tuning parameters at Δϕ_io = 0.08π and Δϕ_x = 0.011π and vary the input and output coupling ratios. Figures 6(a) -6(c) show the reflection spectra for various combinations of t_a ² and t_b ². When the input and output coupling ratios are balanced, i.e., t_a ² = t_b ², the filter spectrum is always symmetric, as shown in Fig. 6(a). The imperfection of the input and output couplers (non-ideal 3-dB couplers) causes the filter passband rounded and broadened. If only one of the couplers is imperfect, then the passband becomes slightly rounded and inclines towards one side, with its central wavelength shifts a little bit, as shown in Fig. 6(b). The most interesting combination is when the couplers are complementary, i.e., t_a ² + t_b ² = 1. In this case, the passband top remains almost flat while it inclines towards one side, as shown in Fig. 6(c). From these plots, it can be seen that if the variation of t_a ² and t_b ² is within ±0.1, the high-order filtering performance is still reasonably good.

 

Fig. 6 (a)-(c) Reflection intensity spectra for various input and output coupling ratios. (d)-(f) Reflection group delay responses for various input and output coupling ratios.

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For the delay line configuration, we fix the phase tuning parameters at Δϕ_io = 0.08π and Δϕ_x = 0.007π. The imperfection of the couplers also degrades the delay line performance with a similar trend as in the filter case. For the complimentary combination of t_a ² and t_b ² (Fig. 6(f)), the larger the difference between t_a ² and t_b ² is, the sharper the group delay curve becomes. As the passband inclines to one side, the corresponding transition band becomes narrower (faster roll-off) and thus the phase change becomes more rapid, leading to a high group delay. It can be seen that the group delay is still within an acceptable level, if the coupling ratio change is within ±0.1.

Besides the input and output couplers, the MZI crossing coupler also plays a significant role in the device performances. The imperfection of the MZI coupler causes the mutual-mode coupling coefficient κ_x to only value in a limited range, or in other words, κ_x cannot be very small and thus the filter and delay line bandwidth cannot be very narrow. However, as long as Eqs. (16) and (18) are satisfied, the flat-top passband and group delay responses still can be obtained.

To reduce the coupling sensitive to fabrication errors, multimode interference (MMI)-based couplers can be used instead of directional couplers. MMI-based couplers are based on self-imaging to achieve the power splitting, and therefore, they are more resilient to the structural dimension change, although they suffer from a higher loss [33].

As like most resonance structures, our structure is also sensitive to the environment temperature fluctuation (if the waveguide is not specially designed). The influence of the temperature change can be modeled by varying n_eff and the coupling coefficients. Due to the symmetry of the structure, the interference paths (except the resonance loop back interference) are all balanced, and therefore, the change in n_eff causes the intensity and group delay spectra to only experience a shift without any change in their profiles. The coupling change has a more negative effect on the device performances as discussed previously. For a typical silicon wire waveguide-based 3-dB directional coupler, our simulation based on beam propagation method (BPM) shows that temperature increases by 10 °C, coupling ratio only changes by <1%, which has an almost negligible effect according to Fig. 6. To stabilize the device over temperature variation, athermal waveguide design can be employed [34].

5. Conclusion

In conclusion, we presented a novel reconfigurable resonance structure, formed by a single self-coupled waveguide, to function as a tunable filter and a tunable delay line. Due to the self-coupling of the waveguide, both the CW and CCW resonance modes are excited with their input/output and mutual mode coupling strengths controlled by two push-pull phase shifters. Using the transfer-matrix method, we got the z-domain transfer functions for the transmission and reflection ports, and their optical characteristics under different configurations were analyzed. In particular, we analyzed its optical performances as a tunable second-order filter and a low-dispersion delay line. The requirements for the two phase shifters to get the flat-top filtering and group delay responses were deduced, providing a guideline for the device design. As the coupled resonances are provided by the CW and CCW modes of one physical resonator, the need for accurate alignment of resonance frequencies is no longer necessary. Compared to conventional coupled microring resonators, our resonance structure is more compact and the power consumption for reconfiguration is lower, which is especially important for future large-scale integrated photonic chips.