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Abstract
We propose and theoretically investigate the coupled concentric ring resonators on a thickness-constrained GaP on insulator (GaP-OI) integrated photonic platform. Achieving anomalous dispersion is fulfilled by mode hybridization in the coupled structure on a 200 nm thick GaP-OI resonator which originally only exhibits normal dispersion for the fundamental mode. The anomalous dispersion profile for the anti-symmetric mode is flattened and broadened in favor of Kerr frequency comb generation by optimizing the waveguide width and the coupling gap size synergistically. We show the flexibility of this design methodology by simultaneously flattening the dispersion profile while anchoring the dispersion peak location at 1550 nm. The optimized design has a flat anomalous dispersion span of 460 nm with a small peak of 160 ps/km/nm, 1.69 times lower than a traditional rectangular waveguide. The engineered dispersion profile enables a broadband Kerr frequency comb generation that has a 3 dB bandwidth of 67 nm and a 20 dB bandwidth of over 250 nm at both 1550 and 1650 nm pump wavelengths. The proposed design proves useful to achieve broad and flat anomalous dispersion on thickness-constrained materials, paving the way towards low-loss GaP-OI frequency comb resonators.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
22 March 2023: A typographical correction was made to the author affiliations.
1. INTRODUCTION
Resonator-based integrated optical frequency combs (OFC) have fostered a variety of applications such as quantum photonics [1,2], chip-scale distance ranging [3,4], low-noise frequency synthesis [5–7], optical atomic clocks [8–10], and precision spectroscopy [11,12]. Those applications often need $f \text{-} {{2}}f$ self-referencing to stabilize the combs for better performance. Although some new approaches [13–16] other than $f \text{-} {{2}}f$ self-referencing, which do not rely on octave-spanning, have been demonstrated recently, the $f \text{-} {{2}}f$ self-referencing still play an indispensable role in comb stabilization, which demand wide spectral bandwidth. For other photonic applications, such as chip-scale distance ranging, spectroscopy, and arbitrary optical/RF waveform generation, wide spectral coverage is also preferred. To satisfy the wide comb bandwidth, the resonator dispersion must be carefully managed [17], since the dispersion not only determines the resonance spacing, which is critical to initiating the comb lines [18] but also governs the ultimate spectral coverage and flatness of combs [19]. Although dark solitons can be generated in resonators with normal dispersion [20,21], which relaxes the required dispersion management, dark solitons have a narrow spectral bandwidth and exhibit a quite small domain of existence in the parameter space as compared to bright Kerr solitons [22]. For bright Kerr soliton formation, slight anomalous dispersion over a wide spectrum range can enhance the comb bandwidth [23]. Thus, it is desirable to design an OFC resonator with a broad and flat anomalous dispersion profile.
The waveguide geometric parameters, such as waveguide thickness [24], width [25], and etching angles [26], have been traditionally used to tune the dispersion. However, this method is subject to the film thickness constrains. Strip/slot multilayer waveguides [19,27] relax the constraints by introducing extra zero-dispersion wavelengths. Despite being more flexible, these multilayer structures mostly work for amorphous dielectric materials but not for crystalline semiconductors, due to a lack of high-quality semiconductor-dielectric alternating film stacks. Besides, post-fabrication trimming [28] becomes less effective in these waveguides, as it only modifies the top layer. Since it is easier to tailor the waveguide lateral dimensions in planar fabrication process, concentric microresonator design works as a promising solution as it only tweaks waveguide lateral geometries without the concern of film thickness [29,30]. These resonators utilize the avoided mode crossing between two adjacent waveguides upon phase matching [31]. The coupling between the two waveguide modes perturbs their original dispersion profiles such that one of the hybrid modes becomes anomalously dispersed locally around the mode crossing point. The excess geometric parameters, such as two waveguide widths, coupling gap width, and gap depth [32], provide extra tuning knobs for dispersion engineering. More importantly, this approach is fabrication-friendly to crystalline semiconductor compounds that have high third-order nonlinearity.
Among the emerging material systems for integrated optical Kerr frequency combs [33], III-V semiconductors are suitable for power-efficient frequency comb generation [34] thanks to their high Kerr nonlinear coefficients and low nonlinear loss. Gallium phosphide (GaP) possesses conspicuous ${\chi ^{(3)}}$ as well as ${\chi ^{(2)}}$ [35] nonlinearities with negligible two-photon absorption above 1.1 µm wavelength. Unlike Si (${\lambda _{\text{TPA}}} \sim {{2250}}\;{\rm{nm}}$ [36]), AlGaAs (${\lambda _{\text{TPA}}} \sim {{1540}}\;{\rm{nm}}$ [37]), and InGaP (${\lambda _{\text{TPA}}} \sim {{1305}}\;{\rm{nm}}$ [38]), whose two-photon absorption wavelengths are above 1300 nm, GaP devices can work in the telecom O-band without the concern of nonlinear loss. GaP also has a broad transparent window from 550 nm to 11 µm and a large refractive index ($\sim {3.05}$ in the C-band [39]), which renders tight mode confinement possible. As long as the lattice mismatch and buffer layer thickness are well controlled in metalorganic chemical vapor deposition (MOCVD) [40], growing GaP suffers less risks from film cracking, which is an advantage over stoichiometric ${\rm{S}}{{\rm{i}}_3}{{\rm{N}}_4}$ film, whose growth often requires countering measures [41,42]. In addition, its nonzero piezoelectric effect and good thermal conductivity enable modulation capabilities via electric field and heat. These optical properties render GaP an excellent candidate for nonlinear photonics. The recent debut of wafer-scale GaP on insulator (GaP-OI) [40,43,44] further releases potentials for nonlinear integrated photonic devices, making it possible to fabricate highly confined waveguides, grating couplers, and ring resonators. Empowered by the GaP-OI architecture, frequency comb resonators have been demonstrated [45], and the required anomalous dispersions were realized by optimizing the waveguide cross sections [26,46]. However, the widening of the anomalous dispersion spectrum is at the expense of the increment of the dispersion peak amplitude, which requires higher nonlinear phase shift to balance the anomalous dispersion. On the other hand, the GaP waveguide thickness may suffer from fabrication constraints, such as defect-free epitaxial growth thickness. Besides, the GaP top surface may need polishing and smoothening to reduce the high scattering loss [35] and film thickness nonuniformity, which thins the usable device layer thickness. Therefore, the fabricated GaP waveguide thickness is expected to be thinner than its original epi layer thickness, which poses even more challenges to obtain a close-to-zero anomalous dispersion amplitude over a wide spectral range in the GaP-OI integrated photonic platform. To address the problem, dispersion engineering techniques that are independent of waveguide thickness—for instance, avoided mode crossing—should be explored.
Here we present the design of a GaP-OI concentric microring resonator for Kerr soliton frequency comb generation with a broad and flat anomalous dispersion profile. The anomalous dispersion is realized by avoided mode crossing, which hybridizes the adjacent waveguide modes and bring the anti-symmetric mode into an anomalous dispersion region [47]. To the best of the authors’ knowledge, few studies have applied avoided mode crossing in the GaP-OI integrated photonic platform yet. Our designed ${\rm{Si}}{{\rm{O}}_2}$-passivated concentric GaP-OI resonator has a device layer thickness of merely 200 nm. Such a thickness only yields normal dispersion for the fundamental mode in strip waveguides in the infrared. Our design, however, can achieve an anomalous dispersion with over 460 nm spanning and a maximum dispersion amplitude of 160 ps/km/nm, 1.69 times lower than a traditional rectangular waveguide with similar anomalous dispersion span. Additionally, we reveal the intracavity field dynamics of mode hybridization and present the geometric dependence on the phase matching condition between the two rings. Unlike previous works [29–32,48] that focused on turning resonators dispersion from normal to anomalous, we take a further step to highlight the flexibility of regulating dispersion peak amplitude and span through synergistical optimization, by means of the waveguide widths and the gap width. Specifically, we reveal the dispersion evolution toward a flattened and broadband profile while maintaining a peak location at 1550 nm. Our design achieves a wider anomalous dispersion span and a lower dispersion maximum compared to other concentric ring resonators. Furthermore, we uncover the influence of the flattened dispersion profile on Kerr soliton frequency comb spectral bandwidth. The bright soliton generated from the dispersion-flattened resonator exhibits a 3 dB bandwidth of 67 nm and a 20 dB bandwidth of over 250 nm at both 1550 nm and 1650 nm pump wavelengths, wider than those from uncoupled single ring resonators. This approach allows unmatched flexibility in dispersion engineering by just tuning the GaP-OI waveguide lateral dimensions. It can also be applied to other material platforms to generate ab arbitrary dispersion profile at desired wavelengths.
2. DEVICE DESIGN AND FIELD DYNAMICS
An artistic illustration of the proposed GaP-OI concentric ring resonator is shown in Fig. 1(a) with its geometric parameters denoted in Fig. 1(b). The GaP device layer is practically assumed to be thin, $\sim 200\; {\rm nm}$, in our design. It is worth noting that the design methodology also works for other waveguide thickness and is not limited to the presented case. The inner and outer waveguides are concentric and share the same center with a radial separation of ${W_{{\rm{gap}}}}$. The radius $R$ is referred to the inner resonator. The inner waveguide width ${W_{{\rm{in}}}}$ is larger than the outer waveguide width ${W_{{\rm{out}}}}$ to facilitate phase matching, which is critical to avoided mode crossing and will be elaborated later. To passivate and protect the GaP waveguide, a 5 nm ALD ${\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}$. layer is deposited after etching, followed by an extra 100 nm ${\rm{Si}}{{\rm{O}}_2}$ cladding layer deposited for further passivation [45] and sidewall smoothening [49]. The inclusion of the ${\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}$ and the ${\rm{Si}}{{\rm{O}}_2}$ passivation layers is considered in simulation to make the design more realistic.
Fig. 1. (a) The concept figure of a concentric coupled GaP-OI resonator. The inset shows the mode profiles of the anti-symmetric mode (upper) and the symmetric mode (lower). (b) The cross section of the surface passivated and direct-bonded GaP-OI waveguide. The waveguide thickness and widths refer to the GaP core dimension. The gap refers to the distance between two GaP cores. (c) The dispersion profile of the concentric coupled rings for symmetric mode (red) and anti-symmetric mode (blue) obtained at ${W_{{\rm{in}}}} = {{750}}\; {\rm nm}$, ${W_{{\rm{out}}}} = {{723}}\; {\rm nm}$, and ${W_{{\rm{gap}}}} = {{355}}\; {\rm nm}$. The orange and purple lines are the dispersion profiles of the ${{\rm{TE}}_0}$ mode in the inner ring and outer ring, respectively. (d) The calculated free spectral ranges (FSRs). The curve legend is the same as in (c).
The fundamental TE modes (${{\rm{TE}}_{0,{\rm in}}}$ and ${{\rm{TE}}_{0,{\rm out}}}$ for the inner and outer waveguide, respectively) of the inner and outer waveguides are coupled with each other, resulting in mode hybridization upon phase matching. It happens when the round-trip optical path lengths (OPL) of the inner and outer rings are equal. The OPL is calculated as ${\rm{OPL}} = 2\pi\! R{n_{{\rm{eff}}}}$ [29,30]. Near the OPL crossing point, the ${{\rm{TE}}_{0,{\rm in}}}$ and ${{\rm{TE}}_{0,{\rm out}}}$ modes evolve into a pair of anti-symmetric mode and symmetric mode. From the coupled-mode theory, one can write the eigenfrequencies of these two hybrid modes as [50,51]
The inner-ring radius is designed to be 140 µm to support a free spectrum range (FSR) of approximate 100 GHz at 1550 nm. Since the radius of the outer ring is larger, the ${n_{\text{eff}}}$ of the inner waveguide should be higher than that of the outer waveguide so as to compensate for the OPL. Our initial design of ${H_{{\rm{wg}}}} = 200\; {\rm nm}$, ${W_{{\rm{in}}}} = 750\; {\rm nm}$, ${W_{{\rm{out}}}} = 723\; {\rm nm}$, and ${W_{{\rm{gap}}}} = {{355}}\; {\rm nm}$ yields an anomalous dispersion with a span from 1.48 to 1.7 µm and a peak amplitude of 466 ps/km/nm as shown in Fig. 1(c). On the contrary, the symmetric mode becomes even more normally dispersed due to the mode hybridization described in Eq. (2). We also add the dispersion profiles of two isolated strip waveguides for comparison. They have the same cross sections and bending radii as the inner (yellow) and outer (purple) waveguides, but neither of them possesses anomalous dispersion at this waveguide thickness. Our earlier work predicted that a minimum of 300 nm thickness is required for the fundamental mode of a single ${\rm{Si}}{{\rm{O}}_2}$-passivated waveguide to realize anomalous dispersion [46].
The existence of mode hybridization could be further verified by looking at the FSRs of each mode in Fig. 1(d). We find that only the anti-symmetric mode (blue) has a negative slope from 1.48 to 1.7 µm, which explains the reason of anomalous dispersion as ${D_\lambda} = \partial ({1/({{\rm{FSR}}(\lambda)L})})/\partial \lambda$. The crossing point location of the FSRs in Fig. 1(d) aligns with the dispersion peak location in Fig. 1(c). Mode hybridization occurs within certain spectral range, out of which the hybrid modes decompose into two individual waveguide modes. We note that the blue line and red line are asymptotic to the single ring ${{\rm{TE}}_0}$ modes at wavelength ${\lt}{1.48}\;\unicode{x00B5}{\rm m}$. However, at the longer wavelengths neither the blue line nor the red line is closely asymptotic to a single ring ${{\rm{TE}}_0}$ modes. This can be attributed to the fact that neither of the strip waveguides can confine each hybrid mode individually at longer wavelengths, and the existence of the neighboring waveguide cannot be neglected.
A close scrutiny on Fig. 1(d) indicates that the distribution of the hybrid modes switches before and after the crossing point. To illustrate the intracavity electric field dynamics, we plot the mode profiles versus wavelengths in Fig. 2, where the anti-symmetric mode evolves from the inner ring to the outer one as the wavelength increases, while the symmetric mode turns out to have the opposite trend. The two hybrid modes can not only be distinguished by their eigenfrequencies based on Eq. (1) but also by the phase of their electric fields. The anti-symmetric mode is out of phase in two coupled resonators, while the symmetric mode is in phase. Near the OPL crossing point, both hybrid modes distribute energy in the two waveguides, and thus their mode distributions look similar, as shown in the inset in Fig. 2. The similarity of the field distribution can be verified by calculating the field overlap integral $\eta$, following the expression as [31]
Fig. 2. Field overlap integral between the two hybrid modes, along with the mode distribution and field dynamics evolving along with the wavelength. The left and right rectangles correspond to the inner and outer waveguides, respectively.
3. DISPERSION ENGINEERING
Following the mechanism described above, we explore the influence of the gap size ${W_{{\rm{gap}}}}$, the outer waveguide width ${W_{{\rm{out}}}}$, and inner waveguide width ${W_{{\rm{in}}}}$ on the resonator dispersion profiles as illustrated in Fig. 3. Given fixed waveguide widths, a widening ${W_{{\rm{gap}}}}$ leads to a sharper and narrower anomalous dispersion region with increased peak values. The dispersion peak undergoes a redshift, which is originated from the movement of the OPL crossing point shown in Fig. 3(b). By contrast, widening ${W_{{\rm{out}}}}$ gives rise to a smoother and broader dispersion profile with decreased peak values. A redshift of dispersion peak location and OPL crossing point is also observed as ${W_{{\rm{out}}}}$ grows. Widening ${W_{{\rm{in}}}}$ gives rise to a sharper and narrower dispersion profile with increased peak values, along with a blueshift of dispersion peak location as well as the OPL crossing point. In principle, the gap size controls the coupling strength $\kappa$, which controls the dispersion flatness and peak amplitude, whereas the waveguide width and radius regulate the phase matching condition that determines the dispersion peak location. It should be noted that varying one parameter may lead to changes of the others, and the final effect is a combined result of the phase matching mechanism and mode coupling mechanism together. The combination of these three parameters opens a wide design space for dispersion engineering and enables complex dispersion profiles that could not be achieved on a single strip waveguide.
Fig. 3. (a) The dispersion profiles of anti-symmetric modes of different values of ${W_{\text{gap}}}$, along with fixed ${W_{\text{out}}} = {{723}}\; {\rm nm}$ and ${W_{\text{in}}} = {{750}}\; {\rm nm}$. (b) The optical path lengths of the inner (blue) and outer (red) rings corresponding to different values of ${W_{\text{gap}}}$. (c) The dispersion profiles of anti-symmetric modes of different values of ${W_{\text{out}}}$, along with fixed ${W_{\text{in}}} = {{750}}\; {\rm nm}$ and ${W_{\text{gap}}} = {{355}}\; {\rm nm}$. (d) The optical path lengths of the inner (blue) and outer (red) rings corresponding to different values of ${W_{\text{out}}}$. (e) The dispersion profiles of anti-symmetric modes of different values of ${W_{\text{in}}}$, along with fixed ${W_{\text{out}}} = {{723}}\; {\rm nm}$ and ${W_{\text{gap}}} = {{355}}\; {\rm nm}$. (f) The optical path lengths of the inner (blue) and outer (red) rings corresponding to different values of ${W_{\text{in}}}$. The red lines in (a), (c), and (e) represent the structure mentioned in Fig. 1(b). (b), (d), and (f) share the same legend. An enlarged edition of (b), (d), (f) is provided (see Supplement 1).
Fig. 4. (a) The evolution of the dispersion profiles for anti-symmetric modes whose dispersion peaks are precisely located at 1550 nm. (b) The comparison of the anomalous dispersion span and peak amplitude when anchoring the dispersion peak location at 1550 nm. (c) The optimized anomalous dispersion profile compared to the anomalous dispersion that comes from the ${{\rm{TE}}_1}$ mode in the single ring. The insets show mode profiles of the anti-symmetric mode and the ${{\rm{TE}}_1}$ mode in the single ring at 1650 nm.
We utilize the design space to fix the dispersion peak location at 1550 nm while flattening the dispersion profile in favor of frequency comb generation. Figure 4(a) depicts several combinations of ${W_{{\rm{out}}}}$ and ${W_{{\rm{gap}}}}$ with fixed ${W_{{\rm{in}}}}$, all of which have peaks at 1550 nm. To anchor the peak location, ${W_{{\rm{out}}}}$ and ${W_{{\rm{gap}}}}$ must be tuned synergistically. Under the condition of inner waveguide’s width and inner radius fixed, we observe a trend that an increased ${W_{{\rm{gap}}}}$ leads to a decreased ${W_{{\rm{out}}}}$. As ${W_{{\rm{gap}}}}$ increases, the outer ring radius is also enlarged. Thus, the ${n_{{\rm{eff}}}}$ of the outer waveguide needs to become smaller to maintain the phase matching condition, which indicates that a narrower outer waveguide should be used. Besides, the reduction of the outer waveguide width also helps counteract the increment of outer ring radius, further reinforcing the relation of ${W_{{\rm{gap}}}}$ and ${W_{{\rm{out}}}}.$ As ${W_{{\rm{out}}}}$ increases and ${W_{{\rm{gap}}}}$ decreases, the dispersion profile becomes broader and flatter, because the dispersion turns from normal to anomalous at longer wavelengths. For convenience, our final optimized structure is referred to as Flattened Coupled Design, and the initial design in Fig. 1 is referred to as Coupled Design. The Flattened Coupled Design has dimensions of ${W_{{\rm{in}}}} = {{750}}\; {\rm nm}$, ${W_{{\rm{out}}}} = {{730}}\; {\rm nm}$, and ${W_{{\rm{gap}}}} = {{230}}\; {\rm nm}$, which exhibits a wide and flat anomalous dispersion spanning from 1.45 to 1.91 µm with a peak amplitude of 160 ps/km/nm. In comparison, the original design has an anomalous dispersion span from 1.48 to 1.7 µm with a peak amplitude of 466 ps/km/nm. The anomalous dispersion span of the optimized design is 2.09 times wider than that of the original design. The peak amplitude of the anomalous dispersion is 2.91 times lower than that of the original one. The dispersion profile of the Flattened Coupled Design is shown by the crimson line in Fig. 4(c). It is intriguing to note that when the coupling gap diminishes, the anti-symmetric mode will ultimately evolve into a ${{\rm{TE}}_1}$ mode at longer wavelength in a single waveguide whose width is the sum of the inner and outer waveguides. To illustrate this point, we add the dispersion profile of the ${{\rm{TE}}_1}$ mode of a single ring in Fig. 4(c) and its mode profile at the wavelength of 1650 nm in the Fig. 4(c) insets for comparison. The single waveguide has a height of 200 nm and width of 1480 nm, the same as the total waveguide width of the coupled waveguides. Despite their mode distributions looking similar, the anomalous dispersion profile of the ${{\rm{TE}}_1}$ mode in the single waveguide is 65 nm narrower in spectral span and 1.69 times larger in peak amplitude than that of the anti-symmetric mode in the Flattened Coupled Design, which undoubtedly highlights the broad and flat feature of the anti-symmetric mode dispersion. This anti-symmetric mode evolution trend toward a higher-order mode was also observed in a coupled resonator with a partially etched gap [32,52]. Dispersion engineering using higher-order modes have been extensively studied in literature and will not be elaborated in this work.
We compare the anomalous dispersion profiles of our Flattened Coupled Design with several reported resonators in Fig. 5. Due to a lack of previous research of mode hybridization in the emerging GaP-OI resonators, our comparison has to target ${\rm{S}}{{\rm{i}}_3}{{\rm{N}}_4}$ resonators whose dispersion engineering has been extensively studied. Two coupled structures are included: the fully etched concentric rings (FECR) and the partially etched concentric rings (PECR) [48,53]. The single rings (SR) are also added for comparison. Compared to other concentric ring resonators, our Flattened Coupled Design has a wider anomalous dispersion span ${{\Delta}}\lambda$ and a lower peak amplitude ${D_{{\max}}}$, representing a broad yet flat anomalous dispersion profile.
Fig. 5. Comparison of the anomalous dispersion between our structure with several reported resonators.
4. KERR FREQUENCY COMB GENERATION
We numerically investigate Kerr soliton frequency comb generation using the anti-symmetric mode in our Flattened Coupled Design. For comparison, we set two groups of structures for comb generation. One group is pumped at 1550 nm (Fig. 6 left column), and the other is pumped at 1650 nm (Fig. 6 right column). New comb lines are generated in the microresonator through the cascaded four-wave mixing (FWM) effect. The resonant frequency ${\omega _\mu}$ of the anti-symmetric mode can be expanded in Taylor series as
where $\mu$ is the mode order, ${D_1}/2\pi$ refers to the FSR around the pump angular frequency ${\omega _0}$, ${D_2}$ is related to the group velocity dispersion ${}{\beta _2}$ by ${D_2} = \frac{{- {\beta _2}}}{{{R^2}\beta _1^3}}$, ${D_3}$ is related to the ${}{\beta _2}$ and third-order dispersion ${}{\beta _3}$ by ${D_3} = \frac{{- {\beta _3}}}{{{R^3}\beta _1^4}} + 3\frac{{\beta _2^2}}{{{R^3}\beta _1^5}}$, and $R$ is the ring radius [33,34]. The integrated dispersion relative to the pump mode is given by ${D_{{\rm{int}}}}(\mu) = {\omega _\mu} - ({{\omega _0} + {D_1}\mu})$. The waveguide integrated dispersions ${D_{{\rm{int}}}}$ for both the initial and optimized designs are drawn in Fig. 6(b). In the optimized dimensions, the second-order dispersion ${D_2}/2\pi$ is extracted to be 1.11 MHz through a parabolic fitting, confirming that the resonator is operating at the anomalous dispersion region. Such a small ${D_2}$ indicates a relatively flat anomalous dispersion curve, which is an enabling factor for a broad frequency comb spectrum.Fig. 6. The left column shows the (a) dispersion, (b) integrated dispersion, and (c) frequency combs pumped at 1550 nm. The right column shows the (d) dispersion, (e) integrated dispersion, and (f) frequency combs pumped at 1650 nm. Flattened Coupled Design: ${W_{{\rm{in}}}} = {{750}}\; {\rm nm}$, ${W_{{\rm{gap}}}} = {{230}}\; {\rm nm}$, ${W_{{\rm{out}}}} = {{730}}\; {\rm nm}$; Coupled Design: ${W_{{\rm{in}}}} = {{750}}\; {\rm nm}$, ${W_{{\rm{gap}}}} = {{355}}\; {\rm nm}$, ${W_{{\rm{out}}}} = {{723}}\; {\rm nm}$; Single Ring: ${W_{{\rm{total}}}} = {{750}}\; {\rm nm}$; Wide Single Ring: ${W_{{\rm{total}}}} = {{1480}}\; {\rm nm}$.
We assume the waveguide propagation loss to be 3 dB/cm, yielding an intrinsic Q factor of $\sim 2 \times 10^5$. This assumption is valid since the state-of-the-art of GaP-OI waveguide propagation loss ranges from 1.2 to 52 dB/cm in literature [40,45,54]. The resonator is critically coupled at the pump wavelength. With proper selection of pump power and laser detuning rate, the frequency combs are solved by a Lugiato–Lefever Equation [55]:
Fig. 7. (a) Group velocity dispersion curves considering different high-order dispersion terms for the Flattened Coupled Design are shown by ${\rm{GVD}} = \mathop \sum \limits_{m = 2}^n {\beta _m}{({\omega - {\omega _0}})^{m - 2}}/({m - 2})!$, where ${\omega _0}$ is the pump angular frequency corresponding to the pump wavelength 1550 nm, and $n$ is the maximum order of dispersion; $n$ is set to be 23 in our polynomial fitting. (b) High-order dispersion values. The pump wavelength is 1550 nm.
Table 1. Comparison of the Anomalous Dispersion Span, Peak Amplitude, and Comb Bandwidth of Several Ring Resonators Mentioned Above
The simulated frequency comb of the Flattened Coupled Design has a tooth spacing of 99 GHz. The spectral profile exhibits a ${\sec}{{\rm{h}}^2}$ envelope, which indicates that there is a single soliton pulse that circulates through the cavity in the time domain, sitting on a continuous-wave background. When stable solitons are formed in the cavity, the relationship between waveguide dispersion and the minimum soliton temporal width is described as [23]
For the group that has pump wavelength 1550 nm, the Flattened Coupled Design yields a ${D_{{\max}}}$ of 160 nm as compared to that of 466 nm of the Coupled Design. Given similar pump power, the generated frequency comb from the Flattened Coupled Design has a 3 dB bandwidth of 67 nm, 16 nm wider than the unoptimized design, and a 20 dB bandwidth of 270 nm, 60 nm wider than its unoptimized counterpart. For the group that has pump wavelength 1650 nm, the Flattened Coupled Design yields a ${D_{{\max}}}$ of 160 nm as compared to that of 270 nm of the single ring with ${W_{{\rm{total}}}} = {{1480}}\; {\rm nm}$. Given the same pump power, the generated frequency comb from the Flattened Coupled Design has a 3 dB bandwidth of 67 nm, 13 nm wider than the single ring, and a 20 dB bandwidth of 253 nm, 27 nm wider than its single ring counterpart (see Table 1).
5. CONCLUSION
In summary, we proposed a dispersion engineering technique based on avoided mode crossing on the emerging GaP-OI integrated photonic platform. This technique is proved to be effective with unprecedented design flexibility. It helps relax the design constraints imposed by the GaP layer fabrication process. Furthermore, it enables a flat and broad anomalous dispersion profile, which cannot be readily obtained from a strip waveguide. Kerr frequency comb generation was numerically investigated on the optimized resonator. The generated Kerr soliton comb has a 3 dB bandwidth of 67 nm and a 20 dB width of 270 nm. Using this technique, we can manipulate the dispersion profile with lateral coupling parameters, such as the two rings’ widths, as well as the gap size. An example of anchoring the dispersion peak location while flattening the dispersion profile has been demonstrated. This work does not aim to boast the superior optical property of a specific design using the GaP-OI concentric coupled resonator, but to demonstrate the dispersion engineering flexibility of the proposed methodology, which is especially useful for thickness-constrained materials or high-loss materials, where a thinner waveguide layer is preferred.
Funding
Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515120074, 2023A1515012141); Open Project Program of Wuhan National Laboratory for Optoelectronics (2021WNLOKF001); National Natural Science Foundation of China (62205137).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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