Abstract

Integrated-optic cavity resonators, such as Fabry–Perot microcavities and microrings, are key building blocks of photonics integrated circuits and are used extensively in applications such as optical communications and microwave photonics. For a single, conventional, optical-cavity resonator, resonance peaks appear periodically in frequency and have Lorentzian shapes in nature, which generally cannot be broken. Here, we report on fully tailorable, integrated-optic resonators that allow for independent control of individual resonance or spectral peaks as regards their presence, linewidths and extinction ratios, resonant wavelengths, and shapes and bandwidths. The response shapes can be set to be Lorentzian, Gaussian-like, or square. The resonators are based on chirped waveguide Moiré gratings developed on a silicon-on-insulator platform. We also demonstrate that they can be implemented on compact Archimedean spiral shapes to have sizes comparable to microring and microdisk resonators, with no spectral degradation. The unprecedented spectral flexibility of these resonators makes them attractive for a variety of fields and will enable new avenues for exploration in relevant areas such as optical waveform synthesis and microwave photonics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Optical cavity resonators, such as Fabry–Perot (FP) resonators, are fundamental optical devices and have attracted enormous attention in science and technology [1]. In a generic cavity resonator, the light injected into the cavity will circulate and interfere with itself. A resonance occurs when the round-trip optical path length of the resonator is exactly a whole number of wavelengths. As a result, resonance peaks appear in the resonator response in a periodic manner. In the area of integrated optics, silicon-based, monolithically integrated resonators, such as waveguide grating-based FP microcavities [2], microrings [3], and microdisks [4], have been extensively studied and used. Their features of miniature size, high extinction ratio, narrow linewidth, low insertion loss, and active tuning capability make them ideally suited for on-chip filters [5], routers [6], switches [7], modulators [8], and laser engineering [9] in optical telecommunications. The periodic nature of the spectral responses also makes it easy to extend these functionalities to the multi-wavelength regime for wavelength-division multiplexing (WDM) systems, for use in comb switches [10] and filters [11], interleavers [12], multichannel signal processing [13], and multiwavelength laser and frequency-comb generation [14].

The strict periodic nature of the resonances in frequency for optical resonators is particularly useful in precision optical frequency synthesis and metrology [15]. On the other hand, however, an ambitious and attractive vision is to break this periodicity and even to independently tailor the characteristics of individual resonances such as their wavelengths, presence, etc. This will bring immediate benefits to a variety of applications where non-repetitive and controlled comb-like responses are of great interest, such as optical telecommunications [1618], passive WDM routing [19,20], astrophotonics [21], and microwave photonics [22]. In addition, providing this novel capability to optical resonators will add significant new spectral flexibility to devices, e.g., for developing flexible-grid multiwavelength devices, such as lasers, filters, and transmitters, for next-generation flexible optical networking [23]. Furthermore, it can be envisioned that controlled resonator responses will open up new avenues for exploration in areas where traditional, periodic, resonator or comb-like responses have been exploited, including optical arbitrary waveform generation [15], microwave photonics [24,25], and spectroscopy [26].

Another intrinsic spectral feature of optical resonators is the Lorentzian shapes of their resonance peaks [3]. In many filtering applications, however, a response with a flattop instead of a narrow Lorentzian behavior is preferred. As an example, in optical communication systems, a flattop response can largely alleviate wavelength drift issues due to, for example, temperature variations, and is also much more suitable for high-speed signals that have large bandwidths [3]. A flattop response cannot be achieved in a single, conventional resonator, and is typically synthesized by using a high-order resonator filter, such as by cascading a series of microring resonators [5,27], or through a complex circuit comprising several different optical elements [12,28]. In addition to the high complexity, real-time active tuning is critically needed in those solutions to compensate for parameter mismatches between elements due to fabrication non-uniformity and environmental changes.

An integrated-optic resonator can be formed within waveguide Bragg gratings by, for example, phase-shifting the grating by $\pi$ to form a microcavity [2]. Grating-based resonators, compared with microring resonators, contain the advantages of less difficulty in obtaining large free spectral ranges (FSRs) and controllable operation bandwidths [29]. In addition to $\pi$-phase shifted gratings, another type of grating resonator is the Moiré grating, which is formed by superimposing two gratings with slightly different periods [30,31]. This will cause the overall grating strength to follow a Moiré or concave profile along the length, and a $\pi$-phase shift will be naturally produced at the crossover point, thereby forming a resonance peak at the Bragg wavelength. Advantages of Moiré gratings compared with $\pi$-phase shifted gratings stem mainly from their inherent Moiré apodization profiles. Examples of such advantages are weaker spectral sidelobes and ripples (see Supplement 1, Section 1), and suppressed spatial hole burning and enhanced single longitudinal mode properties in laser engineering [32]. Moiré gratings were realized on optical fibers by double-exposure with two interference patterns of slightly different periods [30,31], and, later, period chirps were imposed on the gratings to further broaden the operation bands to enable multiple resonance peaks [33,34]. Some recent work also studied the implementations of such devices on integrated waveguides [32,3537].

In this paper, we first report on fully tailorable, integrated-optic resonators in which independent control of parameters of individual resonance peaks, including their presence, linewidths and extinction ratios, resonant wavelengths, and shapes and bandwidths can be realized. The response shapes can vary between Lorentzian, Gaussian-like, and square. The resonators are based on chirped waveguide Moiré gratings (CWMGs) developed on a silicon-on-insulator (SOI) platform. It is shown that each CWMG resonator can be identified by its complex (amplitude and phase) spatial Moiré profile. The grating chirp maps a specific location on the grating to a specific wavelength in the spectral response. This mapping connects each crossover point of the spatial Moiré profile to a resonance peak of the resonator response, which, in turn, allows us to control specific resonances in the wavelength domain by tailoring the Moiré profile along the length of the CWMG. Then, we demonstrate the proposed CWMG resonators and response tailoring capability. Notably, a five-channel square filter with an ultra-high overall extinction ratio of ${\sim}45\;{\rm dB}$ is achieved. Finally, we also demonstrate that these promising CWMG resonators can be implemented on compact Archimedean spirals to have sizes comparable to microring and microdisk resonators with no performance degradation. Compared with conventional, sophisticated, high-order, integrated resonators, such as Vernier ring resonators [3], our CWMG resonators provide unprecedented design freedom as regards the resonator responses, which offers new flexibility to device/system design and potential new applications, as discussed above. In addition, when serving as filters, CWMGs provide superior filtering characteristics, including being sidelobe-free, having little in-band dispersion, containing easily designable passband shapes and bandwidths, having perfect flattops (when shaped to squares), and possessing high extinction ratios. Finally, since CWMG resonators are based on single waveguide gratings, they have simpler and more compact configurations. This also avoids the need for real-time, active tuning to compensate for parameter mismatches between elements, as is typically required in high-order resonators.

2. BASIC PRINCIPLE

The proposed CWMG resonators are based on the well-known Moiré effect [30]. A Moiré pattern is an interference pattern that can be obtained when two periodic profiles with a small difference in their periods but otherwise identical parameters [y1 and y2 in Fig. 1(a)] are superimposed. Due to the beating of the two profiles, the superimposed pattern will have a slowly varying envelope [Fig. 1(a), blue] following a cosine behavior with a large Moiré period, ${\Lambda _M}$, given by

$${\Lambda _M} = \frac{{2{\Lambda _1}{\Lambda _2}}}{{\Delta \Lambda}},$$
where ${\Lambda _1}$ and ${\Lambda _2}$ are the spatial periods of the two patterns, and $\Delta \Lambda$ is the difference in their periods.
 

Fig. 1. (a) Illustration of how superimposing two periodic patterns with the same parameters except for a small difference in their periods can generate a Moiré profile. (b) Schematic representation of a waveguide Moiré Bragg grating. (c) Grating strength and phase profiles; the grating strength follows a Moiré profile along the length, and a $\pi$-phase shift occurs at the crossover point. (d) Spectral responses of the Moiré grating, where a resonance peak is opened at the Bragg wavelength due to the $\pi$-phase shift at the crossover point. For the grating in (c) and (d), ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 nm and ${\sim}311.3\;{\rm nm}$, respectively, the grating length, $L$, is ${\sim}0.3\;{\rm mm}$, and the corrugation width, $\Delta W$, is 13 nm, which is defined as the width difference between the inner and outer grating sidewalls on a single side of the waveguide.

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To apply this Moiré effect to a sidewall waveguide Bragg grating, we impose a small difference on the periods of two gratings formed on either side of the waveguide [Fig. 1(b)]. In this way, the offset between these two “sub-gratings” will vary periodically along the length of the waveguide from zero to its maximum value of one half of a grating period. Correspondingly, the strength distribution of the overall grating will present a Moiré pattern with a period equal to half of the Moiré period, ${\Lambda _M}/2$, and the phase of the grating changes by $\pi$ at the crossover point [Fig. 1(c)]. As a consequence of this $\pi$-phase shift, a sharp resonance peak will occur at the Bragg wavelength of the Moiré grating spectrum [Fig. 1(d)]. The grating Moiré profile, $M(z)$, is defined as a complex function, and the amplitude and phase of $M(z)$, denoted by $|M(z)|$ and $\angle M(z)$, respectively, represent the overall grating strength (i.e., grating coupling coefficient) and phase profiles, respectively. Note that all of the Moiré gratings designed in this work are developed on SOI strip waveguides with cross sections of $220 \times 500\;{\rm nm}$ and designed for the fundamental TE mode.

When the waveguide Moiré grating is long enough, the grating Moiré profile will contain multiple periods and crossover points, and a $\pi$-phase shift will occur at each crossover point [Figs. 2(a) and 2(b)]. In this case, the resonance peak will be largely strengthened and broadened, but will still be centered at the single Bragg wavelength [Fig. 2(c)]. This grating is similar to a uniform grating with $\pi$-phase shifts at various positions [38] or a high-order resonator filter [27], both of which have been used to obtain a flattop passband.

 

Fig. 2. Illustration of the situation of a long, un-chirped, waveguide Moiré grating. (a) Sub-grating period profiles; ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 and ${\sim}311.9\;{\rm nm}$, respectively, and $L$ is ${\sim}0.84\;{\rm mm}$. (b) Grating Moiré profile, $M(z)$, defined as a complex function whose amplitude and phase [denoted by $|M(z)|$ and $\angle M(z)$, respectively] represent the overall grating strength (i.e., grating coupling coefficient) and phase profiles, respectively. (c) Grating spectral responses; $\Delta W$ is 5 nm.

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Now, consider that the two sub-gratings of the waveguide Moiré grating described above are linearly chirped along the length with the same chirp rates ($C$) [Fig. 3(a)]. In such a CWMG, the period difference between the sub-gratings ($\Delta {\Lambda _G}$) will remain constant throughout the grating, and $|M(z)|$ will be the same as that of the un-chirped Moiré grating described above. However, the linear chirp of the grating now maps a specific position on the grating to a wavelength [Fig. 3(b)]. Now, the spatial Moiré grating profile can also be expressed as a function of wavelength, $M(\lambda)$, and a $\pi$-phase shift at each crossover point of $M(z)$ is mapped to a different resonance of the grating spectral response [Figs. 3(c) and 3(d)]. As a result, similar to a classical optical-cavity resonator, periodic and equidistant resonance peaks will be formed in the spectral response of our CWMGs. As opposed to traditional optical resonators, each resonance peak in the spectral response for our CWMGs is uniquely associated with a crossover point of $M(z)$. This allows for independent control of specific resonance peaks by modifying the corresponding crossover points of the grating Moiré profile. The Bragg wavelength of the overall CWMG as a function of the position can be expressed as

$${\lambda _B} = {\lambda _0} + \gamma z,$$
where ${\lambda _0}$ is the initial grating Bragg wavelength, which is determined by the initial grating period, ${\Lambda _{G0}}$, and $\gamma$ is the space-to-wavelength mapping coefficient (see Supplement 1, Section 2):
$$\gamma = \frac{{2Cn_{\rm eff}^2}}{{{n_g}}}.$$
The FSR of CWMG resonators is the wavelength range that maps to adjacent crossover points, and thus can be written as
$${\rm FSR} = \frac{{{\Lambda _M}}}{2}\gamma \approx \frac{{2Cn_{\rm eff}^2\Lambda _{G0}^2}}{{{n_g}\Delta {\Lambda _G}}},$$
where ${n_{\rm eff}}$ and ${n_g}$ are the effective refractive and group indices of the waveguide, respectively. The resonance linewidths and extinction ratios will depend on the overall grating strength or the amplitude of $|M(z)|$, which is determined by the grating corrugation width, $\Delta W$, defined as the width difference between the inner and outer grating sidewalls on a single side of the waveguide. More design and implementation details of CWMG resonators are provided in Supplement 1, Section 3.
 

Fig. 3. Illustration of the case where the two sub-gratings of the waveguide Moiré grating in Fig. 2 are applied by the same linear chirp (${\sim}18.5\;{\rm nm/mm}$). (a) Sub-grating period profiles. (b) Bragg wavelength of the overall Moiré grating against the length. (c) Spatial (top $x$ axis) and wavelength (bottom $x$ axis) complex grating Moiré profile. (d) Grating spectral responses.

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An important point to note is that a CWMG resonator may have the same $M(\lambda)$, and thus the same basic response properties, as another one having a shorter length but a higher $C$ and thus a higher $\gamma$ (see Supplement 1, Section 4), which will be referred to as “$M(\lambda)$-equivalence.”

The above analyses are based on CWMGs with linear chirps, which ensures a constant $\gamma$, and thus a constant FSR, which is required for a regular resonator. Other chirp profiles, such as parabolic profiles, can contain spatially varying chirp rates, and can be used in special cases such as the design of resonator responses with varying FSRs. Also, the starting offset between the sub-gratings will determine the initial phase of $M(z)$ [or $M(\lambda)$]. When the two sub-gratings are aligned at $z = 0$, $M(z)$ will start from an amplitude peak, while if the sub-gratings are misaligned by a half grating period at the input, $M(z)$ will start from a zero crossing. Since the latter can lead to a complete Moiré period for the first crossover point and thus a complete “mirror” for the first resonance, it will be chosen for most of the CWMGs designed below.

Note that to study how the inherent Moiré strength profiles of CWMGs can affect their resonant responses, we have compared the spectral response of a Moiré grating to that of a $\pi$-phase shifted grating (see Supplement 1, Section 1). A $\pi$-phase shifted grating is similar to a Moiré grating but has a constant strength throughout the grating. The results of this comparison suggest that the inherent strength profiles of Moiré gratings can actually give them weaker spectral sidelobes and ripples as compared to those of $\pi$-phase shifted gratings.

Finally, it should be noted that in addition to the coupling between forward and backward TE0 modes, which is used to create our resonator responses, there would also be coupling between forward TE0 and backward TE1 modes at shorter wavelengths due to the asymmetric grating structures of CWMGs [39]. For one of our typical CWMGs [designed in Fig. S2(b) in Supplement 1], we have calculated that the wavelength band of such unintended coupling is far enough away from the band that we will use (the wavelength distance is ${\gt}130\;{\rm nm}$) and thus will not affect the resonator response. Nevertheless, when designing CWMGs with broader resonant responses or on other waveguides, care should be taken to ensure that the band of such undesired coupling does not overlap with the band that will be used.

3. RESPONSE TAILORING OF CWMG RESONATORS

A. Resonance Presence Control

As described above, for a CWMG resonator, each resonance peak in the wavelength domain is due to the $\pi$-phase shift at the corresponding crossover point in $M(z)/M(\lambda)$. Therefore, to achieve a resonator response without the presence of specific resonance peaks, or to suppress particular resonance peaks, we can eliminate or compensate for the $\pi$-phase shifts at the corresponding crossover points. This can be achieved by adding an additional compensation phase profile, ${\phi _C}(z)$, into the overall grating. Figures 4(a) and 4(b) illustrate the process to suppress the second resonance of a CWMG resonator, which has $C \approx 12\; {\rm nm/mm}$, ${\Lambda _{G0}} = 306 \;{\rm nm}$, $\Delta {\Lambda _G} \approx 0.9\; {\rm nm}$ and a total grating length $L$ of ${\sim}1\;{\rm mm}$. The ${\phi _C}(z)$ profile [yellow, right axis, Fig. 4(a)] contains a $\pi$-phase shift at the second crossover point ($z \approx 0.21\;{\rm mm}$) to eliminate its $\pi$ phase-shift. To apply ${\phi _C}(z)$ over the overall grating, ${\phi _C}(z)$ needs to be added into the phase profiles of the two sub-gratings, denoted by ${\phi _{G1}}(z)$ and ${\phi _{G2}}(z)$, which can be obtained from their period profiles [through Eq. (S8), Supplement 1]. The new sub-grating phase profiles, ${\phi ^\prime _{G1}}(z)$ and ${\phi ^\prime _{G2}}(z)$, are then used to create the modified grating structure [see Supplement 1, Section 3]. ${\phi ^\prime _{G1}}(z)$ and ${\phi ^\prime _{G2}}(z)$ are also plotted in Fig. 4(a) (red, left axis). The inset of Fig. 4(a) shows a zoomed-in view of the area near the second crossover point, where a $\pi$-phase jump can be seen for both ${\phi ^\prime _{G1}}(z)$ and ${\phi ^\prime _{G2}}(z)$ due to the incorporation of ${\phi _C}(z)$. The complex $M(z)$ profile of the grating obtained based on ${\phi ^\prime _{G1}}(z)$ and ${\phi ^\prime _{G2}}(z)$ is shown in Fig. 4(b). As can be seen, the $\pi$-phase shift at the second crossover point of the new $M(z)$ now has been eliminated, which will lead to significant suppression of the second resonance in the resonator response.

 

Fig. 4. Resonance peak suppression of a CWMG resonator via applying a compensation phase profile into the overall grating to eliminate the $\pi$-phase shifts at the corresponding crossover points of $M(z)$. (a)–(d) Suppression of the second resonance peak. (a) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (b) Complex Moiré profile of the modified CWMG; the $\pi$-phase shift at the second crossover point has been compensated for/eliminated. (c) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs. (d) Response comparison of two “$M(\lambda)$-equivalent” CWMGs with different values of $\gamma$, designed based on (a) and (b). (e), (f) Suppression of the second and the fourth resonance peaks. (e) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (f) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs.

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Figure 4(c) shows the reflection (left) and transmission (right) responses of the modified CWMG designed above, calculated using the transfer matrix method based on coupled-mode theory (CMT-based TMM) [40], where the response of the original (unmodified) grating is also shown (dashed gray) for comparison. For the reflection response, the second resonance is almost completely eliminated, while in the transmission response, the resonance peak is significantly suppressed (by more than 20 dB). The new sidelobes around the suppressed resonance wavelength in the transmission response should be caused by the sharp variation (a discontinuous first derivative) of $|M(z)|$ (i.e., of the grating strength profile) at the second crossover point when the $\pi$-phase shift has been removed. Note that when the $\pi$-phase shift exists, $M(z)$ essentially follows a smooth cosine behavior, and no sharp change occurs at the crossover point.

In addition, the resonance suppression ratio is related to $\gamma$ and will become smaller as $\gamma$ becomes larger. This is demonstrated in Fig. 4(d), which compares the transmission responses of two $M(\lambda)$-equivalent CWMGs (see Supplement 1, Section 4), each having a different $\gamma$, designed based on Figs. 4(a) and 4(b). This dependence of the suppression ratio on $\gamma$ is likely due to the fact that the Bragg wavelength varies more slowly along the length of the lower-$\gamma$ CWMG, which decreases the impact of the sharp grating strength variation on the spectral response. As a smaller $\gamma$ leads to a longer grating length, there is a trade-off between the resonance suppression ratio and the grating length.

Figure 4(e) shows a case in which the second and fourth resonance peaks are simultaneously suppressed. In this case, ${\phi _C}(z)$ contains $\pi$-phase jumps at both the second and fourth crossover points to compensate for/eliminate their $\pi$-phase shifts. Figure 4(f) plots the calculated responses of the modified CWMG, where the second and fourth resonance peaks are significantly suppressed in both reflection and transmission responses.

B. Resonance Linewidth and Extinction Ratio Control

The resonance linewidths and extinction ratios depend on the amplitude of $|M(z)|$, which is determined by $\Delta W$ [see Supplement 1, Section 3]. Thus, we can control the linewidths and extinction ratios of individual resonance peaks by modulating $\Delta W$ along the grating, or by apodizing the grating. As a proof of concept, we design a CWMG with nine resonances in which the first through fourth resonance peaks have narrower linewidths and higher extinction ratios than the fifth through ninth resonance peaks [Fig. 5(a)]. This is achieved by applying $\Delta W$s of 30 nm and 24.6 nm over the first and second halves of the grating, respectively. In the resulting $|M(z)|$, the first and second halves have amplitudes of 53.6 and $44\;{{\rm mm}^{- 1}}$, respectively. The CWMG has $C \approx 27.7\; {\rm nm/mm}$, leading to $\gamma \approx 80.5 \;{\rm nm/mm}$, ${\Lambda _{G0}} = 311 \;{\rm nm}$, $\Delta {\Lambda _G} \approx 1.4 \;{\rm nm}$, and $L \approx 0.7\;{\rm mm}$.

 

Fig. 5. Resonance linewidth and extinction ratio control of a CWMG resonator via grating apodization. (a) $|M(z)|$ and ${\Delta}\!{W}$ distribution of the CWMG and (b) simulated spectral responses of the CWMG designed in (a). (c) Simulated responses of a $M(\lambda)$-equivalent CWMG with a smaller $\gamma$. The dashed lines in (b) and (c) indicate the change in the extinction ratios.

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The simulated responses of the CWMG designed above are plotted in Fig. 5(b), which are consistent with the design. Also, there is a transition of the resonance extinction ratio in the wavelength region between the fourth and sixth resonance, as indicated by the yellow dashed line. The transition is due to this wavelength range, in the spatial domain, being near the boundary of the two grating regions with different $\Delta W$ s. Thus, the resonances within this wavelength range are impacted by both of the grating regions, creating spatial interference effects that grow smaller for wavelengths, in the spatial domain, that are farther from the boundary. This transition essentially imposes a limited wavelength resolution for controlling the linewidths and extinction ratios of individual resonance peaks. The wavelength resolution, however, can be improved by using a smaller $\gamma$, as indicated in Fig. 5(c), which shows the responses of a $M(\lambda)$-equivalent CWMG with a smaller $\gamma$ compared with $\gamma$ used for Fig. 5(b). As can be seen, the transition for this smaller-$\gamma$ CWMG is shorter than that for the larger-$\gamma$ CWMG. This is because in the spatial domain, the neighboring resonances for the smaller-$\gamma$ CWMG are farther from each other and thus are less subject to the aforementioned spatial interference effects. Thus, smaller values of $\gamma$ will bring about better wavelength resolutions to control the linewidths and extinction ratios of individual resonance peaks, at the expense of requiring longer gratings. When $\gamma$ is sufficiently small, such that the wavelength resolution is smaller than the FSR, the linewidth and extinction ratio of each resonance can be independently controlled.

C. Resonant Wavelength Control

The wavelengths of individual resonance peaks of a CWMG resonator can be controlled by tailoring the positions/wavelengths of the crossover points in $M(z)/M(\lambda)$. This can be accomplished by modifying the sub-grating period profiles of a regular CWMG. This section will illustrate the designs of two different resonant wavelength-tailored CWMGs, while two additional design examples can be found in Supplement 1, Section 5.

Figure 6(a) presents the design of a resonator response containing two four-channel bands, each of which has a different channel spacing or FSR. The channel spacing for the band at the shorter wavelengths (band 1) is ${\sim}5.5\;{\rm nm}$, while that for the band at the longer wavelengths (band 2) is ${\sim}7.3\;{\rm nm}$. Such a resonator response should be of interest in flexible-grid optical networks [23]. To achieve such a dual-band resonator response, we can design $\Delta {\Lambda _G}$ and thus the period of $M(\lambda)$, to be constant over each of the two grating regions but to be different from one region to the other. This can be realized by introducing an abrupt period jump within ${\Lambda _{G2}}(z)$. The designed ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$, and the resulting $|M(z)|/|M(\lambda)|$ profile, are plotted at the top of Fig. 6(a); $C$ is ${\sim}17.9\;{\rm nm/mm}$, ${\Lambda _{G0}}$ is 306 nm, $L$ is ${\sim}1.2\;{\rm mm}$, and a jump of ${\sim}0.22\;{\rm nm}$ is introduced within ${\Lambda _{G2}}(z)$ at $z \approx 0.52\;{\rm mm}$, causing $\Delta {\Lambda _G}$ to be ${\sim}0.89\;{\rm nm}$ or ${\sim}0.67\;{\rm nm}$ over the grating regions before or after the period jump, respectively. The simulated resonator responses are shown at the bottom of Fig. 6(a). The resonant wavelengths agree well with the wavelengths of their corresponding crossover points in the $M(\lambda)$ profile.

 

Fig. 6. Resonant wavelength control of CWMG resonators. (a) Design of a resonator response containing two four-channel bands, each of which has a different FSR. (b) Design of a resonator response with the fourth resonance eliminated by equivalently doubling the spacing between the corresponding resonance peaks. In each of (a), (b), the top figure shows $|M(z)|/|M(\lambda)|$ (blue, left axis) and sub-grating period profiles (red, right axis), while the bottom plot presents the simulated grating reflection and transmission responses.

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Figure 6(b) shows the design of a seven-resonance resonator response with the spacing between the third and fourth resonances doubled compared with all the other spacing (the spacing between the third and fourth resonances is ${\sim}6.8\;{\rm nm}$, while the spacing between the resonances everywhere else is ${\sim}3.4\;{\rm nm}$). This response is essentially equivalent to an eight-resonance resonator response with the fourth resonance eliminated. This design requires one to double the fourth period of $M(z)/M(\lambda)$, which can be realized by decreasing $\Delta {\Lambda _G}(z)$ within an appropriate grating region by modifying ${\Lambda _{G2}}(z)$. The designed ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$ with the corresponding $|M(z)|/|M(\lambda)|$ are plotted at the top of Fig. 6(b); $C$ is ${\sim}9\;{\rm nm/mm}$, ${\Lambda _{G0}}$ is 306 nm, $L$ is ${\sim}1.14\;{\rm mm}$, and $\Delta {\Lambda _G}$ is ${\sim}0.34\;{\rm nm}$ over the region from ${\sim}0.38$ to ${\sim}0.63\;{\rm mm}$ and is ${\sim}0.75\;{\rm nm}$ over the rest of the grating. The simulated responses are presented at the bottom of Fig. 6(b), which agree well with $M(z)/M(\lambda)$.

Finally, it should be noted that for the resonant wavelength-tailored CWMG resonators studied in this section, we again noticed that a low $\gamma$ is beneficial for tailoring resonant wavelengths of a CWMG resonator. When $\gamma$ is too high, the period modification within a certain region to control the wavelength of a specific resonance will also slightly change the wavelengths of the neighboring resonances. This is because, similar to the resonance linewidth and extinction ratio control described above, when $\gamma$ is too high, the neighboring resonances in the spatial domain are located very close to each other, and thus, resonances near the tailored resonances can be affected easily by the $\Delta {\Lambda _G}$-modification regions. This interference effect, however, can be ignored when a small enough $\gamma$ is used.

D. Response Shaping

In this section, we show that spectral shapes and bandwidths of the transmission peaks for our CWMG resonators can also be controlled by engineering their wavelength-domain Moiré profiles, $M(\lambda)$. The response shaping, as schematically illustrated in Fig. 7(a), is realized by introducing jumps in both ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$ at the crossover point. This will open a gap at the crossover point in $M(\lambda)$ and thus introduce a gap in the operation band of the resonator. Such a gap at the crossover point of $M(\lambda)$ will broaden the transmission peak, which will become Gaussian-like or square shaped, depending on the width of the gap, which is decided by the period jump value. Note that the change in the width of the gap will result in a variation of the wavelength center of the shaped passband response. Importantly, having concave grating strength profiles on the two sides of the crossover point naturally smooths the transition between the operation band and the operation gap of the grating. This eliminates undesired spectral properties that could arise due to the grating profile discontinuity, such as spectral sidelobes and ripples. Furthermore, since the response shaping is based on opening a gap in the operation band of the grating, the shaped response will inherently possess ideal filtering characteristics such as having little in-band dispersion, being sidelobe-free, and, when shaped to a square, having a perfect flattop.

 

Fig. 7. (a) Schematic illustration of response shaping of a CWMG resonator, in which (i), (ii), and (iii) illustrate the original case and the cases in which the transmission peaks are shaped to present Gaussian-like and square behaviors, respectively. (b), (c) Designs of five-channel Gaussian and square filters, respectively, via response shaping of CWMG resonators; the upper (i), middle (ii), and lower (iii) figures in (b) and (c) are ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$, $|M(\lambda)|$, and the simulated transmission response of the resonator, respectively. (d), (e) Comparisons of shaped resonator responses with different values of $\Delta W$ for the five-channel Gaussian and square filters, respectively. (f) Response comparison of five-channel square filters developed on two $M(\lambda)$-equivalent CWMGs with different values of $\gamma$. The right insets in (d)–(f) are enlarged views of the first channel responses.

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By introducing period jumps at all the crossover points, all the transmission peaks will be shaped, thus leading to multichannel Gaussian or square filters. Figure 7(b) shows the design of a five-channel Gaussian filter via response shaping of a CWMG resonator; ${\Lambda _{G0}}$ is 315 nm, $C$ is ${\sim}4.2\;{\rm nm/mm}$, $L$ is ${\sim}1.5\;{\rm mm}$, $\Delta {\Lambda _G}$ is ${\sim}0.39\;{\rm nm}$, and a period jump of ${\sim}1\;{\rm nm}$ is introduced in both ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$ at each crossover point. As can be seen in the $M(\lambda)$ profile [Fig. 7(b)-(ii)], wavelength gaps are opened at all of the crossover points. Now, the channel spacing of the filter response is determined by the spacing between the centers of adjacent gaps in $M(\lambda)$, which is ${\sim}6.1\;{\rm nm}$ in the current case. The transmission response of the resonator, calculated from the complex $M(z)$ profile using the CMT-based TMM, is plotted in Fig. 7(b)-(iii) ($\Delta W = 12\; {\rm nm}$). The transmission peaks have been shaped to present Gaussian-like behaviors. As expected, no sidelobe exists near the passbands.

Figure 7(c) presents the design of a five-channel square filter based on a response-shaped CWMG resonator. The CWMG resonator has the same parameters as the five-channel Gaussian filter designed above except that larger period jumps of ${\sim}2\;{\rm nm}$ are used. Thereby, the wavelength gaps in $M(\lambda)$ [Fig. 7(c)-(ii)] become wider. Correspondingly, the channel spacing increases to ${\sim}9.2\;{\rm nm}$. The simulated transmission response is shown in Fig. 7(c)-(iii) ($\Delta W = 17\; {\rm nm}$). The passbands now present square behaviors, with perfect flattop characteristics, large extinction ratios (${\gt}{35}\;{\rm dB}$), high roll-off rates, and no sidelobes. The passbands are also dispersion-less (see Supplement 1, Section 6).

Figures 7(d) and 7(e) compare the transmission responses with various $\Delta W$s for the CWMG-based, five-channel, Gaussian-like and square filters, respectively. The shaped passband responses become narrower as $\Delta W$ increases. This is because different $\Delta W$s lead to different grating strengths, which change the grating operation bandwidths that, in turn, vary the widths of the operation gaps and thus the response bandwidths.

In addition, similar to regular CWMG resonators [see Fig. S3(d) in Supplement 1], the roll-off rates at the band edges of the shaped passband responses will increase as $\gamma$ decreases. This is demonstrated in Fig. 7(f), which compares the transmission responses of five-channel square filters based on two $M(\lambda)$-equivalent CWMGs with different values of $\gamma$. Also, higher roll-off rates lead to higher extinction ratios. Therefore, there is a trade-off between a device’s length and its multichannel filtering performance.

Finally, it is worth noting that our CWMGs may allow for highly attractive reconfigurability of their responses by exploiting silicon waveguide-based programmable photonics. For example, a CWMG may be divided into multiple sections, each of which incorporates an independent lateral PN junction [2]. This would allow for electrical control of the phases, effective indices, and Bragg wavelengths of individual grating sections. In this way, one could be able to reconfigure the resonator response of a CWMG, such as the presence of particular resonance peaks, FSRs, operation wavelength ranges, and passband shapes.

4. EXPERIMENTAL RESULTS

A. Regular CWMG Resonators

Figures 8(a) and 8(b) show measurement results of two regular CWMG resonators without any response tailoring ($\Delta W$s are 27 and 22 nm, respectively). The top plot in each figure shows the ${\Lambda _{G1}}(z)$, ${\Lambda _{G2}}(z)$, and $|M(z)|/|M(\lambda)|$ profiles of the CWMG. As seen in Figs. 8(a) and 8(b), the measured responses of the two resonators show good agreement with their $M(\lambda)$, as indicated by the dashed lines that connect crossover points of $|M(z)|/|M(\lambda)|$ to their associated resonance peaks in the spectral response. The FSRs of the two CWMGs in Figs. 8(a) and 8(b) are measured to be about 7 nm and 2.8 nm, respectively, which agree well with the designed values of 6.9 nm and 2.8 nm. Note that the wavelengths of the resonance peaks in the measured responses do not exactly match those of their corresponding crossover points in $M(\lambda)$, which is due to the fabrication variations that cause ${n_{\rm eff}}$ and ${n_g}$ of the actual waveguides to deviate from the values used in our simulations. The average 3 dB linewidth and the overall extinction ratio extracted from the measured response shown in Fig. 8(a) are about 88 pm and 29 dB, respectively, which are close to the ideal values of about 69 pm and 30 dB predicted by the CMT-based TMM. This suggests very low losses of the CWMG resonators. The actual loss of a typical CWMG resonator is estimated to be $ {\lt} 1\; {\rm dB}$, by comparing its transmission response with the response of a calibration circuit that is identical to the grating testing circuit except that the grating is replaced by a uniform waveguide. The spectral ripples of the reflection responses come mainly from back reflections that occur in the circuit, including those from grating couplers, directional couplers, fiber interfaces, etc.

 

Fig. 8. Experimental data of [(a), (b)] two regular CWMG resonators and (c) a CWMG resonator with tailored resonance linewidths and extinction ratios. The top plots in (a)–(c) show ${\Lambda _{G1}}(z)$, ${\Lambda _{G2}}(z)$, and $|M(z)|/|M(\lambda)|$ of the corresponding resonators.

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B. Resonance Linewidth and Extinction Ratio Control

As a proof-of-concept demonstration of the capability to tailor the resonance linewidths and extinction ratios through grating apodization, a CWMG resonator $M(\lambda)$-equivalent to that in Fig. 5(a) was fabricated and experimentally characterized. ${\Lambda _{G1}}(z)$, ${\Lambda _{G2}}(z)$, and $|M(z)|/|M(\lambda)|$ profiles of the resonator are presented at the top of Fig. 8(c). The resonator response contains a total of nine resonance peaks, and the first through fourth resonance peaks (referred to as the first band) are designed to have narrower linewidths and higher extinction ratios than those of the fifth through ninth resonance peaks (referred to as the second band). This is realized by applying larger $\Delta W$ ($\Delta W = 26\; {\rm nm}$) over the first half grating than that of the second half ($\Delta W \approx 21\; {\rm nm}$). Consequently, in $|M(z)|/|M(\lambda)|$ the first five periods have higher amplitudes than the last five periods. The measured transmission response, shown at the bottom of Fig. 8(c), agrees well with the design. Specifically, from the first to the second band, the average 3 dB linewidth is increased from ${\sim}275\;{\rm pm}$ to ${\sim}785\;{\rm pm}$, and the overall extinction ratio is decreased from ${\sim}19\;{\rm dB}$ to ${\sim}12\;{\rm dB}$.

C. Resonance Presence Control

The ability to independently control the presence of individual resonance peaks for CWMG resonators is demonstrated. Two CWMG resonators designed in Figs. 4(a) and 4(b) and Fig. 4(e) with $\Delta W = 24\; {\rm nm}$ were fabricated and characterized. These resonators were deigned to have the second resonance and both the second and fourth resonances suppressed, respectively. The measured transmission responses are shown in Fig. 9. For both CWMGs, the intended resonances are suppressed, with resonance suppression ratios of up to 20 dB.

 

Fig. 9. Measurement results of CWMG resonators designed to have (a) the second and (b) second and fourth resonance peaks suppressed, which are designed based on Figs. 4(a)4(b) and Fig. 4(e), respectively.

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D. Resonant Wavelength Control

To validate the resonant wavelength control of CWMG resonators, the resonant wavelength-tailored CWMG resonators designed in Figs. 6(a) and 6(b) and Figs. S4(a)–S4(b) in Supplement 1 were fabricated and experimentally characterized. The measurement results of the four resonators are shown in Figs. 10(a)10(d) ($\Delta W$s are 28, 20, 23, and 19 nm, respectively), with the corresponding $|M(z)|/|M(\lambda)|$ profiles presented at the top of the figures for reference. As can be seen, for all of the CWMGs, the tailored resonance wavelengths agree well with the design values, as indicated by the dashed lines that connect the crossover points of $|M(z)|/|M(\lambda)|$ to their corresponding resonance peaks in the measured responses. Note that again the absolute wavelengths of the resonance peaks in the measured responses do not exactly match those of their associated crossover points because of fabrication variations.

 

Fig. 10. Measurement results of various resonant wavelength-tailored CWMG resonators. The resonators shown in (a)–(d) are designed based on Figs. 6(a) and 6(b) and Figs. S4(a)–S4(b) in Supplement 1, respectively.

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E. Response Shaping

To investigate the response shaping of CWMG resonators, we fabricated and experimentally characterized two response-shaped resonators designed in Figs. 7(b) and 7(c), which act as a five-channel Gaussian filter and a five-channel square filter, respectively. The measured transmission responses are shown in Figs. 11(a) and 11(b) ($\Delta W$s are 12 and 17 nm, respectively). The two response-shaped resonators present high-quality multichannel filtering responses with large overall extinction ratios (about 25 dB and 34 dB for the Gaussian and square filters, respectively), high roll-off rates, and no sidelobes. The average 3 dB channel bandwidths of the Gaussian and square filters are calculated to be about 1.3 nm and 3.4 nm, respectively, which show good agreement with the simulated values of 1.4 nm and 3.1 nm. The small differences between the measured and simulated bandwidths are attributed mainly to (1) the deviations of ${n_{\rm eff}}$ and ${n_g}$ of the actual waveguides from the values used in our simulations, which lead to different values of $\gamma$ and thus different widths of the introduced operation gaps in $M(\lambda)$, and (2) fabrication uncertainties.

 

Fig. 11. Measurement results of CWMG resonators with the responses shaped to be (a) a five-channel Gaussian and (b)–(d) five-channel square filters. The resonators shown in (a) and (b) are designed based on Figs. 7(b) and 7(c), respectively. (c) Comparison of the fourth channel responses of the five-channel square filters with different values of $\Delta W$, based on the same design as that in (b). The CWMG resonator shown in (d) uses a smaller $\gamma$ compared with that in (b) to demonstrate the potential of achieving ultra-high extinction ratios and roll-off rates of multichannel square filters using CWMGs; the gray curve in (d) is the noise floor of the detector.

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To explore the effect of $\Delta W$ on the shaped response bandwidths, we fabricated two additional five-channel square filters based on the same design as that used to obtain Fig. 11(b) but each having a different $\Delta W$. Figure 11(c) compares the measured fourth channel responses of the five-channel square filters. As expected, the passband becomes broader as $\Delta W$ decreases. Specifically, the extracted 3 dB bandwidths for the CWMGs with $\Delta W$s of 19 nm, 17 nm, and 14 nm are about 2.4 nm, 3.5 nm, 4.2 nm, respectively.

It has also been shown [Fig. 7(f)] that the roll-off rates at the edges of the shaped passband responses are dependent on $\gamma$. A smaller $\gamma$ will lead to higher roll-off rates, which will bring about larger extinction ratios. Therefore, by using smaller values of $\gamma$, together with their inherent ideal filtering characteristics, our CWMG resonators can potentially act as ultra-high performance, multichannel, square filters. To explore this, we designed, fabricated, and characterized a CWMG-based, five-channel, square filter in which a smaller $C$, and thus $\gamma$, is used; $C$ is ${\sim}3.6\;{\rm nm/mm}$, ${\Lambda _{G0}}$ is 309 nm, $L$ is ${\sim}1.8\;{\rm mm}$, $\Delta {\Lambda _G}$ is ${\sim}0.32\;{\rm nm}$, the period jumps are ${\sim}2.2\;{\rm nm}$, and $\Delta W$ is 20 nm. The measured transmission response of the CWMG is presented in Fig. 11(d). To better present the extinction ratio, the raw measurement data, without the grating coupler baseline shape correction, are shown (blue), and the noise floor, obtained from another detector without connecting any device, is also presented (gray). As can be seen, an overall extinction ratio of as high as ${\sim}45\;{\rm dB}$ is measured. Also, as the levels of the out-of-band measurements have reached the noise floor, the achieved extinction ratio was limited by the fiber-to-chip coupling losses (${\gt}{18}\;{\rm dB}$ in our cases), the limited bandwidths of the grating couplers, and the dynamic ranges of the detectors. The results obtained clearly demonstrate the great potential of CWMG resonators for developing extremely high-performance multichannel filters, interleavers, and demultiplexers/multiplexers for WDM networks.

F. Spiral CWMG Resonators

As shown above, the length of a CWMG resonator can easily exceed 1 mm, which can make it susceptible to fabrication issues such as chip non-uniformity and stitching errors. Also, in all cases, a trade-off exists between the resonator performance and the device length. To overcome these issues, here we demonstrate that long, straight, CWMG resonators can be implemented in much more compact Archimedean spiral shapes with no performance degradation. The straight CWMG resonators previously demonstrated, with their measured data shown in Figs. 8(a), 11(b), and 11(d) (here, referred to as Straight CWMGs 1–3, respectively), were implemented as spiral CWMG resonators (referred to as Spiral CWMGs 1–3, respectively) and were then fabricated and characterized. Note that Straight CWMG 3 has a ${\Lambda _{G0}}$ of 309 nm, while Spiral CWMG 3 has a ${\Lambda _{G0}}$ of 315 nm. The SEM images of the testing circuit for, and an overall and a zoomed-in view of, Spiral CWMG 1 are shown in Figs. 12(a)12(c), respectively. For all of the spiral CWMGs, the spacing between the closest waveguides is ${\sim}1\;\unicode{x00B5}{\rm m}$, and an S-shaped curve with a radius of ${\sim}10\;\unicode{x00B5}{\rm m}$ is used at the center to connect the inward and outward spirals. Due to their different lengths, the outermost radii, denoted by ${R_{\textit{max}}}$ in Fig. 12(b), of Spiral CWMGs 1–3 are about 24 µm, 30 µm, and 32 µm, respectively. These radii fall within the range of radii of demonstrated microring and microdisk resonators [4,6,10,11,13,26,41].

 

Fig. 12. (a)–(c) SEM images of the testing circuit for, and an overall and a zoomed-in view of, Spiral CWMG 1, respectively. (d)–(f) Measurement results of Spiral CWMGs 1–3, respectively, which are spiral versions of the previously demonstrated straight CWMGs with their measured data shown in Figs. 8(a), 11(b), and 11(d), respectively. The gray curve in (f) is the noise floor of the detector.

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The measured spectral responses of Spiral CWMGs 1–3 are shown in Figs. 12(d)12(f), respectively. Compared with the measured spectra of their straight counterparts [see Figs. 8(a), 11(b), and 11(d) for Straight CWMGs 1–3, respectively], no appreciable spectral degradation due to the straight-to-spiral conversion is observed. This is also confirmed in Table 1, which compares the average channel performance of the straight and spiral CWMG-based, five-channel, square filters. The roll-off bandwidth is defined here as the bandwidth over which the normalized transmission goes from the edge of the flattop down to ${-} 30\;{\rm dB}$, which is used to measure the roll-off rate of the square response. The in-band ripple is defined as the standard deviation of the flattop. Note that spiral gratings could have issues such as waveguide bending loss, phase errors due to non-uniform radii along the waveguide grating, and different grating strengths between the inside and outside sub-gratings. Nevertheless, our comparison results between straight and spiral CWMGs suggest that their impacts on the resonator responses are not significant for the radii used here.

Tables Icon

Table 1. Average Channel Performances of Straight and Spiral CWMG-Based, Five-Channel, Square Filters

5. CONCLUSION

We have proposed and demonstrated the first, fully tailorable, integrated-optic resonators based on CWMGs developed on SOI. This new kind of integrated resonator allows for independent control of individual resonance or spectral peaks as regards their presence, linewidths and extinction ratios, wavelengths, and shapes and bandwidths. The response shapes can be set to be Lorentzian, Gaussian-like, or square. In particular, specific resonances can be suppressed via eliminating the $\pi$-phase shifts at the corresponding crossover points in the grating Moiré profile; resonance linewidths and extinction ratios can be controlled by the grating apodization; resonant wavelengths can be engineered by tailoring the locations/wavelengths of the crossover points; transmission peaks can be shaped by introducing grating period jumps to open gaps at the crossover points in the wavelength-domain grating Moiré profile. For the shaped passband responses, the shapes are decided by the period jump values, the bandwidths are determined by the grating strengths, and the roll-off rates are dependent on the chirp rate (or $\gamma$) of the CWMG.

The response shaping of CWMG resonators is essentially based on the new concept of introducing a gap in the operation band of the grating. The concave grating strength profiles on the two sides of the crossover point also naturally smooth the transition between the operation band and the operation gap. These two facts give the shaped response ideal multichannel filtering characteristics including being sidelobe-free, having little in-band dispersion, and containing perfect flattops when shaped to squares. Owing to these ideal filtering properties, five-channel square filters with extremely large overall extinction ratios of ${\sim}45\;{\rm dB}$ and high roll-off rates have been demonstrated, and the achieved extinction ratios were limited by the fiber-to-chip coupling loss, the grating coupler bandwidths, and the detector performance.

Our study has also suggested that there is always a trade-off between the spectral performance and the resonator length. Specifically, a lower grating chirp rate (or a lower $\gamma$), which will lead to a longer grating length, will bring about higher suppression ratios when suppressing specific resonance peaks, a better wavelength resolution for controlling the linewidths, extinction ratios, and wavelengths of individual resonance peaks, and higher roll-off rates and larger extinction ratios of the shaped responses. To overcome these trade-offs and to alleviate the impact of fabrication issues, we have demonstrated that long, straight, CWMG resonators can be implemented on much more compact spiral shapes with no spectral performance degradation.

We believe that, owing to their high spectral flexibility, the proposed CWMG resonators will find applications in various fields such as optical telecommunications and microwave photonics. The unprecedented tailorability of resonator responses offered by CWMGs will also open up new avenues for exploration in applications where conventional, periodic, resonator or comb-like responses are used, including frequency-comb generation, optical signal processing, and microwave photonics.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

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References

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  1. M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice and Applications (Routledge, 2017).
  2. W. Zhang and J. Yao, “Electrically programmable on-chip equivalent-phase-shifted waveguide Bragg grating on silicon,” J. Lightwave Technol. 37, 314–322 (2019).
    [Crossref]
  3. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
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  18. E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.
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  30. D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
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  37. L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
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  38. R. Zengerle and O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
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  39. H. Qiu, J. Jiang, P. Yu, T. Dai, J. Yang, H. Yu, and X. Jiang, “Silicon band-rejection and band-pass filter based on asymmetric Bragg sidewall gratings in a multimode waveguide,” Opt. Lett. 41, 2450–2453 (2016).
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2020 (1)

L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
[Crossref]

2019 (4)

2018 (2)

2017 (2)

2016 (3)

2015 (1)

2014 (2)

2013 (2)

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
[Crossref]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

2012 (3)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
[Crossref]

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

2011 (3)

2008 (1)

2007 (2)

2006 (2)

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
[Crossref]

M. A. Popovíc, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. 31, 2571–2573 (2006).
[Crossref]

2005 (1)

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

2004 (1)

2003 (1)

Q. Wang and S. He, “Optimal design of a flat-top interleaver based on cascaded M–Z interferometers by using a genetic algorithm,” Opt. Commun. 224, 229–236 (2003).
[Crossref]

1998 (1)

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

1995 (2)

R. Zengerle and O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[Crossref]

L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[Crossref]

1991 (1)

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

1990 (1)

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Adibi, A.

Alic, N.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

Ataie, V.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

Baets, R.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Baker, N.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Barwicz, T.

Bayon, F.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Bennion, I.

L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[Crossref]

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Bergman, K.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Bernage, P.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Biberman, A.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Bland-Hawthorn, J.

Bo, F.

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Burla, M.

Buus, J.

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Chen, L.

Chen, L. R.

L. R. Chen, “Silicon photonics for microwave photonics applications,” J. Lightwave Technol. 35, 824–835 (2017).
[Crossref]

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

Chen, M.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

Chen, P.

Chen, S.

Chen, W.

Chen, X.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
[Crossref]

Cheung, K. C.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Chrostowski, L.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Chu, S. T.

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Cooper, D. J.

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

Corcoran, B.

Dai, D.

Dai, S.

Dai, T.

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Ding, Y.

Doerfler, M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Dong, P.

P. Dong, S. F. Preble, and M. Lipson, “All-optical compact silicon comb switch,” Opt. Express 15, 9600–9605 (2007).
[Crossref]

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Douay, M.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Driessen, A.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Dubé-Demers, R.

Dumon, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Edvell, G.

Englund, M.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

Fard, S. T.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Ferdous, F.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

Fertein, E.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Flueckiger, J.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Foster, J. D.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Foster, M. A.

Fu, Q.

Gabriel Sengo,

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Gaeta, A. L.

Gao, F.

Georges, T.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Gerstel, O.

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
[Crossref]

Geuzebroek, D. H.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Grist, S.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Guan, X.

Hamidi, E.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

He, S.

Q. Wang and S. He, “Optimal design of a flat-top interleaver based on cascaded M–Z interferometers by using a genetic algorithm,” Opt. Commun. 224, 229–236 (2003).
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T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
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Hung, Y.

Ippen, E. P.

Jiang, J.

Jiang, X.

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O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
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Kalkman, J.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
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Kelderman, H.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
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Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
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Kirk, J.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
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E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
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J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
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Kumar Selvaraja, S.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
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Kuo, B. P.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

Kuzucu, O.

LaRochelle, S.

Leaird, D. E.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

Lee, B. G.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Legoubin, S.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Leinse, A.

Leminger, O.

R. Zengerle and O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
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Levy, J. S.

Li, J.

Li, S.

Li, W.

Li, X.

L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
[Crossref]

Li, Y.

Lin, J.

Lipson, M.

Little, B. E.

Liu, L.

L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
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Y. Ding, J. Xu, C. Peucheret, M. Pu, L. Liu, J. Seoane, H. Ou, X. Zhang, and D. Huang, “Multi-channel 40 Gbit/s NRZ-DPSK demodulation using a single silicon microring resonator,” J. Lightwave Technol. 29, 677–684 (2011).
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V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

Liu, S.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
[Crossref]

Long, C. M.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

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O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
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Lu, H.

Lu, J.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
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MacIntyre, D. S.

Mitchell, A.

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L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[Crossref]

Mookherjea, S.

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
[Crossref]

Morandotti, R.

Moss, D. J.

Myslivets, E.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

Nguyen, T. G.

Niay, P.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Nitkowski, A.

Ong, J. R.

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
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Ou, H.

Peucheret, C.

Polman, A.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
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Preble, S. F.

Pu, M.

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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

Qiu, H.

Radic, S.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

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D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
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X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
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Seoane, J.

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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
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Soltani, M.

Sorel, M.

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L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
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V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
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V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

Thoms, S.

Vahala, K. J.

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
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W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
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W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
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X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
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Watts, M. R.

Weiner, A. M.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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Wu, J.

Wu, R.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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Xu, X.

Xu, Y.

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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
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R. Zengerle and O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
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Zhang, G.

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L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
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D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
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S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
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L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
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IEEE Commun. Mag. (1)

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
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L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
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X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
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Laser Photon. Rev. (1)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
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Nat. Photonics (1)

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
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L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
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Q. Wang and S. He, “Optimal design of a flat-top interleaver based on cascaded M–Z interferometers by using a genetic algorithm,” Opt. Commun. 224, 229–236 (2003).
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Optica (1)

Phys. Rev. A (1)

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
[Crossref]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Other (4)

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice and Applications (Routledge, 2017).

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Supplementary Material (1)

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Figures (12)

Fig. 1.
Fig. 1. (a) Illustration of how superimposing two periodic patterns with the same parameters except for a small difference in their periods can generate a Moiré profile. (b) Schematic representation of a waveguide Moiré Bragg grating. (c) Grating strength and phase profiles; the grating strength follows a Moiré profile along the length, and a $\pi$-phase shift occurs at the crossover point. (d) Spectral responses of the Moiré grating, where a resonance peak is opened at the Bragg wavelength due to the $\pi$-phase shift at the crossover point. For the grating in (c) and (d), ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 nm and ${\sim}311.3\;{\rm nm}$, respectively, the grating length, $L$, is ${\sim}0.3\;{\rm mm}$, and the corrugation width, $\Delta W$, is 13 nm, which is defined as the width difference between the inner and outer grating sidewalls on a single side of the waveguide.
Fig. 2.
Fig. 2. Illustration of the situation of a long, un-chirped, waveguide Moiré grating. (a) Sub-grating period profiles; ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 and ${\sim}311.9\;{\rm nm}$, respectively, and $L$ is ${\sim}0.84\;{\rm mm}$. (b) Grating Moiré profile, $M(z)$, defined as a complex function whose amplitude and phase [denoted by $|M(z)|$ and $\angle M(z)$, respectively] represent the overall grating strength (i.e., grating coupling coefficient) and phase profiles, respectively. (c) Grating spectral responses; $\Delta W$ is 5 nm.
Fig. 3.
Fig. 3. Illustration of the case where the two sub-gratings of the waveguide Moiré grating in Fig. 2 are applied by the same linear chirp (${\sim}18.5\;{\rm nm/mm}$). (a) Sub-grating period profiles. (b) Bragg wavelength of the overall Moiré grating against the length. (c) Spatial (top $x$ axis) and wavelength (bottom $x$ axis) complex grating Moiré profile. (d) Grating spectral responses.
Fig. 4.
Fig. 4. Resonance peak suppression of a CWMG resonator via applying a compensation phase profile into the overall grating to eliminate the $\pi$-phase shifts at the corresponding crossover points of $M(z)$. (a)–(d) Suppression of the second resonance peak. (a) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (b) Complex Moiré profile of the modified CWMG; the $\pi$-phase shift at the second crossover point has been compensated for/eliminated. (c) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs. (d) Response comparison of two “$M(\lambda)$-equivalent” CWMGs with different values of $\gamma$, designed based on (a) and (b). (e), (f) Suppression of the second and the fourth resonance peaks. (e) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (f) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs.
Fig. 5.
Fig. 5. Resonance linewidth and extinction ratio control of a CWMG resonator via grating apodization. (a) $|M(z)|$ and ${\Delta}\!{W}$ distribution of the CWMG and (b) simulated spectral responses of the CWMG designed in (a). (c) Simulated responses of a $M(\lambda)$-equivalent CWMG with a smaller $\gamma$. The dashed lines in (b) and (c) indicate the change in the extinction ratios.
Fig. 6.
Fig. 6. Resonant wavelength control of CWMG resonators. (a) Design of a resonator response containing two four-channel bands, each of which has a different FSR. (b) Design of a resonator response with the fourth resonance eliminated by equivalently doubling the spacing between the corresponding resonance peaks. In each of (a), (b), the top figure shows $|M(z)|/|M(\lambda)|$ (blue, left axis) and sub-grating period profiles (red, right axis), while the bottom plot presents the simulated grating reflection and transmission responses.
Fig. 7.
Fig. 7. (a) Schematic illustration of response shaping of a CWMG resonator, in which (i), (ii), and (iii) illustrate the original case and the cases in which the transmission peaks are shaped to present Gaussian-like and square behaviors, respectively. (b), (c) Designs of five-channel Gaussian and square filters, respectively, via response shaping of CWMG resonators; the upper (i), middle (ii), and lower (iii) figures in (b) and (c) are ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$, $|M(\lambda)|$, and the simulated transmission response of the resonator, respectively. (d), (e) Comparisons of shaped resonator responses with different values of $\Delta W$ for the five-channel Gaussian and square filters, respectively. (f) Response comparison of five-channel square filters developed on two $M(\lambda)$-equivalent CWMGs with different values of $\gamma$. The right insets in (d)–(f) are enlarged views of the first channel responses.
Fig. 8.
Fig. 8. Experimental data of [(a), (b)] two regular CWMG resonators and (c) a CWMG resonator with tailored resonance linewidths and extinction ratios. The top plots in (a)–(c) show ${\Lambda _{G1}}(z)$, ${\Lambda _{G2}}(z)$, and $|M(z)|/|M(\lambda)|$ of the corresponding resonators.
Fig. 9.
Fig. 9. Measurement results of CWMG resonators designed to have (a) the second and (b) second and fourth resonance peaks suppressed, which are designed based on Figs. 4(a)4(b) and Fig. 4(e), respectively.
Fig. 10.
Fig. 10. Measurement results of various resonant wavelength-tailored CWMG resonators. The resonators shown in (a)–(d) are designed based on Figs. 6(a) and 6(b) and Figs. S4(a)–S4(b) in Supplement 1, respectively.
Fig. 11.
Fig. 11. Measurement results of CWMG resonators with the responses shaped to be (a) a five-channel Gaussian and (b)–(d) five-channel square filters. The resonators shown in (a) and (b) are designed based on Figs. 7(b) and 7(c), respectively. (c) Comparison of the fourth channel responses of the five-channel square filters with different values of $\Delta W$, based on the same design as that in (b). The CWMG resonator shown in (d) uses a smaller $\gamma$ compared with that in (b) to demonstrate the potential of achieving ultra-high extinction ratios and roll-off rates of multichannel square filters using CWMGs; the gray curve in (d) is the noise floor of the detector.
Fig. 12.
Fig. 12. (a)–(c) SEM images of the testing circuit for, and an overall and a zoomed-in view of, Spiral CWMG 1, respectively. (d)–(f) Measurement results of Spiral CWMGs 1–3, respectively, which are spiral versions of the previously demonstrated straight CWMGs with their measured data shown in Figs. 8(a), 11(b), and 11(d), respectively. The gray curve in (f) is the noise floor of the detector.

Tables (1)

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Table 1. Average Channel Performances of Straight and Spiral CWMG-Based, Five-Channel, Square Filters

Equations (4)

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Λ M = 2 Λ 1 Λ 2 Δ Λ ,
λ B = λ 0 + γ z ,
γ = 2 C n e f f 2 n g .
F S R = Λ M 2 γ 2 C n e f f 2 Λ G 0 2 n g Δ Λ G ,

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