Since Marcatili A J^[1] proposed the concept of microring resonator in 1969, the research on the structure of microring resonator has been carried out for many years^[2-6]. The structure of the ding-shaped microring resonator was first proposed by Canadian scholar Sacher W D^[7]et al. in 2014. Due to the large number of coupling regions of the ding-shaped microring resonator, the transmission loss is large and the increase in the number of coupling regions leads to an increase in the volume of the ding-shaped microring resonator. There are relatively few applications and researches on the ding-shaped microring resonator, mainly focusing on filtering and phase shift.

The ding-shaped microring resonator is the deformation of the dual-channel microring resonator. The difference is that the dual-channel microring resonator has only two coupling regions, and the standard ding-shaped microring resonator is based on the dual-channel microring resonator. In order to increase the number of coupling regions, the shape of the straight waveguide and the ring waveguide is adjusted, so that four coupling regions are formed, which greatly improves the selectivity of the ding-shaped microring resonator to specific wavelengths of light. As shown in Fig.1, C₁, C₂, C₃ and C₄ represent four coupling regions respectively, and CW₁ and CW₂ represent two curved waveguides.

In this paper, by adjusting the number of coupling regions of the standard ding-shaped microring resonator, the influence of the number of coupling regions on the output spectrum is observed. The physical model of the ding-shaped microring resonator with different numbers of coupling regions is analyzed. The output spectrum and field of the ding-shaped microring resonator are analyzed.Since the basic structure of the ding-shaped microring resonator is a directional coupler, the ding-shaped microring resonator model is analyzed by using the transfer matrix^[8-10], and the influence of the number of coupling regions on the output of the ding-shaped microring resonator is studied. The coupler structure is shown in Fig.2.

By the transfer matrix method, the transfer matrix relationship of the coupling region C₁ can be expressed as :

$ \left( {\begin{array}{*{20}{c}} {{E_4}} \\ {{E_2}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {t_1}&{k_1} \\ { - k_1^*}&{t_1^ * } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{E_1}} \\ {{E_3}} \end{array}} \right) \quad,$ (1)

where ${E_2}$, ${E_4}$ represent the complex amplitudes generated when the output coupling region is C₁; $ {E_1} $, $ {E_3} $ represent the complex amplitudes generated when the input coupling region is C₁; $t_1$, $t_1^*$ are the self-coupling coefficients of the coupling regions C₁, and $k_1$, $k_1^*$ are the cross-coupling coefficients of the coupling region C₁.

As for the uncoupled region C₂, there are:

$ {E_5} = {\text{exp}}\left[ { - {\rm{j}}\left( {{\varphi _1} + \Delta {\varphi _1}} \right)} \right]{E_1}\quad, $ (2)

$ {E_7} = {\text{exp}}\left[ { - {\rm{j}}\left( {{\varphi _2} + \Delta {\varphi _2}} \right)} \right]{E_2}\quad. $ (3)

In the above formulas, ${E_5}$ and ${E_7}$ are the complex amplitudes generated when the input coupling region is C₂; ${\varphi _1}$, ${\varphi _2}$ are the phase changes in the phase shifter and they meet ${\varphi _1} + \Delta {\varphi _1} = - ({\varphi _2} + \Delta {\varphi _2})$.

Similarly, the transfer matrix relationship of the coupling region C₂ can be expressed as :

$ \left( {\begin{array}{*{20}{c}} {{E_8}} \\ {{E_6}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {t_2}&{k_2} \\ { - k_2^ * }&{t_2^ * } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{E_5}} \\ {{E_7}} \end{array}} \right)\quad, $ (4)

where ${E_6}$ and ${E_8}$ are the complex amplitudes generated when the output coupling region is C₂; $t_2$ and $t_2^ * $ are the self-coupling coefficients of the coupling regions C₂; $k_2$ and $k_2^ * $ are the cross-coupling coefficients of the coupling region C₂.

In the uncoupled region C₃, the relationship between ${E_9}$ and ${E_6}$can be expressed as :

$ {E_9} = {\alpha _{L1}}{\text{exp}}\left( { - \mathrm{j}{\varphi _{L1}}} \right) \cdot \gamma \cdot {\alpha _{L4}}{\text{exp}}\left( { - \mathrm{j}{\varphi _{L4}}} \right){E_6}\quad, $ (5)

in the above formula, ${\varphi _{L1}}$ and ${\varphi _{L4}}$ are the phase changes on the ring waveguide $L_1$ and $L_{4}$ in Fig.1; ${\alpha _{L_{1}}}$ and ${\alpha _{L_{4}}}$ are representing the transmission coefficients of the ring waveguide $L_1$ and $L_4$; ${E_9}$ denotes the complex amplitude when entering the coupling region C₃; $\gamma = {\alpha _{\rm{R}}}\exp \left( {{\rm{j}}\theta /2} \right)$, where $ {\alpha }_{{\rm{R}}} $ is the transmission loss in the semi-ring waveguide; $\theta $ is the phase change in a semi-circular waveguide.

Similarly, the transfer matrix relationship of the coupling region C₃ can be expressed as :

$ \left( {\begin{array}{*{20}{c}} {{E_{12}}} \\ {{E_{10}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{t_3}}&{{k_3}} \\ { - {k_3}^ * }&{{t_3}^ * } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{E_9}} \\ {{E_{11}}} \end{array}} \right)\quad, $ (6)

where ${E_{10}}$ and ${E_{12}}$ are representing the complex amplitudes generated when the output coupling region is C₃; ${E_{11}}$ is representing the complex amplitude generated when the input coupling region is C₃; ${t_3}$ and ${t_3}^*$ are the self-coupling coefficients of the coupling regions C₃; ${k_3}$ and ${k_3}^*$ are the cross-coupling coefficients of the coupling region C₃.

As for the uncoupled region C₄, there are:

$ {E_{13}} = {\text{exp}}\left[ {{{ - j}}\left( {{\varphi _3} + \Delta {\varphi _3}} \right)} \right]{E_{12}} \quad, $ (7)

$ {E_{15}} = {\text{exp}}\left[ { - j\left( {{\varphi _4} + \Delta {\varphi _4}} \right)} \right]{E_{10}}\quad, $ (8)

in the above formulas, ${E_{13}}$ and ${E_{15}}$ are representing the complex amplitudes generated when the intput coupling region is C₄; ${\varphi _3}$ and ${\varphi _4}$ are the phase changes in the phase shifter and ${\varphi _3} +\Delta {\varphi _3} = - \left( {{\varphi _4} + \Delta {\varphi _4}} \right)$.

Similarly, the transfer matrix relationship of the coupling region C₄ can be expressed as :

$ \left( {\begin{array}{*{20}{c}} {{E_{16}}} \\ {{E_{14}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {t_4}&{k_4} \\ { - k_4^ * }&{t_4^ * } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{E_{13}}} \\ {{E_{15}}} \end{array}} \right) \quad, $ (9)

and there is $ {E_{3}} = {\alpha _{L1}}{\text{exp}}\left( {{\rm{j}}{\varphi _{L1}}} \right) \cdot \gamma \cdot{\alpha _{L4}}{\text{exp}}\left( {{\rm{j}}{\varphi _{L4}}} \right) {E_{16}} $. In the above formulas, ${E_{14}}$ and ${E_{16}}$ are representing the complex amplitudes generated when the output coupling region is C₄; $t_4$ and $t_4^*$ are the self-coupling coefficients of the coupling region C₄; $k_4$, $k_4^ * $ are the cross-coupling coefficients of the coupling region C₄.

After the coupling occurs in the coupling region, the optical signal coupled into the ring waveguide interferes with the optical signal of the ring waveguide. If the two optical signals are in the same direction, they are superimposed during transmission, and the intensity of the optical signal is enhanced when transmitting in the ring waveguide. On the contrary, if the direction of the two optical signals is inconsistent, the two optical signals are canceled during transmission, and the intensity of the optical signal is weakened when transmitting in the ring waveguide.

The Full Width at Half Maximum (FWHM) and quality factor Q are important indicators to measure the performance of microring resonators. The FWHM represents the peak width when the peak height of the output spectral line is half. The smaller the FWHM, the denser the resonance peak at the working wavelength. Since the FWHM is inversely proportional to the quality factor Q, the smaller the FWHM, the larger the quality factor Q, and the stronger the energy storage performance ofthe microring resonator.

From the perspective of energy transfer^[11], the optical signal enters the ding-shaped microring resonator from the Input port, and the energy is transmitted along the curved waveguide CW₁. After entering the coupling region C₁, the energy is split. A part of the energy is coupled into the ring waveguide and transmitted. The remaining energy will continue to be transmitted along the curved waveguide CW₁ to the coupling region C₂, and the energy is split again in the coupling region C₂. Most of the energy is coupled into the ring waveguide and transmitted, and the remaining small part of the energy is output from the Through port. When the energy is transmitted to the coupling region C₃, the shunt occurs again. Part of the energy is coupled from the ring waveguide to the curved waveguide CW₂ and transmitted, and the remaining energy continues to transmit along the ring waveguide. When the energy is transmitted to the coupling region C₄, the shunt occurs again, and most of the energy is coupled to the curved waveguide CW₂ and output from the Drop port. The coupling region makes the optical signal undergo multiple energy shunts during transmission. The optical signal that does not satisfy the resonance condition will be directly output, and the optical signal that satisfies the resonance condition will continue to be transmitted in the ring waveguide, which increases the selectivity of the microring to the optical signal.

When $L_1 = 10$ μm, $L_2 = 2$ μm, $L_3 = 15$ μm, $L_4 = 5$ μm, the coupling distance of the coupling region gap is 0.1 μm, x_span=5 μm，y_span=5 μm，the coupling coefficient is 0.2896, waveguide thickness h=0.22, and the working wavelength is 1.54~1.56 μm, the electric field distribution and transmission spectrum^[12] of the standard ding-shaped microring resonator are shown in Fig. 3(color online). When the optical signal is input from the Input port and passes through the coupling region C₁, the electric field satisfying the resonant condition is coupled into the ring and transmitted along the ring waveguide^[13-15]. At this time, the Through port has not passed the electric field. According to this mechanism, optical devices with switching function are designed. The remaining electric field continues to transmit along the curved waveguide CW₁.When the optical signal is transmitted to the coupling region C₂, the electric field satisfying the resonance condition is coupled into the ring waveguide and transmitted along the ring waveguide, and the remaining electric field is output through the Through port. When the electric field in the ring is transmitted to the coupling region C₃, the electric field satisfying the resonance condition is coupled out of the ring waveguide and transmitted along the curved waveguide CW₂, while the remaining electric field continues to transmit along the ring waveguide into the coupling region C₄, and the electric field satisfying the resonance condition continues to transmit along the ring waveguide, while the remaining electric field is coupled to the curved waveguide CW₂ and output from the Drop port. The resonance condition can be expressed as :

$ m \cdot \lambda_0 = L \cdot n_{{\rm{eff}}} \quad,$ (10)

Where $m$ is the resonant series of the microring resonator, $\lambda_0$ is the wavelength transmitted in vacuum, $L$ is the circumference of the ring waveguide, and $n_{{\rm{eff}}}$ is the effective refractive index. In Fig.3, we can see that after the electric field of the input optical signal passes through the coupling region C₁, most of the electric field is coupled into the ring waveguide and transmitted along the ring waveguide. When passing through the coupling region C₂ and C₃, part of the electric field is transmitted along the ring waveguide, and the other part of the electric field is transmitted to the curved waveguide CW₂.When passing through the coupling region C₄, the electric field in the ring waveguide is partially coupled to the curved waveguide CW₂ and output from the Drop port. The remaining part of the electric field continues to transmit in the ring waveguide until the energy in the ring waveguide tends to be stable, which is consistent with the mechanism of electric field transmission in the micro-ring resonator. From the transmission spectrum of the standard microring resonator, the transmittance of the Through port can reach about 75 %, indicating that the electric field on the curved waveguide CW₁ can be mostly coupled into the ring waveguide in the coupling region C₂. The transmittance of the Drop port can reach 35 %, indicating that part of the electric field continues to resonate in the ring waveguide, and part of the electric field does not satisfy the resonant condition to couple out of the ring waveguide from the Drop port output. From the overall transmission spectrum, the spectral line is relatively stable and smooth, and has good filtering performance.

Fig.4 (color online) shows the electric field distribution of the microring resonator with different numbers of coupling regions. From Fig.4(a), it can be seen that when the ding-shaped micro-ring resonator has only a pair of coupling zones, the electric field is input from the Input port, most of the electric field is coupled into the ring waveguide, and resonance occurs along the ring waveguide. Only part of the unqualified electric field is output from the Drop port. In Fig.4(b), there are two pairs of coupling zones in the ding-shaped microring resonator. The strength of the electric field entering from the Input port decreases. When the coupling occurs in the four coupling zones, most of the electric field can be continuously coupled in the ring waveguide. Only a small part of the electric field does not meet the coupling condition and is output from the Drop port. In Fig.4(c), there are three pairs of coupling zones in the ding-shaped microring resonator. It can be seen from the diagram that when the electric field enters from the Input port, part of the electric field can be coupled into the ring waveguide and resonate all the time, and a small part of the electric field not meeting the resonance condition output from the Drop port.

Fig.5 (color online) shows the output spectra when the wavelength range of the input optical signal is 1.54~1.56 μm, and other parameters are unchanged. When the number of coupling regions is one, two and three pairs respectively, the influence of the number of coupling regions on the output spectral line of the ding-shaped microring resonator is simulated. Fig.5(a) shows that when the ding-shaped microring resonator has only a pair of coupling regions, four resonance peaks are generated in the working band. The transmittance of the Drop port reaches about 40 %, and the transmittance of the Through port reaches about 85 %. The output spectrum lines of the Drop port and Through port are smooth, and the light waves with the wavelength of 1.5428, 1.5445, 1.5532 and 1.5585 μm can be well filtered. It can be seen from Fig.5(b) that when the ding-shaped microring resonator has two pairs of coupling regions, six resonance peaks are generated in the working band. The transmittance of the Drop port reaches about 35 %, and the transmittance of the Through port reaches about 75 %. And the output spectral lines of the Drop port and the Through port are also relatively smooth, which can well filter the light waves with wavelengths of 1.5402, 1.5445, 1.5478, 1.5514, 1.5550, and 1.5587 μm. It can be seen from Fig.5(c) that when the ding-shaped microring resonator has three pairs of coupling regions, seven resonance peaks are generated in the working band. The transmittance of the Drop port reaches about 38 %, and the transmittance of the Through port reaches about 70 %. And the output spectral lines of the Drop port and the Through port are also relatively smooth, which can well filter the light waves with wavelengths of 1.5415, 1.5445, 1.5472, 1.5498, 1.5526, 1.5558, 1.5580 μm. As shown in Fig.5, due to the fluctuation of the light wave during transmission, the refractive index of the material in the light wave of different wavelengths is also different, and the increase of the coupling region affects the occurrence of two adjacent resonances, so the transmittance of the Drop port and the Through port fluctuates within a certain range.

In summary, this paper mainly analyzes the physical model of the ding-shaped microring resonator, and simulates the influence of the number of coupling regions of the ding-shaped microring resonator on the output spectrum. It is found that with the increase of the number of coupling regions of the microring resonator, the number of resonance peaks in the working band of 1.54~1.56 μm increases, which means that the density of resonance peaks in the working band increases, and the FWHM decreases. Due to the inverse relationship between the FWHM and quality factor Q, the larger the quality factor Q, the better the energy storage performance of the microring resonator, and the filtering function on specific wavelengths can be realized.