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Abstract
By using the transfer matrix method, we analyze the optical properties of periodically sparse patterned microring resonators in the resonator-waveguide system. The dispersion relation of a periodically sparse patterned microring is investigated theoretically. Two kinds of modes supported by the periodically sparse patterned microring resonators, traveling modes and stationary modes, are found. We also derive the reflectivity of the microring resonator side-coupled to a waveguide and find that just the stationary modes lead to total reflections. Traveling modes do not reflect, which can be used to extend free spectral range. All the transmission properties are confirmed by the finite-difference time-domain method numerically.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microring (or microdisk) resonators, as a key component in integrated photonic devices, have gained worldwide interest in the past few decades due to the potential use for channel drop filters [1–4], biosensors [5–10] and slow light [11–14]. Periodically patterned microring resonators made out of coupled-resonator optical waveguide (CROW) [15], have been intensively studied numerically and experimentally [3,10–14,16], especially the slow-light dispersion [10–14] and mode splitting properties [11–13,16]. The slow-light dispersion of this kind of microring resonators, originated from the flattening of the photonic bands near the band edge, leads to strong optical confinement which can be used to enhance biosensing [6–10]. For the other kind of microring resonator with or without scattering (or surface roughness), many efforts have been also paid to investigate transmission, filtering and sensing properties [5,17–21]. Among these works, scatterers periodically arranged on the top of microring resonators are designed to suppress most of resonant modes supported by the microring, which are also used to extend the free spectral range and enhance the sensitivity [5].
For periodically patterned microring resonators, each unit segment can be treated as a cavity, and the nearest cavities are coupled with each other via an evanescent optical field [15]. All the modes supported by this kind of resonators are stationary modes, and the generated modes are split when the symmetry of the coupling system is broken [16]. However, for periodically sparse patterned microring resonators, the light propagates along the waveguide between scatterers as a normal guided wave. Due to the lack of systematical analysis of this kind of resonator-waveguide coupling system, many specific and important characteristics, the effects of the rotation of the patterned microring resonators, the existence of the non-reflective traveling modes, are not studied in detail yet.
In this work, we direct our intention to modeling the periodically sparse patterned microring resonators (PSPMR) by using the transfer matrix method and two kinds of modes supported by the PSPMR are found. The filtering properties of the PSPMR side-coupled to a waveguide are investigated analytically, and conditions for reflections which originates from the excitation of stationary modes, are obtained. The free spectral range of the filtering can be extended due to the existence of non-reflective traveling modes located between stationary modes. Our analysis also shows that rotation of the patterned microring resonator does not change the transmission profiles. All the transmission properties are confirmed by the finite-difference time-domain method numerically.
2. Dispersion relations of PSPMR
Figure 1(a) shows the structure of the optical system investigated, which consists of a slab waveguide with sparse patterned scatterers (for example, microholes or surface roughness). When the spacing between scatterers is larger than the optical wavelength considered, the light can propagate along the waveguide segment as a normal guided wave, which is different from other type of periodic dielectric waveguide [22], or CROW [15], where the energy of the guided modes couple and propagate via evanescent wave. In order to analyze the dispersion relations of the system, scattering matrix [23,24] is used to get scalability of the system. The relationships of the amplitudes of the incoming (${P_{ + ik}}$, $k = 1,2$) and outgoing (${P_{ - ik}}$) waves between ${i^{th}}$ and ${(i + 1)^{th}}$ scatterers are expressed as
where and ${M_i}$ is the transfer matrix
Fig. 1. Illustrations of (a) slab waveguide with sparse patterned scatterers indicated by white circles and (b) periodically sparse patterned microring.
For simplicity and without sacrificing the physics, here we just consider the periodically sparse patterned optical waveguide (PSPOW), i.e., ${t_i} = t$, ${r_i} = r$, ${d_i}\textrm{ = }d$, ${\varphi _i} = \varphi$, ${W_i}\textrm{ = }W$ and ${M_i} = M$. For PSPMR made out of PSPOW with N units as shown in Fig. 1(b), we can treat the PSPMR as end-to-end PSPOW, and from Eq. (1), the mode amplitudes can be expressed as
By solving the secular equation of Eq. (6), we get the dispersion relation of the microring with $N$ scatterers
From Eq. (7), we find that the PSPMR dispersion relation is discretized due to finite number of scatterers and there are two kinds of modes supported by the PSPMR: (1) when ${{2m} \mathord{\left/ {\vphantom {{2m} N}} \right.} N}$ is an integer, which means that $\cos \varphi = t$ or $\cos \varphi ={-} t$, the modes are stationary waves located at the edge of the bandgap of the PSPOW; (2) when ${{2m} \mathord{\left/ {\vphantom {{2m} N}} \right.} N}$ is not an integer, the modes supported by the PSPMR are traveling waves rotating along the microring where the backward scattering waves are cancelled due to the period of the scatterers.
3. Transmission analysis
For PSPMR with a scatterers side-coupled to a bus waveguide (the coupling region is free from scatterers), as shown in Fig. 2, the amplitudes of the incoming and outgoing waves can be express by the coupling matrix [23]
Fig. 2. Schematic of the periodically sparse patterned microring resonators side- coupled to waveguide. (a) Structure of the coupling system. The optical system consists of a bus waveguide and microring resonator with periodically sparse patterned air holes. The width of the waveguide and the microring are set to be a ($a$ is the arbitrary unit of length) and the radii of the microring and the air holes are $9a$ and $0.12a$. The gap between the waveguide and the microring is $0.8a$. (b) Modeling of the coupling system shown in (a).
When an electromagnetic wave is incident from the input port, i.e., ${S_{ + 2}} = 0$, as shown in Fig. 2, from Eqs. (8)–(10), the reflection of the coupling system can be expressed as
It should be mentioned that, although in Eq. (10), the coupling between the waveguide and the PSPMR are positioned by the phase shifts ${\varphi _{01}}$ or ${\varphi _{N0}}$, there is no the phase shift information in Eq. (11), i.e., the rotation of the PSPMR does not change the profiles of the reflection spectra. These coupling properties are conserved even the number of reflective elements larger than one [28]. As shown in Fig. 2(b), the coupling plane can be chosen anywhere provided that scatterers are not directly coupled to the bus waveguide, which simplifies the fabrication of the microring-based devices.
Total reflection occurs when ${D_N}(\theta ) = 0$, so we get the condition for total reflections
From Eqs. (13)–(14), we find that (1) the total reflection modes supported by the PSPMR are stationary modes located in the bandgap of the PSPOW due to $|{\cos \varphi } |> t$. For traveling modes, no total reflection occurs since the total reflective condition cannot be met. And (2) three types of reflectivity profiles can be obtained. For an odd number N, when ${\tau ^{\frac{1}{N}}} + {\tau ^{ - \frac{1}{N}}}$ is less than, equal to or larger than $2{t^{ - 1}}$, two, one, or no total reflection dips can be obtained in phase region $(2m - 1)\pi \le \varphi < (2m + 1)\pi$, respectively. While for an even number N, there are four, two or no total reflection dips.
In order to analysis the transmission properties of the optical system as shown in Fig. 2, we plot in Fig. 3 the transmittance of the coupling structure calculated by Eq. (11) with scatterer numbers (a) $N = 1$, (b) $N = 2$, (c) $N = 3$, (d) $N = 4$, where we assume a linear dispersion relation for the mode supported by the waveguide between two scatterers, i.e., $\beta = {\omega \mathord{\left/ {\vphantom {\omega \upsilon }} \right.} \upsilon }$, where $\upsilon$ is the phase velocity of the waveguide mode. Here, we choose ${\upsilon ^{ - 1}} = 4.0{c^{ - 1}}$ ($c$ is the light velocity in vacuum) for the waveguide. In the calculation, we also set the dependence of the parameters t and $\kappa$ on frequency as $t\textrm{ = }1 - 0.01\omega$ and $\kappa = {{0.96} \mathord{\left/ {\vphantom {{0.96} {\omega - }}} \right.} {\omega - }}0.775$, respectively. From Fig. 3(a) where $N = 1$, we find that, there is a resonant dip at each $\varphi = 2m\pi$. At $\varphi = 40\pi$, $({\tau + {1 \mathord{\left/ {\vphantom {1 \tau }} \right.} \tau }} ){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ = }1$, there is one total reflection. And at other resonant dips, from Eq. (13), no total reflections occur due to $({\tau + {1 \mathord{\left/ {\vphantom {1 \tau }} \right.} \tau }} ){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ > }1$. When $N = 2$, there is just one total reflection at $2\varphi = 34\pi$ as shown in Fig. 3(b), from Eq. (14), we find that $\left( {\sqrt \tau + {1 \mathord{\left/ {\vphantom {1 {\sqrt \tau }}} \right.} {\sqrt \tau }}} \right){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ = }1$. When $2\varphi < 34\pi$, $\left( {\sqrt \tau + {1 \mathord{\left/ {\vphantom {1 {\sqrt \tau }}} \right.} {\sqrt \tau }}} \right){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ > }1$, there is no total reflections, while for $2\varphi > 34\pi$, $\left( {\sqrt \tau + {1 \mathord{\left/ {\vphantom {1 {\sqrt \tau }}} \right.} {\sqrt \tau }}} \right){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ < }1$, the modes supported by PSPMR split, and two total reflection dips are obtained near $2\varphi \textrm{ = }36\pi$, $38\pi$ and $40\pi$ as shown in Fig. 3(b). When $N = 4$, $\left( {\sqrt[4]{\tau } + {1 \mathord{\left/ {\vphantom {1 {\sqrt[4]{\tau }}}} \right.} {\sqrt[4]{\tau }}}} \right){t \mathord{\left/ {\vphantom {t 2}} \right.} 2}\textrm{ < }1$ and in the vicinity of $4\varphi = 2m\pi$, the resonant modes are split and there are two total reflections as shown in Fig. 3(d). It should be noted that, between these split modes, there are some traveling modes rotating along the microring, for example, near $32\pi$ and $34\pi$ for $N = 3$, and near $34\pi$ and $38\pi$ for $N = 4$ as shown in Figs. 3(c) and (d), total transmission occurs since, from a mathematical point of view, $\frac{{\sin \theta }}{{\sin N\theta }} \to \pm \infty$ in Eq. (12). This is because the incidence light is coupled into the microring and scattered by the scatterers, and the backward scattering waves are destructively cancelled due to the period of the scatterers. Therefore, these kinds of travelling modes are rotated along the microring.
Fig. 3. Theoretical transmission spectra of the optical systems as shown in Fig. 2 calculated from Eq. (11) with (a) $N = 1$, (b) $N = 2$, (c) $N = 3$, (d) $N = 4$.
It is known that microring resonator without scatterers [23] supports non-reflective traveling modes while periodically patterned microring resonator only supports stationary modes [11,12,16]. It is interesting that PSPMR can support both kinds of modes. From Fig. 3, we also find that, free spectral range of the structure can be adjusted by the number of scatterers, for example, a larger prime number can be used to extend the free spectral range.
4. Numerical validations of the optical transmissions
To validate the results obtained above, we designed a periodically sparse patterned microring resonator (PSPMR) with N scatterers side-coupled to a bus waveguide, as shown in Figs. 2(a) and 2(d) finite-difference time-domain method [29,30] is used to investigate the response of the coupling system. The optical system consists of a bus waveguide and microring resonator with air holes suspended in air. The dielectric constants of the bus waveguide and the microring resonator are 11.56.
Figure 4 shows the normalized transmissions with different number of scatterers. From Fig. 4, we find that the frequency positions of the two kinds of modes are the same as theoretical prediction as shown in Fig. 3, however, from Figs. 4(a) and 4(b), the transmission dips are increased with the frequency. This is due to the dependence of coupling coefficients t and $\tau$ on the frequency. Increasing the frequency leads to the decrease of t and increase of $\tau$, which is preferred to get a real solution of $\varphi$ in Eqs. (13) or (14). For Fig. 4(b), at frequency $0.12207({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$ where $N = 2$ and $2{t^{ - 1}} = \sqrt \tau + {1 \mathord{\left/ {\vphantom {1 {\sqrt \tau }}} \right.} {\sqrt \tau }}$, there is just one total reflection dip and the resonant modes at higher frequencies are split into two resonant dips. Compared the resonant dips near the frequency of $0.12754({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$ in Figs. 4(a)–4(d), we find that with the increase of N, the resonant dips approach zero and begin to split into two dips, and widths of dips become broad, which can also be obtained by Eqs. (13) or (14).
Fig. 4. The transmission spectra of the structure with (a) $N = 1$, (b) $N = 2$, (c) $N = 3$, (d) $N = 4$. The inset in (c) is the same plots except for the frequency range to exhibit further details of the travelling modes near the resonant frequencies.
For the travelling modes, compared the simulation spectra shown in Figs. 4(c) and 4(d) with the theoretical results shown in Figs. 3(c) and 3(d), we find that the transmission cannot reach unity, as shown in the inset of Fig. 4(c). Departures in the transmission between simulation and theory are mostly attributed to theoretical consideration, where we assume the coupling between the bus waveguide and the microring is at the coupling plane, while in the simulation, the coupling occurs in a region of the waveguide near the microring. And therefore, the backward scattering wave cannot be totally cancelled. However, from the inset in Fig. 4(c), there are shoulders indicated the existences of these travelling modes. Figure 4 also shows that, the number of travelling modes between two stationary modes determines the free spectral range, which can be used to design optical filters.
In order to further investigate the optical properties of the two kinds of modes, we plot the field distributions of the coupling system with $N = 3$ as shown in Fig. 5. At a frequency of $0.12210({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$ where the phase shift of the microring $3\varphi \approx 34\pi$, the mode is a travelling mode rotating along the microring, which acts like a microring without scatterers side-coupled to waveguide, while for the mode at frequency of $0.12754({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$ where $3\varphi \approx 36\pi$, it is a stationary mode and all the energy is reflected back.
Fig. 5. Steady-state electric field distributions in the resonator-waveguide structures with $N = 3$ at frequencies (a) $0.12210({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$, (b) $0.12754({{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} a}} \right.} a})$.
Simulation results show that transmission properties of the coupling system are not sensitive to the size of scatterers. While the breaking of the periodicity will destroy the travelling modes since the backward scattering light cannot be cancelled any more.
5. Conclusion
In conclusion, we have modeled and investigated dispersion properties of periodically sparse patterned waveguide and the corresponding microring resonator by using the transfer matrix method. We also theoretically analyzed the transmission coefficients of the microring resonator side-coupled to a waveguide. Results shows that (1) the periodically sparse patterned microring resonators can support two kinds of modes, traveling modes and stationary modes; (2) just the stationary modes lead to total reflections and conditions for two, one, or no total reflection dips are obtained; and (3) the rotation of the microring resonator does not change the transmission profiles. All the mode properties are numerically demonstrated by the finite-difference time-domain method and excellent agreements have been found with the theoretical analysis. Our results can be used to simplify the design and fabrication of the microring-based optical devices.
Funding
Jilin Scientific and Technological Development Program (20180101281JC); National Natural Science Foundation of China (11474041); Education Department of Jilin Province (135 Project (JJKH20190579KJ)); Higher Education Discipline Innovation Project (D17017).
Disclosures
The authors declare no conflicts of interest.
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