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Abstract
A systematic dispersion engineering approach is presented toward designing a III-nitride micro-resonator for a blue frequency comb. The motivation for this endeavor is to fill the need for compact, coherent, multi-wavelength photon sources that can be paired with, e.g., the 171Yb+ ion in a photonic integrated chip for optical sensing, time-keeping, and quantum computing applications. The challenge is to overcome the normal material dispersion exhibited by the otherwise ideal (i.e., low-loss and large-Kerr-coefficient) AlGaN family of materials, as this is a prerequisite for bright-soliton Kerr comb generation. The proposed approach exploits the avoided-crossing phenomenon in coupled waveguides to achieve strong anomalous dispersion in the desired wavelength range. The resulting designs reveal a wide range of dispersion response tunability, which is expected to allow access to the near-UV wavelength regime as well. Numerical simulations of the spatio-temporal evolution of the intra-cavity field under continuous-wave laser pumping confirm that such a structure is capable of generating a broadband blue bright-soliton Kerr frequency comb. The proposed micro-resonator heterostructure is amenable to the current state-of-the-art growth and fabrication methods for AlGaN semiconductors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Frequency standards play an important role both in fundamental science and in a wide range of applications such as sensing, metrology, and position, navigation & timing, to name just a few. The field was revolutionized by the advent of the optical frequency comb [1]. Over the past decade, integrated photonic approaches to optical frequency comb generation have received much attention as a potential technology for enabling ultra-precise measurements in a miniaturized platform, wherein the generation of stable, coherent frequency components (also called comb lines) is achieved through the injection of a continuous-wave (CW) laser into a micro-resonator exhibiting third-order (i.e., Kerr) nonlinearity [2–7]. A dissipative Kerr soliton provides a tool for generating coherent and low-noise comb lines over a wide spectral range, circumventing the challenges associated with sub-comb processes in non-solitonic Kerr combs, which suffer from much higher noise.
There has been significant progress on both theoretical and experimental fronts, to the point where we now have fully developed spectro- and spatio-temporal models [8,9] of the nonlinear dynamical Kerr-comb generation process inside a micro-resonator, as well as optimized fabrication processes [10–12] and advanced dispersion engineering approaches [13–17] for micro-resonator design optimization, allowing the manufacture of near-IR micro-resonators with very high quality factors ($Q$) in a variety of material platforms such as Si [18], SiO$_{2}$ [19], Si$_{3}$N$_{4}$ [20–22], Hydex glass [23], diamond [24], quartz [25], AlGaAs [26], AlN [27], CaF$_{2}$ [28], MgF$_{2}$ [29], and LiNbO$_3$ [30,31]. These advances have, in turn, enabled the laboratory demonstrations of frequency-bin entangled photon sources [32], Doppler-cooling of atoms/ions [33], and light detection, ranging, and tracking [34], among others.
While there has been extensive reporting on efficient bright-soliton Kerr-comb generation at infrared, and more recently at near-visible, wavelengths as highlighted above, there has been no report of direct Kerr-comb generation in the blue and UV end of the spectrum. This is largely due to the significant normal dispersion exhibited by candidate materials that otherwise have desirable features such as large bandgaps and reasonably strong third-order nonlinearity at these wavelengths. Indeed, ternary and quaternary III-V semiconductors, and specifically the group III-nitride materials, are very promising for ultrashort pulse generation at short wavelengths, but are unfortunately normally dispersive.
Anomalous dispersion is a favorable, if not essential, ingredient for generating broadband bright-soliton Kerr combs. This is because soliton formation is greatly aided by the anomalous dispersion of the cold (unloaded) cavity, which can compensate for the (normal) Kerr-nonlinearity dispersion of the loaded resonator. Purely geometric approaches to achieving anomalous waveguide dispersion typically lead to feature sizes that are impractical to fabricate in the blue/UV wavelength regime, thereby necessitating a more advanced dispersion engineering approach.
In this paper, we describe a systematic dispersion engineering approach for achieving anomalous dispersion in an AlGaN micro-resonator at blue wavelengths. At the core of the approach is the avoided-crossing behavior exhibited by coupled weaveguides, which we use to our advantage in designing a hybrid structure that is at once reasonably easy to fabricate and highly tailorable in its dispersion response. Similar ideas have been used before at visible and near-IR wavelengths, working with Si or Si$_{3}$N$_{4}$ where $Q$ can be very high [35–40]. Since AlGaN devices have both a normal material dispersion and comparatively low reported $Q$ values in the target wavelength regime, it is particularly challenging to achieve bright-soliton based comb generation. We then describe an AlGaN heterostructure that embodies these dispersion engineering principles toward the demonstration of a Kerr comb with our micro-resonator design.
Another way to obtain anomalous dispersion in a waveguide made of material with normal dispersion is through the use high-order modes, as was done in [41,42]. The waveguide dispersion becomes progressively more anomalous with increasing mode order, which makes it possible to overcome the normal material dispersion. It is interesting to compare this approach with the one presented here. The clear advantage of using a high-order mode to achieve anomalous dispersion is the avoidance of a more complex heterostructure fabrication, as required by our proposed scheme. However, high-order waveguide modes typically suffer from greater radiation loss, so one would expect a lower $Q$ in a micro-resonator configuration. In addition, the (more) complicated transverse profile of a high-order mode makes it harder to excite with an external source. Overall, we expect the extra degrees of design freedom available in our approach to offer a wider range of tunability of dispersion response, which may justify the additional fabrication steps involved.
As another alternative approach, one may consider frequency comb generation using a dark soliton instead, which can theoretically exist in materials with normal dispersion and would therefore obviate the need for anomalous dispersion engineering. However, a rigorous stability analysis [43] has shown that large pump detuning and power are required to facilitate the onset of modulational instability which subsequently leads to the formation of a dark soliton in a normally dispersive resonator, and that, once formed, dark solitons are only weakly stable. These predictions have indeed been borne out in numerical and laboratory experiments [44,45] that demonstrated this very approach to frequency comb generation. On the other hand, bright-soliton based combs, while necessitating advanced dispersion engineering, seem to be relatively easier to generate and maintain, as well as allowing access to a wider bandwidth.
The utility of an AlGaN-based blue/UV frequency comb design is evidenced by the recent interest in optical timing, trapping, and quantum computing applications employing Ytterbium, which has a number of suitable optical transitions; see Fig. 1 [46]. The application of AlN for optical waveguides and micro-resonators has been explored recently [47,48]. Pumped by a quantum-well UV/visible laser grown and fabricated on a common nitride epitaxial wafer [49], a micro-resonator generating a broadband Kerr comb would be an ideal integrated photonic source for such applications.
Fig. 1. Energy diagram for the $^{171}{\textrm{Yb}}^{+}$ ion (adapted from [46]), showing the blue and UV transitions germane to our discussion: $436$ nm and $467$ nm optical clock transitions (blue arrows) and $370$ nm Doppler-cooling transition (purple arrow).
2. Proposed method
An individual waveguide mode with a propagation constant $\beta (\omega )$ has a group velocity $1/v_{\textrm{g}} = \mathrm {d}\beta /\mathrm {d}\omega \equiv \beta '(\omega )$, whose dispersion is, in turn, given by $\mathrm {d}^{2}\beta /\mathrm {d}\omega ^{2} \equiv \beta ''(\omega )$. The two group-velocity dispersion (GVD) regimes are delineated by the sign of this latter quantity; i.e., $\beta '' > 0$ for normal, and $\beta '' < 0$ for anomalous. The III-nitride materials exhibit strong normal dispersion in the blue/UV part of the spectrum, and therefore are not inherently suitable for bright-soliton generation.
Consider instead two waveguides with intrinsic propagation constants $\beta _{1}$ and $\beta _{2}$ that exhibit coherent, evanescent, co-directional coupling. The mutual coupling strengths $\kappa _{12}$ and $\kappa _{21} = \kappa _{12}^{*}$ can be tailored by adjusting the thickness of the gap between the guides. It is well known (see, e.g., [50], § 8.2) that such a hybrid structure possesses a pair of super-modes with propagation constants
where $\kappa = \sqrt {\kappa _{12} \kappa _{21}} = |\kappa _{12}|$, and we defined These super-modes are further distinguished by their symmetric ($+$) and anti-symmetric ($-$) spatial field profiles over the transverse cross-section of the hybrid structure.Now, let $\omega _{\textrm{c}}$ denote the central frequency around which we wish to engineer anomalous dispersion in the hybrid structure. The geometric parameters of the high-index layers are then designed to achieve phase matching between the two guides at this frequency; i.e.,
The propagation constants $\beta _{\pm }(\omega )$ of the hybrid guide display the so-called “avoided crossing” (also known as “level repulsion”) behavior around $\omega _{\textrm{c}}$. This effect, controlled through the coupling parameter $\kappa$, makes it possible to push $\beta _{-}''$ toward negative values, thus suppressing the normal material dispersion and achieving a net anomalous dispersion in the odd super-mode of the hetero-structure.In order to explore this possibility in detail, we differentiate (1) with respect to $\omega$ to obtain
This effect persists over a band of frequencies around $\omega _{\textrm{c}}$, beyond which the guides become phase-mismatched and the dispersion behavior reverts to the normal regime. This anomalous dispersion band is delimited by the zero-dispersion frequencies $\omega _1$ and $\omega _2 > \omega _1$ at which $\beta _{-}(\omega _{1, 2}) = 0$. In order to facilitate the widest possible frequency comb, it is therefore desirable to maximize the difference $\omega _2 - \omega _1$, and this criterion can be included in the design optimization process. Ultimately, the intrinsic absorption band edge of the material will limit the extent to which a sufficiently wideband anomalous dispersion response can be engineered in this way.
3. Dispersion engineering
We now choose a basic hybrid waveguide structure to demonstrate the approach proposed above. We exclusively focus on the odd super-mode, and drop the subscript on $\beta _{-}$. Defining an effective refractive index through $\beta = (\omega /c) \, n_{\textrm{eff}}$, the GVD parameter of this super-mode will be computed via
Figure 2 shows a vertical stack of two rectangular waveguides with heights $h_{\textrm{f}1}$ and $h_{\textrm{f}2}$ and a common width $w$, separated by a gap of height $h_{\textrm{g}}$ and sitting on a pedestal of height $h_{\textrm{p}}$. The two guides are made of $\textrm{Al}_{x} \textrm{Ga}_{1 - x} \textrm{N}$ (with $x_{1} \neq x_{2}$, in general), whereas the gap and the pedestal are made of AlN. The waveguide heterostructure is shown on an AlN substrate with no (dielectric) encapsulation.
Fig. 2. A representative AlGaN hybrid waveguide structure and the field amplitude profile of its fundamental odd super-mode at $\lambda = 442$ nm. The dimensions are $h_{\textrm{p}} = 150$ nm, $h_{\textrm{f}1} = 440$ nm, $h_{\textrm{g}} = 250$ nm, $h_{\textrm{f}2} = 200$ nm, and $w = 700$ nm. The Al mole fractions are $x_{1} = 0.65$ and $x_{2} = 0.4$.
We studied the eigenmodes of this structure numerically using the Wave Optics Module of the finite-element modeling software COMSOL Multiphysics. With consideration of the blue transitions of the $^{171}{{\textrm{Yb}}}^{+}$ ion in mind and guided by the analytical results of the previous section, we selected the geometric parameters and the material compositions of the structure such that the avoided-crossing wavelength falls around $442$ nm. As shown in Fig. 2, the fundamental odd super-mode at this wavelength has an anti-symmetric transverse field profile, as expected, which is a perturbed superposition (with a phase shift of $\pi$ rad) of the two quasi-TE (HE$_{00}$) modes of the individual rectangular guides. The figure caption lists the geometric parameters of this particular structure, which are readily achievable via standard epitaxial growth techniques.
Figure 3 shows the GVD parameter as well as the effective refractive index of the odd super-mode, which were computed using material parameters reported in [51]. As the figure reveals, both AlN and GaN are normally dispersive at these wavelengths, and yet, this particular design of the hybrid structure is able to achieve anomalous dispersion near the target wavelength interval, as indicated by the portion of the dark solid blue curve for which $D_{\lambda }$ goes positive. The corresponding parameters for the even super-mode are also included in Fig. 3 in order to exhibit the avoided-crossing phenomenon described in Sec. 2.
Fig. 3. The GVD parameter $D_{\textrm{odd}}$ (dark solid blue) and the effective refractive index $n_{\textrm{odd}}$ (dark solid red) of the odd super-mode as functions of wavelength near the avoided-crossing point. The corresponding parameters for the even mode are depicted as well (in dashed blue and red, respectively). The dotted curves show the GVD parameters (cyan and pink) and the (ordinary) refractive indices (blue and red) for bulk AlN and GaN, respectively (data from [51]; the index of bulk GaN is shifted down by $0.3$ for plotting purposes).
We next assessed the sensitivity of the GVD parameter to the key geometric parameters of the hybrid structure, namely, its width, the gap height, and heights of the individual guides. The results of this study are summarized in Fig. 4, where the curves with dots indicate the GVD profile of the optimal design shown in Fig. 2; i.e., they are the same as the dark solid blue $D_{\lambda }$ curve shown in Fig. 3.
Fig. 4. The sensitivity of the GVD parameter $D_{\lambda }$ to the geometric parameters of the hybrid structure shown in Fig. 2. The violet curves with dots correspond to the optimal (underlined) design values. (a) Guide width $w = 680$ (blue), $\underline {700}$, and $720$ nm (magenta). (b) Guide 1 height $h_{\textrm{f}1} = 435$ (brown), $\underline {440}$, and $445$ nm (magenta). (c) Gap height $h_{\textrm{g}} = 245$ (blue), $\underline {250}$, and $255$ nm (cyan). (d) Guide 2 height $h_{\textrm{f}2} = 195$ (brown), $\underline {200}$, and $205$ nm (magenta).
Deviations of the waveguide width $w$ from its target value can arise from imprecise etching of the epitaxial layers. Accordingly, we investigated a representative range of $\pm 20$ nm around the nominal design value of $700$ nm. As can be seen in Fig. 4(a), this has an imperceptible impact on $D_{\lambda }$, which is not surprising since $w$ is appreciably larger than $\lambda$ in all three cases.
Deviations from target values of the height parameters can be expected to be negligible, since the epitaxial layer thickness is usually a well-controlled parameter at the level of a few monolayers of resolution. This fact notwithstanding, even small deviations from an optimal design value in the vertical direction may be expected to significantly impact the GVD parameter of the resulting hybrid structure, since this is the direction along which crucial coupling takes place. With this in mind, we investigated a modest $\pm 5$ nm range of variation around the target values of $h_{\textrm{f}1}$, $h_{\textrm{f}2}$, and $h_{\textrm{g}}$. As can be seen in Figs. 4(b) and (d), increasing $h_{\textrm{f}1}$ results in a gradual shifting of the $D_{\lambda }$ peak up and to the left (in wavelength), while increasing $h_{\textrm{f}2}$ shifts the $D_{\lambda }$ peak much more dramatically and in the opposite direction (down and to the right). The reason for this is the considerable difference in the slopes of the dispersion curves $\beta _{1}$ and $\beta _{2}$ near the avoided-crossing point, which are, in turn, controlled by the aspect ratios of the individual guides [50]. Finally, an increase in $h_{\textrm{g}}$ causes little change in the position of the peak but does enhance the value of $D_{\lambda }$ significantly, as seen in Fig. 4(c). This can be understood from our results in the previous section, as follows. The coupling coefficient $\kappa$ is given approximately by an overlap integral between the modes of the individual guides [50]. Consequently, a larger $h_{\textrm{g}}$ translates into a smaller $\kappa$, which, in turn, makes the second term in (6) larger, with the overall effect of driving $\beta ''$ more negative.
As mentioned before, the run-to-run processing variability in these geometric parameters is likely to be smaller than the $\pm 5$ nm range considered here. The observed sensitivity is therefore more relevant as a demonstration of the range of tunability of the anomalous dispersion response. By suitably optimizing the thicknesses of the high-index regions as well as the thickness of the gap, it should be possible to increase or decrease the peak $D_{\lambda }$ value while shifting the positive-$D_{\lambda }$ band toward shorter or longer wavelengths. Another available “knob” in design optimization is the Al composition $x$, which controls the refractive index contrast between neighboring homogeneous regions within the heterostructure, and may be tweaked slightly in order to reduce the burden on geometric parameters in tailoring the anomalous dispersion response of the structure into the desired regime.
4. Comb generation
With anomalous dispersion in the blue part of the spectrum achievable in a III-nitride material system through advanced dispersion engineering as described in the previous section, we are motivated to investigate the formation of a Kerr cavity soliton in a microring resonator with radius $R$ and transverse cross-section as shown in Fig. 2. Closing the waveguide upon itself in this manner results in a (countably infinite) set of cavity modes whose resonant frequencies satisfy $\beta (\omega _{l}) L = 2 \pi l$ for positive integer $l$, where $L = 2 \pi R$ is the cavity length. Inverting this equation for a resonator with radius $R = 40~\mu$m, the cavity free-spectral range is found to be $\nu _{\textrm{F}} \simeq 460$ GHz.
Strictly speaking, one must apply a “curvature” correction to $\beta (\omega )$ of the linear waveguide in order to obtain the corresponding propagation constant of the circular resonator. In addition, there will be a distortion of the transverse mode profile and an increase in the radiation loss due to this curvature. A perturbation-theory solution shows that this correction is of the order $w/R$ (see, e.g., [52,53]); for the geometric parameters chosen here ($w = 700$ nm, $R = 40~\mu$m), this is indeed a negligibly small correction, as Fig. 5 illustrates. For a more comprehensive exploration of the design space, however, it would be advisable to pursue full 3D finite-element eigensolutions for the resonator modes. This is borne out nicely in [51], where resonators with considerably smaller radii were studied and the curvature-induced distortion of the mode fields can be seen clearly.
Fig. 5. The GVD parameters of an infinite straight waveguide and a waveguide with the same cross section (depicted in Fig. 2) that has been bent to form a circular resonator with radius $R = 40~\mu$m.
Assume now that this resonator is pumped by a continuous-wave (CW) laser with amplitude $A_0$ and frequency $\omega _{0}$ that is near a particular cavity resonance at $\omega _{\ell }$. The phase detuning between the pump and the cavity mode is therefore $\delta = 2 \pi \ell - \beta _0 L = (\omega _{\ell } - \omega _{0}) t_{\textrm{r}}$, where $\beta _0 = \beta (\omega _0)$ and $t_{\textrm{r}} = n_{\textrm{eff}}L/c$ is the cavity round-trip time. In an integrated design, the laser light will typically be delivered to the resonator via an evanescently coupled bus waveguide. Let the power transmittance of this input/output port be denoted $T$, whose value depends on the proximity of the waveguide and the resonator as well as on their respective transverse mode profiles. Finally, let $\alpha$ denote the total per-unit-length power attenuation due to material absorption and radiation loss in the cavity. Then, the photon lifetime in the cavity is $t_{\textrm{p}} = 2 t_{\textrm{r}}/(\alpha L + T)$, in terms of which the resonator quality factor is defined as $Q = \omega _0 t_{\textrm{p}} /2$.
Taking the main pumped mode as a frequency reference, the intra-cavity electric field can be written in the form $\mathbf {E}(\mathbf {r}, t) = \Re \left [\mathbf {F}(\rho , z) \, A(\phi , t) \, \mathrm {e}^{-\mathrm {i} \omega _{\ell } t}\right ]$ [9]. With the transverse mode profile $\mathbf {F}(\rho , z)$ determined through the numerical solution of a suitable Helmholtz equation, our attention here is focused on the slowly-varying amplitude $A(\phi , t)$. An equation of motion for $A$ can be obtained by combining the nonlinear Schrödinger equation satisfied by the mode amplitude in a uniform, (third-order) nonlinear waveguide with the infinite-cavity Ikeda map suitable for a circular resonator, as was originally done by Wabnitz and co-workers [54] (also see [55]). Transforming to a frame that is rotating with the group velocity in the cavity by defining $\theta = \phi - (v_{\textrm{g}}/R) \, t$, normalizing time via $\tau = t/t_{\textrm{p}}$, and defining a normalized amplitude through $\psi = \sqrt {\gamma L t_{\textrm{p}}/t_{\textrm{r}}} \, A$, one obtains
For the numerical solution of (7) via the standard split-step Fourier method [57], we take $n_{2} = 3 \cdot 10^{-19}$ m$^{2}/$W [41] and $Q = 10^5$ as conservative estimates of what is achievable with AlGaN (see Section 5 below), compute $A_{\textrm{eff}}$ for the mode profile shown in Fig. 2 and $d_k$ from the exact propagation constant $\beta (\omega )$, and finally assume a real pump power of $A_{0}^{2} \simeq 150$ mW with critical coupling. With the corresponding parameter values $\varDelta = 3.4$ and $S^2 = 3.85$, the solution yields the bright Kerr soliton and the corresponding frequency-comb spectrum shown in Fig. 6. As can be seen, the comb spans a wide spectral range in excess of $75$ nm at the $-70$ dB window, thus providing strong comb lines at the optical clock transitions $E_2$ and $E_3$ of $^{171}{\textrm{Yb}}^{+}$. The presence of dispersive waves (i.e., Čerenkov radiation), evident in both the time- and the frequency-domain plots, indicates that the comb has extended beyond the zero-dispersion wavelengths of the structure. Under critical coupling, $A_{0}^{2}$ scales as $Q^{-2}$, which suggests that the required pump power can be reduced significantly through improved fabrication techniques that can yield higher resonator quality factors.
Fig. 6. Intra-cavity Kerr soliton (top) and associated normalized frequency-comb spectrum (bottom) in a microring resonator with radius $R = 40~\mu$m and transverse cross-section shown in Fig. 2, driven by a CW laser at $\lambda _0 = 442$ nm. Dispersive waves are evident in both figures. The two insets highlight the specific comb lines matching the $^{171}{\textrm{Yb}}^{+}$ ion optical clock transitions at $436$ nm ($E_2$) and $467$ nm ($E_3$), respectively.
In addition to anomalous dispersion, another important consideration for Kerr comb generation is the competition between the four-wave mixing and the stimulated Raman scattering processes. The former is the very mechanism that leads to comb formation while the latter tends to be stronger, and therefore it is crucial to ensure their separation in the frequency domain in order to maintain comb coherence. This seems possible, thanks to the extremely narrow Raman lines in crystalline AlGaN. Focusing attention on the strongest of the Raman-active phonon transitions, the Raman shift and bandwidth are about $590~\textrm{cm}^{-1}$ and less than $5~\textrm{cm}^{-1}$, respectively, for an annealed crystal with the Al composition in the range $0.4 < x < 0.65$ [58] (also see [59,60]). This puts the anti-Stokes Raman line about $\nu _{\textrm{R}} = 17.7$ THz away from the $442$ nm pump, with a bandwidth of $\varDelta \nu = 150$ GHz. With $R = 40~\mu$m, the cavity free-spectral range $\nu _{\textrm{F}} = 460$ GHz comfortably exceeds this Raman bandwidth. In fact, the Raman line falls nearly halfway between two comb lines: $l \nu _{\textrm{F}} < \nu _{\textrm{R}} < (l+1) \nu _{\textrm{F}}$ with $l = 38$. For $Q = 10^5$, the cavity linewidth is only about $\delta \nu = 6.96$ GHz near the Raman line, ensuring that there will indeed be negligible overlap between the tails of the Raman gain spectrum and those of the adjacent cavity lines.
5. Fabrication considerations
The fabrication of the type of structure described in this paper is challenging, but quite realistic considering the recent advances in III-nitride materials development for more conventional opto-electronic devices and high-resolution patterning techniques.
For example, the proposed III-nitride heterostructure can be epitaxially grown on low defect-density native AlN substrates using Metal-Organic Vapor Phase Epitaxy (MOVPE). Lowest absorption levels in the micro-resonator, as it is relevant for realizing highest $Q$ factors, can be achieved with epitaxially grown, crystalline films. With an optimized and well controlled AlGaN growth process, highest optical grade materials have been demonstrated, as shown by active laser operation in the deep-UV (up to $237$ nm) [61] and individual layer thicknesses with sub-nanometer precision (see, e.g., [62]).
Due to the inherently robust material properties of III-nitrides, only dry etching techniques are typically chosen for lateral structuring. Established etching processes of III-nitrides have been shown to produce $90^{\circ }$ sidewalls for $2.5~\mu$m deep etches as it is implemented, for example, for the formation of laser mirror facets (see, e.g., [63]). The presented concept of dispersion engineering with a layered heterostructure has also been successfully simulated using waveguides with tilted side facets, and the results will be shared in future work. Thus, there are fewer constraints on the selected fabrication methods for a successful implementation of the proposed concept [64].
Regarding the achievable quality factors, AlN micro-resonators with $Q = 210,000$ have been demonstrated at $390$ nm in [48], and with $Q > 2 \cdot 10^6$ at IR wavelengths [64]. Despite our use of $Q = 10^5$ in the simulations, we expect to approach $Q \sim 10^6$ for the proposed AlGaN micro-resonator after the fabrication procedure is completely optimized, thus furthermore validating our proposition.
Finally, the transverse mode profile of the heterostructure shown in Fig. 2 will require careful consideration and control of the mode coupling to the resonator. A recent example of successful work in this area with AlN waveguides can be found in [47], where coupling of UV-A light at $369$ nm to an on-chip micro-ring resonator was demonstrated. A second example [65] demonstrated evanescent coupling of blue light from a micro-disk laser to a closely spaced integrated bus waveguide. The short emission wavelength and small waveguide dimensions necessarily require precise control of gap dimensions and waveguide profiles. The authors are confident that state-of-the-art lithography and the above discussed dry etching methods can provide the necessary tools for a successful implementation of our proposed design.
6. Conclusion
In this paper, our goal was to leverage the state-of-the-art photonic materials and fabrication techniques toward designing an integrated optical frequency comb in the short-wavelength-visible and near-UV regions of the spectrum. With the relevant transitions of the $^{171}{\textrm{Yb}}^{+}$ ion targeted for concreteness in the design exercise reported here, we demonstrated the design of an anomalously dispersive hybrid micro-resonator structure using III-nitride family materials that exhibit normal dispersion in this spectral regime. The design approach exploits the avoided-crossing phenomenon in coupled waveguides, and leads to practicably realizable resonator dimensions while allowing for a wide range of tunability of the dispersion response. Numerical solutions of the Lugiato–Lefever equation demonstrated the capability of the designed micro-resonator to support the formation of a broadband Kerr-soliton blue frequency comb under CW pumping. The observed range of tunability of the dispersion response and the ability to vary material parameters, such as Al composition, offer the means to access other bands in the visible and UV-A spectra for other applications.
Disclosures
The authors declare no conflicts of interest.
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