1. Introduction

Silicon photonics has become one of the most commonly used integration platforms due to the advantages of high bandwidth, low cost, high integration density, and compatibility with complementary metal-oxide-semiconductor (CMOS) fabrication technology [1,2]. Optical resonators play an important role in the success of silicon photonics because the high refractive index contrast of silicon waveguide makes the optical resonators extremely compact [3,4]. The silicon optical resonators can be employed in many applications including optical filters, electro-optical modulators, add-drop multiplexers, optical delay lines and label-free biosensors. As a traveling-wave resonator, microring resonator (MRR) is the most popular geometry for its advantages including simple structure, small footprint and no intrinsic reflection [5]. However, the bending loss of waveguide becomes large when the radius is small. Therefore, it’s hard for a single MRR to achieve a wide spectrum range (FSR) with high quality factor (Q-factor) [5–7]. On the other hand, the Fabry-Perot (F-P) resonator based on the photonic crystal structures or Bragg gratings can easily realize a wide FSR or an FSR-free response with little effect on the internal loss. But the conventional F-P resonator is a standing-wave resonator so the backward wave will cause the reflection to the input port [8–10]. Recently, the multi-mode Bragg reflectors enabling the coupling between two different transverse modes have attracted great interest and have been well studied [11–15]. For example, the Bragg reflector with anti-symmetric periodic structure can make the incident fundamental transverse-electric mode (TE₀) reflected and converted to the first order transverse-electric mode (TE₁), and vice versa [11,12]. Using such mode-conversion based Bragg reflectors, several traveling-wave-like F-P resonators have been proposed and demonstrated as optical filters, add-drop multiplexers and modulators [16–30]. Taking the F-P resonator with anti-symmetric Bragg reflectors as an example, when the input light is directly coupled to the resonator, the reflected TE₁ mode can be radiated by adding an adiabatic taper [19,29], or dropped by adding an asymmetric directional coupler [26,27]. When the input light in the bus waveguide is side-coupled to the forward TE₁ mode in the resonator, the backward TE₀ mode in the resonator cannot be coupled to the bus waveguide due to the large effective index difference [23,24]. Therefore, this type of F-P resonator can avoid the reflection to the input port. It is an attractive configuration to realize an FSR-free response with high Q-factor.

However, it’s still challenging to realize a narrow bandwidth, i.e., high Q-factor, for the transverse-mode-conversion based F-P resonator because of the coupling between the two degenerated resonant modes, which is caused by the incomplete transverse mode conversion, i.e., part of the incident transverse mode is reflected to the same mode. In this case, the resonator becomes a second order filter. The coupling between the resonant modes will bring additional loss and reflection to the input port, and even cause resonance splitting, which has been observed in both simulated and experimental results [12,16,19,21–24]. This phenomenon becomes much more severe when using the Bragg reflector with strong perturbation of refractive index, i.e., large coupling coefficient, such as etching circular holes in the multimode waveguide. Up to now, there is few research on the suppression of coupling between the resonant modes [15]. In this paper, the coupled mode theory (CMT) is used to model the incomplete transverse mode conversion in the Bragg reflector. It can be found that the coupling between the resonant modes can be greatly suppressed by carefully designing the phase shifter length of the resonator and adding smaller or tapered holes at the interface between the phase shifter and Bragg reflector. It should be noted that the focus of this paper is on the resonant mode coupling caused by the structure itself rather than the fabrication error.

2. Principle

2.1 Description of the structure

The F-P resonator discussed in this paper utilizes anti-symmetric nanobeam Bragg reflectors to realize the coupling between TE₀ and TE₁ modes, which is one of the most common configurations of transverse-mode-conversion based resonators [12,19,20,23,27]. It consists of one phase shifter and two Bragg reflectors, as shown in Fig. 1. The input light can be directly coupled to the resonator or side-coupled to the resonator through the directional coupler.

 

Fig. 1. Schematics of the (a) direct-coupled and (b) side-coupled F-P resonators with anti-symmetric nanobeam Bragg reflector. The inserted figure is the zoom-in view of the Bragg reflector. Here w, r and Λ denote the multimode waveguide width, hole radius and grating period, respectively.

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The schematic of the direct-coupled resonator is shown in Fig. 1(a), which can be used as a narrow pass-band filter. The input and through ports with single mode waveguides are connected to the resonator with multimode waveguide via two adiabatic tapers. The input light will excite the resonance formed by the conversion between the forward TE₀ mode and backward TE₁ mode. The light at resonant wavelength can nearly fully travel through the resonator. It should be noted that the reflected TE₁ mode is radiated in the adiabatic taper and single mode waveguide so the reflection to the input port can be avoided.

The schematic of the side-coupled resonator is shown in Fig. 1(b), the resonator with multimode waveguide is side-coupled to two single mode waveguides so as to form an add-drop filter. The taper on each end of the resonator is used to radiate the residual transmitted light and avoid reflection at the facet. In addition, the waveguide widths should be designed following the phase match condition that the effective index of the TE₀ mode in the single waveguide is equal to that of the TE₁ mode in the multimode waveguide to make the two modes efficiently coupled. The input light will excite the resonance formed by the conversion between the forward TE₁ mode and the backward TE₀ mode. It should be noted that the backward TE₀ mode cannot be coupled to the single mode waveguide due to the large effective index difference. The light at resonant wavelength can be removed from the bus and nearly fully dropped.

2.2 Coupled mode analysis

Next, we discuss the coupling of transverse modes in the Bragg reflector. In ideal condition that the perturbation is very weak, we can assume that the modal profile of each transverse mode remains constant. Thus only the contra-directional coupling between the TE₀ and TE₁ modes exists in the Bragg reflector. However, in practice the index variation in the Bragg reflector is large so the modal profiles severely change in the propagation direction. The mismatches of modal profiles lead to the other unwanted coupling terms, i.e., the contra-directional coupling between the forward and backward TE₀ modes, the contra-directional coupling between the forward and backward TE₁ modes, as well as the directional coupling between the TE₀ and TE₁ modes. As a consequence, the coupled mode equations can be written as [31–33]

The solution of the coupled mode equations can be expressed in the following form [31,34]:

Thus the response of Bragg reflector can be obtained using the transfer matrix method (TMM).

Due to the presence of κ₀₀, κ₁₁ and μ₀₁, the transverse mode conversion becomes incomplete and part of the incident transverse mode is reflected to the same transverse mode, which will lead to the coupling between the two degenerated resonant modes. One resonant mode contains the forward TE₁ and backward TE₀ modes (denoted as RM1), and the other contains the forward TE₀ and backward TE₁ modes (denoted as RM2), as shown in Fig. 2(a). Here r_ij denotes the amplitude reflection coefficient between TE_i and TE_j modes. The resonant wavelength λ_r satisfies the following phase match condition:

 

Fig. 2. Schematics of (a) the coupling between two resonant modes, (b) the resonant mode coupling induced by the reflection between the forward and backward TE₀ modes and (c) the resonant mode coupling induced by the reflection between the forward and backward TE₁ modes.

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3. Resonators with uniform anti-symmetric Bragg reflector

3.1 Reflection spectrum of Bragg reflector

The reflection spectrum of the Bragg reflector is firstly investigated. The length, waveguide width, hole radius and grating period used in the simulation are 10 μm, 837 nm, 40 nm and 326 nm, respectively. The effective indices and coupling coefficients in the Bragg reflector are calculated by simulating the Bragg grating with one period via finite-difference time domain (FDTD) method with the perfectly matched layer (PML) boundaries, as shown in Fig. 3. When the TE₀ mode is incident, we record the phase response of the transmitted TE₀ mode (Δϕ₀), and then the effective index of TE₀ mode (n_{eff 0}) can be deduced by $\Delta {\phi _0} = {n_{eff0}}\frac{{2\pi }}{\lambda }\Lambda $. Similarly, we can obtain n_{eff 1} when the TE₁ mode is incident. The amplitude reflection and transmission coefficients including |t₀₁|, |r₀₀| and |r₀₁| can be obtained when the TE₀ mode is incident, as shown in Fig. 3(a). |t₀₁|, |r₁₁| and |r₀₁| can also be obtained when the TE₁ mode is incident, as shown in Fig. 3(b). Then the coupling coefficients can be deduced by |t₀₁| = μ₀₁Λ, |r₀₀| = κ₀₀Λ, |r₀₁| = κ₀₁Λ and |r₁₁| = κ₁₁Λ, respectively. The calculated Δϕ₀, Δϕ₁, |t₀₁|, |r₀₀|, |r₀₁| and |r₁₁| using FDTD method and the deduced n_{eff 0,} n_{eff 1,} μ₀₁, κ₀₀, κ₀₁ and κ₁₁ at 1550 nm are shown in Table 1. Such large values of κ₀₀, κ₁₁ and μ₀₁ cannot be neglected in the coupled mode analysis. Figure The effective indices and coupling coefficients width different wavelength are also calculated, as shown in Fig. 4(a) and (b), respectively.

 

Fig. 3. Schematics of the calculation of effective indices and coupling coefficients in the Bragg reflector when (a) the TE₀ mode is incident and (b) the TE₁ mode is incident.

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Fig. 4. Calculated (a) effective indices of TE₀ and TE₁ modes in the Bragg reflector and strip waveguide, and (b) coupling coefficients of Bragg reflector.

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Table 1. Calculated effective indices and coupling coefficients with 40-nm hole radius at λ = 1550 nm

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The normalized reflection spectra calculated by the FDTD method and CMT are shown in Fig. 5. Here the modal loss is neglected in the calculation using CMT. It can be seen the two curves in each spectrum are well matched, which validates that CMT can be used to model the incomplete transverse mode conversion in the Bragg reflector, though the perturbation is very strong. |r₀₁|² reaches 0.99 at the Bragg wavelength of 1550 nm. Moreover, |r₀₀|² and |r₁₁|² are almost equal within the stopband of Bragg reflector. Both of them are dramatically increased around 1520 nm, which indicates the incomplete transverse mode conversion is much more severe for the shorter wavelengths.

 

Fig. 5. Calculated normalized reflection spectra of (a) |r₀₁|², (b) |r₀₀|² and (c) |r₁₁|² using 3D FDTD method and CMT.

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The impacts of the hole radius on the coupling coefficients are also investigated. Figure 6(a) shows the coupling coefficients at 1550 nm with different hole radius. Here the grating period is varied with the hole radius to keep the Bragg wavelength at 1550 nm. Figure 6(b) and (c) show the calculated |r₀₁|² and |r₀₀|² with different hole radius using CMT, respectively. For easy comparison, the length of Bragg reflector (L_B) is varied with the hole radius to keep κ₀₁LB = 3.5. As the hole radius increases, |r₀₀|² increases and the maximum value of |r₀₁|² slightly decreases. Moreover, when the hole radius is 50 μm and 60 μm, the maximum point of |r₀₁|² is apparently red shifted due to the severe incomplete mode conversion. Therefore, to increase |r₀₁|² and decrease |r₀₀|², the phase shifter length of the resonator should be designed to make the resonant wavelength red shifted from the Bragg wavelength when using a large hole radius. Using a smaller hole radius can also effectively decrease |r₀₀|². This is because the perturbation is weakened, which reduces the impact of the unwanted couplings between the transverse modes. However, a small hole radius also leads to a small κ₀₁, which requires a large Bragg reflector length to achieve a sufficient power reflection of |r₀₁|². Therefore, the hole radius is chosen to be 40 nm in the following simulations.

 

Fig. 6. Calculated (a) coupling coefficients with different hole radius at 1550 nm, as well as (b) |r₀₁|² and (c) |r₀₀|² with different hole radius using CMT when keeping κ₀₁LB = 3.5.

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3.2 Performance of the resonator

The resonant wavelength and the resonant mode coupling coefficient (k_rm) with different phase shifter length (L_P) are calculated by Eqs. (8) and (11), respectively, as shown in Fig. 7. The Bragg reflector length used in the calculation is 10 μm. The resonant wavelength equals 1550 nm when L_p equals m·0.96Λ, where m is an integer, since the effective indices of the TE₀ and TE₁ modes in the phase shifter are higher than that in the Bragg reflector. When L_P equals 0, the value of (φ₀₀−φ₁₁) is −0.1π, so the value of k_rm is up to 0.1. When L_P increases to 1.88 μm (6 × 0.96Λ), the value of (φ₀₀−φ₁₁) is 1.05π, and k_rm decreases to 0.01. When L_P further increases to near 2.0 μm, the resonant wavelength is red shifted and k_rm can decrease to a small value close to 0. However, |r₀₁|² may decrease when the resonant wavelength deviates from the Bragg wavelength, which will lead to the decrease of Q-factor.

 

Fig. 7. Calculated resonant mode coupling coefficient (k_rm) and resonant wavelength versus phase shifter length (L_p).

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The spectra of direct-coupled and side-coupled resonators are simulated using 3D FDTD method. In the side-coupled resonator, the two bus waveguides are made in arc shape in the directional coupling region with the same bending radius of 10 μm and minimum waveguide spacing of 250 nm. The Bragg reflector lengths of the direct-coupled and side-coupled resonators are 6.5 μm and 10 μm, respectively. Such difference of Bragg reflector length is used to keep the bandwidths or the Q-factors of the two resonators approximately equal. When L_P is 0, the calculated |r₀₀|² and |r₀₁|² of the 6.5-μm long Bragg reflector are both about 2.6 × 10⁻³, and the calculated (φ₀₀−φ₁₁) is about −0.1π. These values are very close to that of the 10-μm long Bragg reflector. Therefore, the calculated resonant mode coupling coefficients (k_rm) shown in Fig. 7 is also suitable for the direct-coupled resonator with 6.5-μm long Bragg reflectors.

The calculated spectra of the direct-coupled resonator with different L_P are shown in Fig. 8. Due to the large value of k_rm, an apparent resonance splitting can be observed when L_P is 0, as shown in Fig. 8(a). When L_P is 1.57 μm (5 × 0.96Λ), the resonance splitting becomes weak, as shown in Fig. 8(b). The resonance splitting vanishes when L_P is 1.88 μm (6 × 0.96Λ), the insertion loss and 3-dB bandwidth of through port are 0.51 dB and 0.89 nm, respectively, as shown in Fig. 8(c). The corresponding Q-factor is 1.74 × 10³. When L_P is 1.96 μm (6.25 × 0.96Λ), the resonant wavelength is shifted to around 1565 nm, the insertion loss and 3-dB bandwidth of through port are 0.23 dB and 1.21 nm, respectively, as shown in Fig. 8(d). The corresponding Q-factor is 1.29 × 10³.

 

Fig. 8. Calculated spectra of the direct-coupled resonators with uniform Bragg reflector at the two ports when L_p are (a) 0, (b) 1.57 μm (5 × 0.96Λ), (c) 1.88 μm (6 × 0.96Λ) and (d) 1.96 μm (6.25 × 0.96Λ), respectively.

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The calculated spectra of the side-coupled resonator with different L_P are shown in Fig. 9. Resonance splitting can be observed when L_P is 0 and 1.57 μm, as shown in Fig. 9(a) and (b), respectively. The resonance splitting vanishes when L_P is 1.88 μm. The extinction ratio of through port, as well as the insertion loss and 3-dB bandwidth of drop port are −16.6 dB, −1.31 dB and 0.91 nm, respectively, as shown in Fig. 9(c). The corresponding Q-factor is 1.70 × 10³. Besides, the transmission at add port and the reflection at input port are both decreased to around −22 dB. When L_P is 1.96 μm, the resonant wavelength is shifted to around 1565 nm, the extinction ratio, insertion loss and 3-dB bandwidth are 16.0 dB, 1.48 dB and 1.03 nm, respectively, as shown in Fig. 9(d). The corresponding Q-factor is 1.52 × 10³. The transmission at add port and the reflection at input port are further decreased to around −28 dB.

 

Fig. 9. Calculated spectra of the side-coupled resonator at the four ports when L_p are (a) 0, (b) 1.57 μm (5 × 0.96Λ), (c) 1.88 μm (6 × 0.96Λ) and (d) 1.96 μm (6.25 × 0.96Λ), respectively.

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From the simulations we can see that the coupling between the two resonant modes can severely affect the shape, linewidth and insertion loss of the spectrum, and lead to crosstalk at other ports. The resonant mode coupling can be greatly suppressed by carefully designing the phase shifter length, which agrees with the calculation of k_rm shown in Fig. 7.

4. Resonators with tapered anti-symmetric Bragg reflector

Adding tapered holes in the initial position of Bragg reflector can effectively suppress the unwanted reflections of |r₀₀|² and |r₁₁|², and further decrease the coupling between the two resonant modes [15,16,22]. However, the previously reported papers didn’t explain whether and how the optimal solution of the tapered holes is obtained. Here we add the holes with 20-nm radius and holes with 30-nm radius, with period numbers of n₁ and n₂, respectively, as shown in the inserted figure of Fig. 10(a). The length of the holes with 40-nm radius is 10 μm. The periods of the holes with 20-nm radius and 30-nm radius are 316.5 nm and 320.5 nm respectively to keep the Bragg wavelength at 1550 nm. The TMM based on CMT is used to calculate |r₀₀|² at 1550 nm with different n₁ and n₂, as shown in Fig. 10(a). It has much higher efficiency than FDTD method due to the lower time consumption. In order to achieve a low |r₀₀|² with a shorter length of tapered holes, n₁ and n₂ are both chosen to be 3, which is marked with a red dot in Fig. 10(a). Then |r₀₀|² and |r₁₁|² are calculated using FDTD method, as shown in Fig. 10(b). It can be seen that |r₀₀|² and |r₁₁|² at 1550 nm decreases to 3.2 × 10⁻⁵ and 2.7 × 10⁻⁵ (∼ −45 dB), respectively. According to Eq. (11), the maximum value of k_rm at resonant wavelength of 1550 nm is only 0.012, which is much lower than that of the resonator with uniform Bragg reflector. Due to the tapered holes, the resonant wavelength equals 1550 nm when L_p = 20 nm + m·0.96Λ, where m is an integer and Λ is the period of holes with 40-nm radius. The calculated (φ₀₀−φ₁₁) is about −0.3π when L_P is 20 nm.

 

Fig. 10. (a) Calculated |r₀₀|² at 1550 nm with different n₁ and n₂ using TMM bases on CMT, and (b) calculated |r₀₀|² and |r₁₁|² spectra using FDTD method when n₁ = n₂ = 3, which is marked with a red dot in (a).

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Except for the added smaller holes, the other structures of the direct-coupled and side-coupled resonators are the same as that of the resonators with uniform anti-symmetric Bragg reflector. Considering the strongest and weakest coupling between the resonant modes, the two cases that L_P equals 0.65 μm (20 nm + 2 × 0.96Λ) and 2.22 μm (20 nm + 7 × 0.96Λ) are calculated, with the corresponding (φ₀₀−φ₁₁) of 0.08π and 1.03π, respectively. The calculated spectra of the direct-coupled and side-coupled resonators with different L_P are shown in Fig. 11 and Fig. 12, respectively.

 

Fig. 11. Calculated spectra of the direct-coupled resonators with tapered Bragg reflector at the two ports when L_p are (a) 0.65 μm (20 nm + 2 × 0.96Λ) and (b) 2.22 μm (20 nm + 7 × 0.96Λ), respectively.

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Fig. 12. Calculated spectra of the side-coupled resonators with tapered Bragg reflector at the four ports when L_p are (a) 0.65 μm (20 nm + 2 × 0.96Λ) and (b) 2.22 μm (20 nm + 7 × 0.96Λ), respectively.

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For the direct-coupled resonator, when L_P is 0.65 μm, the insertion loss and 3-dB bandwidth are −2.12 dB and 0.63 nm, respectively, as shown in Fig. 11(a). The corresponding Q-factor is 2.38 × 10³. The reflection at the input port is −8.29 dB. When L_P is 2.22 μm, a better performance with insertion loss of −0.29 dB and 3-dB bandwidth of 0.36 nm is achieved, as shown in Fig. 11(b). The corresponding Q-factor is 4.31 × 10³. The reflection to the input port is also greatly decreased, which indicates the coupling between the resonant modes is effectively suppressed.

For the side-coupled resonator, when L_P is 0.65 μm, the extinction ratio, insertion loss and 3-dB bandwidth are −19.7 dB, −0.89 dB and 0.98 nm, respectively, as shown in Fig. 12(a). The corresponding Q-factor is 1.58 × 10³. The transmission at add port and the reflection at input port are both around −15 dB. When L_P is 2.22 μm, the extinction ratio, insertion loss and 3-dB bandwidth are −22.5 dB, −0.55 dB and 0.65 nm, respectively, as shown in Fig. 12(b). The corresponding Q-factor is 2.38 × 10³. The transmission at add port and the reflection at input port are both decreased to around −28 dB.

From the simulations we can see that adding the tapered holes is an effective way to suppress the coupling between the resonant modes. Due to the small values of |r₀₀|² and |r₁₁|², the resonance splitting is not observed even in the worst case that (φ₀₀−φ₁₁) is close to 0. However, an optimal response still requires the careful design of L_P to further suppress the coupling between the resonant modes.

5. Conclusion

In summary, we have studied the coupling between the two degenerated resonant modes in the transverse-mode-conversion based resonator with anti-symmetric nanobeam Bragg reflector, which is induced by the incomplete mode-conversion of the Bragg reflector. Such incomplete mode-conversion can be modeled by the coupled mode equations in which all the coupling coefficients are considered. We have also explained that the coupling between the two resonant modes can be effectively suppressed by carefully designing the phase shifter length and adding the tapered holes. This study is believed to benefit the design of transverse-mode-conversion based resonator filters which have great potential for applications in wavelength-division multiplexing networks.