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Abstract
Fano resonance is considered to be a promising approach for integrated sensing. However, achieving and controlling Fano resonance lineshapes on ultra-compact chips remains a challenge. In this article, we propose a theoretic model based on the transfer matrix method (TMM) to quantitatively interpret the impact of a micro-reflective unit (MRU) etched in the straight waveguide of a microring resonator (MRR). Numerical calculations and FDTD simulations indicate that the size and position of the MRU can be used to control the Fano resonance lineshape. Since the MRU is etched in the coupling region, the reflection caused by the MRU will significantly enhance the intensity of the counter-clockwise (CCW) mode in the microring. When applied to a single nanoparticle sensing, clockwise (CW) and CCW modes will couple due to a single nanoparticles or rough cavity walls, resulting in a sharp shift and split of the Fano lineshape. The proposed model for single nanoparticle sensing is described by the scattering matrix, and the calculations show a well matches with FDTD simulations. The results show that the model proposed in this paper provides a new theoretical basis for controlling Fano resonance lineshape and presents a new approach for the integrated sensing of silicon photonic devices with high sensitivity.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As one of the most important components in on-chip integrated optical technology, microring resonator (MRR) has shown a wide range of application prospects in optical communication, filtering, sensing, non-reciprocal transmission, optical switching, nonlinear optics and, more recently, quantum optics [1–6]. The transmission spectrum of the MRR usually shows a symmetric Lorentzian resonance lineshape, and devices based on such lineshape have the disadvantages of low extinction ratio, small resonant slope, and low tuning level. If the narrowband Lorentzian response is superimposed to a broadband background field, an asymmetric resonance lineshape occurs, and this misaligned resonance lineshape is also called Fano resonance lineshape [7]. Compared with the symmetric Lorentzian lineshape, the asymmetric Fano lineshape has a higher quality-factor and a steeper slope-rate [8], and its transmission spectrum coefficient has a larger range of variation and a sharper change trend. These excellent characteristics determine that Fano resonance has great potential for applications in high-contrast low-power fast optical switches [9], modulators with high modulation lineshape [10], narrow bandwidth filters [11], non-reciprocal optical devices [3] and high-sensitivity sensors [8,12–14].
Fano resonances can be realized in many configurations, such as photonic crystal cavity MRR [13], cascading and nested MRRs [14–17], as well as MRR coupled with a Fabry-Perot cavity [18–20] or a Mach-Zehnder interferometer [21,22]. However, the device structure in these schemes is complex and the size is not conducive to compact integration. Besides, the fabrication accuracy of these devices is very high. Very recently, Gan et al proposed a compact configuration, by simply etching an air hole in the straight waveguide of a MRR, which is possible to form an asymmetric Fano resonance lineshape [8,23,24]. However, due to the existence of the micro-reflective unit (MRU) formed by an air hole, the structure will have light transmitted in both directions. The modeling in previous literatures only considered the forward propagation of light, and ignored the impact of reverse light. Besides, the quantification of the effect of the location and size of etched hole on resonance intensity and lineshape remains unexplored [8,18,23,24]. However, the reverse light generated by the MRU will greatly affects other devices, and cause damage to the light source [25]. It is also essential to analyze the clock-wise (CW) and counter-clockwise (CCW) modes in the microring cavity. Some studies have shown that scattering will cause the CW and CCW mode to couple in the cavity, resulting in the splitting of the resonance lineshape, which can be used for nanoparticle detection [26,27].
In this article, we established a refinement theoretic model based on the transfer matrix method (TMM) to describe the effect of etching an MRU on the straight waveguide of an MRR. The theoretic model quantifies the changes in transmission and reflection lineshape and intensity caused by the MRU position and size. Compared with the single nanoparticle detection of MRR, the MRU induces a strong CCW mode in the cavity. This makes the proposed model very sensitive to the reflections, both the single nanoparticle and rough cavity walls can lead to sharp changes in resonance lineshapes. Consequently, the structure can be applied to highly sensitive single nanoparticle detection. We predict and observe the shifting and splitting of Fano lineshape through numerical calculation and FDTD simulation. The model proposed in this paper provides a new basis for the control and application of microcavity Fano resonance lineshape.
2. Theoretic model
To facilitate a comparison with existing methods, the MRU in our model is also formed by a circular hole located on a straight waveguide, as shown in Fig. 1(a). In a real device, the MRU can be a rectangular, square, elliptical, or other micro structures of arbitrary length and width. To make a generalized theoretic model, we assume that the width of the waveguide is $w$, the length and width of the MRU structure are $d$, and $h$ respectively. ${E_{\textrm{cw - in}}}$ is the amplitude of the light partially incident onto the MRU structure, ${E_{\textrm{cw - out}}}$ is the amplitude of the outgoing light. Due to the presence of MRU, there will actually be light propagating in both directions in the waveguide. Therefore, ${E_{\textrm{ccw - in}}}$ is used to denote the amplitude of the reverse incident light and ${E_{\textrm{ccw - in}}} = 0$, ${E_{\textrm{ccw - out}}}$ denotes the amplitude of the backward outgoing light.
Fig. 1. (a) Microring resonator with an MRU; (b) The MRU is located on the coupling region; (c) The MRU is located on the left side of the coupling region; (d) The MRU is located on the right side of the coupling region.
According to the TMM method, when the MRU is located on the coupling region, as shown in Fig. 1(b), the relationship between the input and output amplitudes is given by [18]
To evaluate the coupling affected by the MRU, as shown in the white shaded region in Fig. 1(b), the transmission and reflection lineshapes of the light can be calculated as follows:
$\varphi = {\varphi _1} + {\varphi _2} = 2\pi d{n_{\textrm{MRU}}}/\lambda$ is the total phase shift caused by the MRU, ${n_{\textrm{MRU}}}$ is introduced to describe the refractive index of the MRU. In a straight waveguide, when the width of the MRU is less than the straight waveguide ($h < w$), there will inevitably be some light that does not flow into the MRU but directly couples with the MRR. Therefore, it is crucial to distinguish the ratio of energy that passes through the MRU, which has not been discussed before. To address this, a dimensionless parameter $\xi$ is introduced to characterize this ratio, and the transmission and reflection lineshapes of the overall structure can be calculated as:
As shown in Fig. 1(c), when the MRU locates on the left side of the coupling region, the TMM method gives the following relations:
As shown in Fig. 1(d), when the MRU locates on the right side of the coupling region, the TMM method gives the following relations:
3. Calculations and simulations
3.1 Influence of MRU location and size on the lineshape
The parameters in the proposed model were approximately set as follows:${r_1} = {r_2} = 0.5$, $\tau = {a_{\textrm{rt}}} = 0.95$, $R = 5\mathrm{\ \mu m}$, ${n_{\textrm{eff}}} = 4.2$. In order to determine the relationship between $\xi $ and $d$, FDTD was used to simulate two structures which shown in the illustration of Fig. 2(a). The waveguide width and height were set to $500\textrm{ nm}$ and $220\textrm{ nm}$, a continuous light with a wavelength of 1550 nm was fed from $S$, ${\textrm{M}_1}$ and ${\textrm{M}_2}$ are frequency domain field and power monitors. As shown in Fig. 2(a), ${\textrm{M}_1}$ obtains the $|E{|^2}$ at different diameters, it is evident that the energy will be divided into three parts in the etched hole region, each of which will be coupled separately to the microring. In the model that proposed in this article, we only need to determine the energy incident in the white shaded region of the waveguide. For this purpose, we made another monitor ${\textrm{M}_2}$ on the waveguide that coincides with the size of the hole, and as the monitor length increases, the energy incident into the etched region is determined.
Fig. 2. (a) The relationship between $\xi $ and $d$; (b) MRU was etched in the left of coupling region; (c) MRU was etched in the coupling region; (d) MRU was etched in the right of coupling region.
The fitted relations between $\xi $ and $d$ are substituted into previously derived formulas, and the corresponding transmission and reflection lineshapes are shown in Fig. 2(b-d), where the changes in resonance lineshape and intensity can be clearly observed. According to Fig. 2(c), when the hole is etched in the coupling region, with the gradual increase of the etched hole diameter, the phase delay between the discrete resonance mode and the continuum propagating mode also increased. Thus, the transmission lineshape gradually transitions from a Lorentzian lineshape to a Fano lineshape, then degenerate to a Lorentzian lineshape, and once again converts to a Fano lineshape. In addition, we observed a split in the reflection lineshape and found that the reflection lineshape can also transition between Fano and Lorentzian resonant lineshape. When the hole is not etched in the coupling region, there is no phase delay between the discrete resonance mode and the continuum propagating mode. Consequently, their transmission both are Lorentzian lineshape, but the reflection lineshape are different. As shown in Fig. 2(b) and Fig. 2(d), when the hole is etched in the left side of the coupling region, the reflection lineshape are straight, when the hole is etched in the right side of the coupling region, the reflection lineshape are Lorentzian lineshape, respectively.
To validate the correctness of our numerical calculations, Lumerical software was used for FDTD simulation verification. The diameter of the micro-reflective hole ranged from $0\textrm{ nm}$ to $500\textrm{ nm}$, with incremental steps of $50\textrm{ nm}$. The distance away from the coupling center is set at $\textrm{5 }\mu \textrm{m}$ when micro-reflective hole locates in the left and right sides. Figure 3(b) and Fig. 3(c) demonstrate the simulation results when the micro-reflective hole was etched in the coupling region and in the right side of the coupling region, respectively. These results verify all our calculation lineshapes, including the transition between Lorentzian lineshape and Fano resonance lineshape, which is caused by different locations and sizes of the MRU. However, when the micro-reflective hole was etched in the left side of the coupling region, the reflection lineshape at the resonance position deviated from the predicted straight line, showing a slight resonance lineshape. This deviation can be attributed to the presence of reflected light within the cavity due to non-fine meshes in FDTD simulation. Theoretically, when the grid is infinitely fine, the simulation results can infinitely approach the theoretical results. Notably, this finding is consistent with practical limitations in processing and manufacturing, as achieving a completely smooth side wall is technically unfeasible, we have discussed this point in Ref. [28].
Fig. 3. FDTD simulation result. (a) The micro-reflective hole is etched in the left of coupling region; (b) The micro-reflective hole is etched in coupling region; (c) The micro-reflective hole is etched in the right of coupling region.
The refinement model we constructed shows more detailed lineshape changes in the MRR, quantitatively describing the relationship between transmission (reflection) intensity and MRU diameter. When the MRU is etched in coupling region, due to the changes in discrete and continuous mode phase caused by different diameters of etched MRU, the Fano lineshape also exhibits corresponding changes. According to Fano formula, the asymmetric parameter q can describe the interaction between these two states, and $q = \cot (\varphi )$ [7,28]. Thus, the q factor and the phase shift of the continuum $\varphi $ should follow changes with $\pi $ as the period. When $\varphi = 0$, that means there is no etched hole in the straight waveguide and no phase delay between two states, which caused the lineshape is Lorentzian. When $\varphi = \pi /2$, interference cancellation occurs between the discrete and continuous states, and the lineshape should also be Lorentzian. As shown in Fig. 4(a), we can observe destructive interference and the transformation of Fano $q$ value in our model, which has not been mentioned clearly in previous research. We plotted the slope (SR) and extinction ratio (ER) the Fano lineshape in $\lambda = 1552.3\textrm{ nm}$ via Eq. (4). It is evident that when the phase shift $\varphi $ is $\pi /4$, there is a maximum interference between two states and SR reaches its maximum value. When the phase shift $\varphi $ is $\pi /2$, SR reaches its minimum value and the Fano resonance lineshape degenerates into Lorentzian resonance lineshape. ER indicates changes in the strength of the resonance lineshape. As characterizations of the resonance line shape, SR and ER can associate the resonance lineshape with the $d$ and $\varphi $, allowing us to select the appropriate etch size to control the Fano resonance lineshape based on the desired SR and ER values.
Fig. 4. (a) Fano parameter q is cotangent of the phase-shift $\varphi $ with a period of $\pi $, the insets show the Fano profiles at different $\varphi $ as well as the extracted q factors. SR and ER is calculated in different diameter circular holes; (b) The influence of MRU location on Fano lineshape.
As mentioned earlier, the position of MRU may not be accurately etched at the center of the coupling region due to machining limitations. Therefore, we utilized FDTD simulations to analyze the influence of MRU horizontal deviation from the coupling center on the resonance lineshape. Figure 4(b) illustrates this situation, where the deviation of the MRU (450 nm) from the coupling center leads to the degeneration of the Fano resonance line into a Lorentzian resonance lineshape, but a distance of more than $\mathrm{2\ \mu m}$ is required to achieve this. Considering the separate analysis of coupling energy in the vertical direction, vertical movement will not significantly affect the resonance lineshape, and there is less room for vertical movement. The above analysis indicates that the horizontal movement of the MRU in the straight waveguide can control the resonance lineshape and the device demonstrates a high manufacturing tolerance rate in achieving the Fano resonance lineshape.
3.2 Single nanoparticle sensing based on Fano resonance
Previous studies have demonstrated that rough cavity walls and nanoparticles can induce the generation of CCW mode and cause the coupling between CCW and CW modes in the cavity [26,27]. However, the CCW mode generated by the rough cavity walls and nanoparticles is usually weak, resulting in non-intuitive splitting of the resonance lineshape, which limits the sensitivity of MRR based sensors. When the MRU is etched in the coupling region, the reflection lineshape shown in Fig. 2(c) and Fig. 3(b) indicates that the reflection resonance intensity increases with the MRU size, which means the intensity of the CCW mode in the microring is also significantly enhanced. When the device is applied for sensing, CW and CCW modes will be coupled due to the rough cavity walls and nanoparticles, resulting in a sharp change in Fano resonance lineshape. This finding reveals that etching an MRU in the MRR coupling region can improve the device performance in nanoparticles detection, expanding the application of Fano resonance for highly sensitive sensing.
Setting a larger mesh size in the FDTD simulation can achieve a characterization of cavity wall roughness and the splitting of the resonance lineshape can be observed. As shown in Fig. 5(a), which means that there is a coupling between CW and CCW modes in the cavity due to reflection of rough cavity walls. This allows our model to be applied to highly sensitive single nanoparticle detection, as the particles also cause reflections and coupling between CW and CCW. So here we describe the influence of single nanoparticle and the rough cavity wall on the Fano lineshape according to the nanoparticle scattering matrix, which can expend the application of microring resonator with etched MRU in single particle sensing. In our scattering matrix formalism, the influence of a single nanoparticle and rough cavity wall is quantified by its reflection ${r_\textrm{p}}$ and transmission ${t_\textrm{p}}$, respectively. In fact, the appearance of nanoparticle only affects ${T_{\textrm{MRR}}}$ in Eq. (1), Fig. 5(b) shows this model and ${T_{\textrm{MRR}}}$ will be corrected accordingly [29,30].
Fig. 5. (a) Lineshape splitting due to reflection; (b) Microring resonator with MRU in nanoparticle sensing.
Fig. 6. (a) The relationship between the ${r_\textrm{p}}$ and the diameter of the nanoparticle ${d_p}$; (b) The effect of single nanoparticle on the Fano lineshape numerical calculation; (c) The effect of single nanoparticle on the Fano lineshape FDTD simulation.
For this purpose, we can numerically calculate the effect of single nanoparticle on the Fano lineshape under different diameters, as shown in Fig. 6(b), where the diameter of the micro-reflective hole was chosen to be at $200\textrm{ nm}$. It can be clearly observed that the resonance spectral line shifts when the nanoparticle is small, but splits when it is large. The simulation is shown in Fig. 6(c), which is consistent with our theoretical calculations. It should be noted that, due to the size limitation of the simulation grid and the approximation of numerical calculation parameters, SR and ER in Fig. 6(b) and Fig. 6(c) cannot achieve the absolute consistency.
4. Conclusion
In conclusion, a refinement theoretic model has been proposed to interpret the impact of an MRU etched in the straight waveguide of an MRR. The model comprehensively considers the influence of the light transmitted in both directions within the structure, addressing the limitations of previous theories. Through numerical calculations and FDTD simulations, we have analyzed the control of Fano resonance lineshape by the MRU and illustrated the mechanism of Fano resonance lineshape shifting and splitting caused by a single nanoparticle. We confirmed that the MRU structure can significantly enhance the intensity of the CCW mode in the cavity and demonstrated the application of Fano resonance lineshape in single nanoparticle sensing. It is hoped that the theoretical work can pave the way for using MRR with an MRU as a precise sensing platform and provide a new basis for controlling the Fano resonance lineshape on ultra-compact chips.
Funding
National Key Research and Development Program of China (2017YFF0107502, 2018YFE0204000); National Natural Science Foundation of China (11874132, 12074094).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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