Optical Transmission Properties of Si3N4 Add-Drop Micro-Ring Resonator Induced by a Fabry–Perot Resonance Effect

To resolve the problem of miscellaneous peaks and improve the accuracy of data processing in micro-ring resonators (MRRs), we propose an optical transmission model based on a Fabry–Perot (F-P) resonance effect in a Si3N4 add-drop MRR-waveguide structure, which is analyzed using a coupled mode theory and stationary interference method. The analysis indicates the experimentally obtained miscellaneous peaks are mainly induced by the multiple reflections between the two end grating couplers, which form a F-P cavity. In addition, an anti-reflection film on the interface of the grating couplers is proposed to reduce the F-P resonance effect. This work could be useful to analyze optical transmission properties of other MRR-based structures.


Introduction
Silicon photonics has become one of the most promising photonic integration platforms in recent years [1,2]. This can be mainly attributed to the combination of a very high index contrast and the availability of complementary metal-oxide-semiconductor (CMOS) fabrication technology, which allows the use of electronics fabrication facilities to make photonic circuitry [3][4][5]. As a typical integrated on-chip optical device, the micro-ring resonator (MRR) is widely used as a wavelength selective device in the field of sensing, e.g., temperature [6][7][8], humidity [9][10][11], and refractive index [12][13][14], due to its highquality factor, narrow bandwidth, high integration, and other advantages. There are high requirements for the accuracy of the selected wavelength, including the peak intensity and position of the resonant peaks of MRR transmission. At this stage, there are a lot of works and experiments about the wavelength selection characteristics of MRR, and we notice that there are different degrees of miscellaneous peaks in the resonance spectrum of MRR transmission [15][16][17]. Compared with the resonance peaks of MRR, the miscellaneous peaks are disordered and relatively dense. The existence of these miscellaneous peaks inevitably affects the accuracy of wavelength selection, which leads to errors in the data analysis and processing results [18][19][20]. In the related experiments of MRR, Wirth et al. analyzed the source of the miscellaneous peaks and verified the existence of the Fabry-Perot (F-P) cavity [17]. Chao et al. analyzed the experimental results and showed that there is the F-P cavity effect, and compared with the transmission port, the background oscillation of the download port is smaller [18]. However, the corresponding work only states that the cause of the miscellaneous peaks may the formation of an F-P cavity between the waveguide grating couplers, and it lacks detailed theoretical analysis and experimental verification [17][18][19][20]. 2 of 10 In this paper, we investigate the optical transmission properties of a Si 3 N 4 adddrop MRR affected by a F-P cavity effect. The corresponding theoretical analysis and experimental measurements are carried out in detail. Meanwhile, we propose that an anti-reflection film on the interface (air and Si 3 N 4 ) of the grating couplers can be useful to reduce the F-P cavity effect to the MRR output.

Analysis of an Add-Drop MRR without F-P Cavity Effect
An add-drop MRR consists of a looped and a straight optical waveguide, as shown in Figure 1a. The optical signals are transmitted between the waveguides through evanescent wave coupling. The coupling resonance condition is expressed as: where n eff is the effective refractive index of the waveguide mode; R m is the micro-ring radius; m is the resonance series, taking a positive integer; λ m is the resonant wavelength. states that the cause of the miscellaneous peaks may the formation of an F-P cavity between the waveguide grating couplers, and it lacks detailed theoretical analysis and experimental verification [17][18][19][20].
In this paper, we investigate the optical transmission properties of a Si3N4 add-drop MRR affected by a F-P cavity effect. The corresponding theoretical analysis and experimental measurements are carried out in detail. Meanwhile, we propose that an anti-reflection film on the interface (air and Si3N4) of the grating couplers can be useful to reduce the F-P cavity effect to the MRR output.

Analysis of an Add-Drop MRR without F-P Cavity Effect
An add-drop MRR consists of a looped and a straight optical waveguide, as shown in Figure 1a. The optical signals are transmitted between the waveguides through evanescent wave coupling. The coupling resonance condition is expressed as: where neff is the effective refractive index of the waveguide mode; Rm is the micro-ring radius; m is the resonance series, taking a positive integer; λm is the resonant wavelength. In is the optical signal input port; Tran is the direct output port; Drop is the download port; Add is the load port. The spacing between looped optical waveguide and straight optical waveguide is Gap, and the length of straight waveguide is L. (b) Optical transmission schematic diagram based on waveguide F-P cavity. Es1, Es2…Esn are optical amplitude at different stages. r and t are the reflection coefficient and transmission coefficient of grating coupler interface, respectively.
E1, E2, E3, and E4 are the amplitude of signal In, Tran, Add, and Drop, respectively. E1a, E2a, E3a, and E4a are the amplitude in MRR. k1, k2, t1, and t2 are the coupling coefficients and transmission coefficients of the two coupling regions, respectively, and they are assumed In is the optical signal input port; Tran is the direct output port; Drop is the download port; Add is the load port. The spacing between looped optical waveguide and straight optical waveguide is Gap, and the length of straight waveguide is L. (b) Optical transmission schematic diagram based on waveguide F-P cavity. Es 1 , Es 2 . . . Es n are optical amplitude at different stages. r and t are the reflection coefficient and transmission coefficient of grating coupler interface, respectively. E 1 , E 2 , E 3 , and E 4 are the amplitude of signal In, Tran, Add, and Drop, respectively. E 1a , E 2a , E 3a , and E 4a are the amplitude in MRR. k 1 , k 2 , t 1 , and t 2 are the coupling coefficients and transmission coefficients of the two coupling regions, respectively, and they are assumed where φ = 2ϕ is the phase of a cycle.

Analysis of an Add-Drop MRR with a F-P Cavity Effect
In real condition, the reflection coefficient of Si 3 N 4 is not zero, there is inevitable multireflection phenomenon between two grating couplers. For the convenience of calculation, we only consider the reflection of the first interface of the grating coupler, that is, the signal only produces one reflection in the grating coupler. The optical transmission during the coupling region could be different from that mentioned in Section 2.1. Herein, we thought a F-P cavity is formed between two grating couplers. For simplicity, there are two suppositions: (1) the coupling occurs at point O; (2) the amplitude before point O and after point O is affected by parameter K, where K is the ratio of E 2 /E 1 in the model without F-P cavity effect. According to the Equation (5), the ratio K is expressed as: The transmission loss of the slab waveguide made of Si 3 N 4 is 0.045 dB/m from ultraviolet (UV) to infrared radiation (IR) [21,22], which is almost transparent, so the transmission loss of the straight waveguide can be ignored. As shown in Figure 1b, there are two steps to calculate the real transmission properties. Firstly, the combined field at point O generated by multiple reflections (induced by the waveguide F-P cavity) is the actual incident field E 1 of MRR. Secondly, we get the Drop signal according to the MRR transfer matrix mentioned in Section 2.1. Es 1 , Es 2 . . . Es i are the transmission amplitudes at different reflections, respectively. The amplitude at point O coupled to the MRR can be expressed as: where A is the amplitude of the incident light, δ 0 is the initial phase, and δ is the phase difference of the transmission field at two consecutive O points, where δ 1 = 2 π(nL)/λ. The sum of the combined field of O 1 , O 3 , O 5 . . . O i (coupled to micro-ring) can be calculated by: Therefore, the equivalent incident intensity is expressed as follows: where, ξ 1 , ξ 2 and ξ 3 is expressed as follows: where R = r 2 is the reflectivity and T = t 2 is the transmissivity. According to Equations (7) and (8), the intensities of MRR Drop and Tran are expressed as: Under ideal conditions, R = r 2 = 0 and T = t 2 = 1, there is no F-P cavity effect. nEquations (15), (19) and (20) can be considered as the traditional MRR model mentioned in Section 2.1.
To clearly show the analysis, simulations are carried out. According to the experimentally fabricated device, the parameters used in the simulation are as follows. The effective refractive index of Si 3 N 4 is 2.2, MRR radius R m is 10 µm, t 1 = t 2 = 0.8, L = 300 µm, the transmissivity T is 0.7, the MRR loss factor α = 0.6, and the reflectivity R is 0.3 when the absorption loss of the interface is ignored. We set the wavelength resolution (∆λ) in the simulation to be 0.  that when there is no F-P cavity effect, the wavelength selection is less affected by the wavelength resolution. When the F-P cavity effect exists, there are miscellaneous peaks in the Drop signal. By improving the wavelength resolution as ∆λ = 0.1 nm, the F-P cavity effect will be more obvious.
simulation to be 0.1 nm and 0.4 nm (the wavelength resolution of the ocean spectrometer (USB-4000) used in our experiment in Section 3 is 0.4 nm). The incident light has a spectrum of a Gaussian line shape with a central wavelength of 785 nm, as shown in Figure  2a. Based on Equations (8) and (19), the output of Drop without and with F-P cavity effect (Δλ = 0.1 nm and 0.4 nm) is shown in Figure 2b. It can be clearly seen from Figure 2c that when there is no F-P cavity effect, the wavelength selection is less affected by the wavelength resolution. When the F-P cavity effect exists, there are miscellaneous peaks in the Drop signal. By improving the wavelength resolution as Δλ = 0.1 nm, the F-P cavity effect will be more obvious.

Experimental Results and Discussion
To verify the above theoretical analysis, add-drop MRR samples with different radii and coupling gaps with the bus-waveguide were fabricated with electron beam lithography and reaction ion etching. Figure 3a-e present the scanning electron microscope (SEM) images of the fabricated Si3N4 waveguide structure. The length of the waveguide L is 300 µ m ± 3 µ m and the radius Rm is 10 µ m ± 0.1 µ m. Figure 3f-h are the energy depressive spectroscopy (EDS) images of the fabricated Si3N4 waveguide structure material. The results show that the waveguide material is Si3N4 and the substrate material is SiO2. The mass percentages of N, O, and Si in the measured grating region are 13%, 18%, and 69%.

Experimental Results and Discussion
To verify the above theoretical analysis, add-drop MRR samples with different radii and coupling gaps with the bus-waveguide were fabricated with electron beam lithography and reaction ion etching. Figure 3a-e present the scanning electron microscope (SEM) images of the fabricated Si 3 N 4 waveguide structure. The length of the waveguide L is 300 µm ± 3 µm and the radius R m is 10 µm ± 0.1 µm. Figure 3f  The experimental system is shown in Figure 4a. A broadband light source is coupled to the waveguide grating coupler using a focused fiber. The output is also coupled to the spectrometer orderly using a waveguide grating coupler and a focused fiber. The spectrometer is an ocean spectrometer with a wavelength resolution of 0.4 nm. A charge coupled device (CCD) microscope is used to determine the position information of focused fiber and waveguide. The experimental system is shown in Figure 4a. A broadband light source is coupled to the waveguide grating coupler using a focused fiber. The output is also coupled to the spectrometer orderly using a waveguide grating coupler and a focused fiber. The spectrometer is an ocean spectrometer with a wavelength resolution of 0.4 nm. A charge Sensors 2021, 21, 6370 6 of 10 coupled device (CCD) microscope is used to determine the position information of focused fiber and waveguide.
The experimental system is shown in Figure 4a. A broadband light source is coupled to the waveguide grating coupler using a focused fiber. The output is also coupled to the spectrometer orderly using a waveguide grating coupler and a focused fiber. The spectrometer is an ocean spectrometer with a wavelength resolution of 0.4 nm. A charge coupled device (CCD) microscope is used to determine the position information of focused fiber and waveguide.  Figure 4b-e. We can observe the miscellaneous peaks phenomenon through the experimental results, but the details cannot be clearly observed due to the insufficient resolution of the spectrometer used in the experiment. Therefore, based on Equation (15), we simulate the equivalent incident light of F-P cavity effect with higher wavelength resolutions for further analysis.
The wavelength resolution is improved to 0.01 nm. The simulation results of the equivalent incident light of F-P cavity effect in the wavelength range of 770-800 nm are shown in Figure 5a. Figure 5b shows the detailed information of the wavelength range of 782.5-787.5 nm when Δλ is 0.01 nm. The simulation results show that the equivalent incident light of F-P cavity effect has free spectral range (FSR) and is the same as the standard  Figure 4b-e. We can observe the miscellaneous peaks phenomenon through the experimental results, but the details cannot be clearly observed due to the insufficient resolution of the spectrometer used in the experiment. Therefore, based on Equation (15), we simulate the equivalent incident light of F-P cavity effect with higher wavelength resolutions for further analysis.
The wavelength resolution is improved to 0.01 nm. The simulation results of the equivalent incident light of F-P cavity effect in the wavelength range of 770-800 nm are shown in Figure 5a. Figure 5b shows the detailed information of the wavelength range of 782.5-787.5 nm when ∆λ is 0.01 nm. The simulation results show that the equivalent incident light of F-P cavity effect has free spectral range (FSR) and is the same as the standard F-P cavity formed by the two end grating couplers. With the same parameters, the FSR of the equivalent incident light and standard F-P cavity is equal to 0.46 nm (λ 1 = 783.83 nm).  The F-P cavity effect will be different due to different interface reflectivities of grating couplers. To characterize the F-P cavity effect, we set γ(λ) to represent the intensity error value at the resonance wavelength, γ(λ) = (IMRR − IFP-MRR)/IMRR, where IFP-MRR and IMRR are the signal intensity with and without F-P cavity effect, respectively. The intensity error The F-P cavity effect will be different due to different interface reflectivities of grating couplers. To characterize the F-P cavity effect, we set γ(λ) to represent the intensity error value at the resonance wavelength, γ(λ) = (I MRR − I FP-MRR )/I MRR , where I FP-MRR and I MRR are the signal intensity with and without F-P cavity effect, respectively. The intensity error value of multiple resonance wavelengths is calculated by γ(R) = (γ(λ 1 ) + γ(λ 2 ) + . . . γ (λ n ))/n. We simulate the Drop signal under different reflectivities R (0/0.14/0.5/0.7), where R = 0.14 is the reflectivity of the interface between air and Si 3 N 4 , and the results are shown in Figure 6. The F-P cavity effect will be different due to different interface reflectivities of grating couplers. To characterize the F-P cavity effect, we set γ(λ) to represent the intensity error value at the resonance wavelength, γ(λ) = (IMRR − IFP-MRR)/IMRR, where IFP-MRR and IMRR are the signal intensity with and without F-P cavity effect, respectively. The intensity error value of multiple resonance wavelengths is calculated by γ(R) = (γ(λ1) + γ(λ2) + … γ (λn))/n. We simulate the Drop signal under different reflectivities R (0/0.14/0.5/0.7), where R = 0.14 is the reflectivity of the interface between air and Si3N4, and the results are shown in Figure  6. Here, we select five resonance wavelengths in the signal center, which are λ = 776.57, 780.96, 785.40, 789.89, and 794.43 nm. From the simulation results, it can be seen that when R = 0.14, 0.5, and 0.7, γ (0.14) = 27.07%, γ (0.5) = 70.55%, and γ (0.7) = 87.61%. When R = 0.5 and R = 0.7, we can hardly distinguish the position of its resonance wavelength, and MRR cannot play a role in wavelength selection. When the reflectivity decreases, the influence of the waveguide F-P cavity on the MRR transmission signal decreases. According to the analysis, the F-P cavity has a great influence on the wavelength selection characteristics of the MRR. When the interface of the coupled grating has a large reflectivity, the position of the resonance peak cannot even be distinguished. Here, we select five resonance wavelengths in the signal center, which are λ = 776.57, 780.96, 785.40, 789.89, and 794.43 nm. From the simulation results, it can be seen that when R = 0.14, 0.5, and 0.7, γ (0.14) = 27.07%, γ (0.5) = 70.55%, and γ (0.7) = 87.61%. When R = 0.5 and R = 0.7, we can hardly distinguish the position of its resonance wavelength, and MRR cannot play a role in wavelength selection. When the reflectivity decreases, the influence of the waveguide F-P cavity on the MRR transmission signal decreases. According to the analysis, the F-P cavity has a great influence on the wavelength selection characteristics of the MRR. When the interface of the coupled grating has a large reflectivity, the position of the resonance peak cannot even be distinguished.

Optimization Scheme
A layer of anti-reflection film with a refractive index between air and Si 3 N 4 is introduced on the side wall of the grating coupler (there is the same procession to CG2), as shown in Figure 7a. The grating period is d, and the thickness of the anti-reflection film is Λ. Figure 7b shows that the phase difference between the two reflected lights is δ (δ = 2π (2nΛ)/λ). The reflection coefficients at the two interfaces are r 1 and r 2 , which are expressed as:

Optimization Scheme
A layer of anti-reflection film with a refractive index between air and Si3N4 is introduced on the side wall of the grating coupler (there is the same procession to CG2), as shown in Figure 7a. The grating period is d, and the thickness of the anti-reflection film is Λ. Figure 7b shows that the phase difference between the two reflected lights is δ' (δ' = 2π (2nΛ)/λ). The reflection coefficients at the two interfaces are r1 and r2, which are expressed as: − − nn nn r = , r = n + n n + n According to the air refractive index (n0 = 1) and Si3N4 refractive index (n2 = 2.4), the ideal refractive index of anti-reflection film is 1.549. However, in fact, we cannot find the material with exactly the same refractive index, so we use SiO2 as the material to prepare the film, and nSiO2 is 1.45. According to Equation (19), we set the thickness of SiO2 antireflection film as 150 nm, and the simulation results of its transmission spectrum are shown in Figure 8a. When the wavelength range is 0.448~0.911 µ m, the R' is less than 0.05. The intensity combination I of the reflected light I 1 and I 2 can be expressed as: When the optical thickness of the anti-reflection film is one quarter of a certain wavelength, δ = π, then the directions of the two vectors are opposite, and the vector combination is the smallest. Under ideal conditions, r 1 = r 2 , the combined reflection coefficient is the smallest. At this time r = 0 (R = 0), n 1 can be expressed as: According to the air refractive index (n 0 = 1) and Si 3 N 4 refractive index (n 2 = 2.4), the ideal refractive index of anti-reflection film is 1.549. However, in fact, we cannot find the material with exactly the same refractive index, so we use SiO 2 as the material to prepare the film, and n SiO2 is 1.45. According to Equation (19), we set the thickness of SiO 2 anti-reflection film as 150 nm, and the simulation results of its transmission spectrum are shown in Figure 8a. When the wavelength range is 0.448~0.911 µm, the R is less than 0.05. According to Equation (16), when R = 0.05, the simulation results of Drop signal with and without F-P cavity effect are shown in Figure 8b, and the partial enlarged view is shown in Figure 8c. According to Section 3.1, the resonance wavelength intensity error value γ (0.05) is 11.08%, and the influence of waveguide F-P on the MRR output signal is almost nonexistent.
When the optical thickness of the anti-reflection film is one quarter of a certain wavelength, δ' = π, then the directions of the two vectors are opposite, and the vector combination is the smallest. Under ideal conditions, r1 = r2, the combined reflection coefficient is the smallest. At this time r' = 0 (R' = 0), n1 can be expressed as: According to the air refractive index (n0 = 1) and Si3N4 refractive index (n2 = 2.4), the ideal refractive index of anti-reflection film is 1.549. However, in fact, we cannot find the material with exactly the same refractive index, so we use SiO2 as the material to prepare the film, and nSiO2 is 1.45. According to Equation (19), we set the thickness of SiO2 antireflection film as 150 nm, and the simulation results of its transmission spectrum are shown in Figure 8a. When the wavelength range is 0.448~0.911 µ m, the R' is less than 0.05. According to Equation (16), when R' = 0.05, the simulation results of Drop signal with and without F-P cavity effect are shown in Figure 8b, and the partial enlarged view is shown in Figure 8c. According to Section 3.1, the resonance wavelength intensity error value γ (0.05) is 11.08%, and the influence of waveguide F-P on the MRR output signal is almost nonexistent.

Conclusions
An optical transmission model based on a waveguide Fabry-Perot (F-P) cavity effect in an add-drop MRR waveguide structure was proposed and analyzed in detail. The experimental results show the phenomenon of miscellaneous peaks induced by the multiple reflections between the two end grating couplers, and the FSR can be observed after the resolution is improved. In addition, the F-P cavity effect will be different due to the different interface reflectivities. Meanwhile, according to the principle of interference cancellation, we propose that an anti-reflection film on the interface of the grating couplers can be useful to reduce the F-P cavity effect to the MRR output. The simulation results show that when a SiO 2 film with an optical thickness of 150 nm is used, a transmittance greater than 95% can be achieved in the wavelength range 0.448~0.911 µm. Combined with the optical transmission model based on a waveguide F-P cavity effect, the intensity error value at resonance wavelength γ = 11.08%, and the spectral signal of MRR Drop is almost not affected by the waveguide F-P cavity. This work could be useful to improve optical transmission properties of other waveguide structures.
Author Contributions: This research was conceptualized by J.Z. as a supervisor. The experiment and analysis were implemented by X.C., P.H. and L.G. In addition, X.C. wrote the first draft of this paper. This paper was reviewed and edited by X.G., N.W., Y.Z. and J.Z. All authors have read and agreed to the published version of the manuscript.