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Abstract
We study nonlinear mode interaction in the cylindrical microresonator with nanoscale radius variation. We obtain a model which describes the interplay between azimuthal modes and investigate the nonlinear dynamics of axial-azimuthal modes. We also define regimes of generation depending on the pump parameters. Using a properly designed radius longitudinal profile, we obtain broadband generation of stable low-repetition-rate frequency combs based on axial-azimuthal modes.
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1. Introduction
Generation of optical frequency combs (OFC) is in particular interest for the last few decades due to its diverse range of applications including precise measuring of distance in astronomy, atomic clock, GPS navigation, precise spectroscopy, chemical and biological sensing, and other [1–5]. The OFC can be considered as a precise rule in optics. The main comb characteristics are spectral width, coherence, and Free Spectral Range (FSR) - the distance between lines in the comb. In applications such as dual-comb spectroscopy [6] the smaller FSR is preferable as it ensures higher resolution of spectroscopic measurements.
There are two alternatives to generate the frequency combs mode-locking lasers and microresonators. The first platform (see e.g. [7]) allows to generate OFC with small FSR, but it requires significant power consumption and possesses a relatively large size. In its turn, microresonators possess high Q-factor, small dimensions but their FSR is limited by the resonator dimension. In ring-type microcavities, the frequency interval is determined by the spectral distance between the azimuthal modes and is 10 - 1000 GHz [3,8]. Generation of optical frequency combs in microresonators with a frequency interval of less than 10 GHz is an important task.
A promising platform for generating low-repetition-rate OFC is Surface Nanoscale Axial Photonics (SNAP) [9]. The platform is a microcavity of whispering gallery modes (WGM) propagating along the boundary of a standard optical fiber with a specially designed effective radius variation on the nanometer scale (see Fig. 1). The source of the WGM is the input/output microfiber located perpendicular to the fiber axis $z$. In this case, WGMs experience slow axial propagation and can be localized along the fiber axis in the region of the effective radius variation [10]. In a cylindrical microcavity, the azimuthal mode is split into a set of axial modes depending on the shape of the radius variation profile. The free dispersion region for axial modes can be hundreds of megahertz or less. This is a promising characteristic of this type of microcavity for generating a low-repetition-rate optical comb [11,12]. However, in these works the comb generation was considered in the approximation of self-action of one azimuthal mode. In practice, there is an interplay between different azimuthal modes that should be taken into account.
Fig. 1. SNAP platform: fiber resonator with nanoscale radius variation coupled to the microfiber, which serves as a source as well as for the detection.
Note that such type of mode interaction was considered in [12–14] in the framework of 2D Lugiato–Lefever model. In the work [13] the authors studied bifurcation of nonlinear modes from a set of axial modes and then investigated azimuthal modulational instability that leads to the formation of soliton solution. An experiment was carried out in [14] demonstrating the interplay between different azimuthal modes where additionally the Raman effect played its role.
We investigate dynamics of comb formation following the typical experimental conditions, including pump tuning, perturbations introduced by the exciting waveguide. For this, we expand a model considered in [11] and take into account azimuthal mode interactions. With the model, we perform first principle investigation of the idea proposed in [15] where they offer concept of a low repetition rate broadband comb generator based on SNAP technology. Note that the interplay of only two azimuthal modes was also considered in [16]. In this work, though, a special case with one mode being much weaker than another was studied. The similar approach based on a system of nonlinear Schrodinger equations was introduced in [17] for two azimuthal modes and in [18] for many modes case, though the proposed models didn’t comprise localization effects due to effective radius variations.
In the first section, we present the master model of nonlinear mode interaction, next we design the potential for comb generation and then numerically investigate different regimes of nonlinear mode generation depending on source parameters.
2. Nonlinear mode interaction in the SNAP fiber resonator
The microfiber perturbs an electrical field inside the SNAP resonator that can be decomposed into the set of eigenmodes as was shown in [19]:
where $\omega _{m}$ – eigenfreqency of an azimuthal mode, $m$ and $p$ – azimuthal and radial quantum numbers, correspondingly, $e_{m,p}(r,\varphi )=e_{m,p}(r)e^{im\varphi }$ – space distribution of the mode in the crossection of the fiber, $A_{m}(z,t)$ – slowly varying mode amplitude such that $\omega _mA_m(z,t)\gg \partial _tA_m(z,t)$. This approach is valid if the radius variation of the fiber satisfies the condition $\Delta r(z)\ll r_0$ and the field slowly propagates near the resonator surface in the axial direction. Note that we consider only the linear polarization of light, thus $e_{m,p}$ is scalar function. We also dwell on consideration of one radial mode $p=1$ which has a distribution maximum near the fiber surface since this mode has the largest overlap integral with the radiation source.Starting from Maxwell’s equations and using ansatz (1) for the electric field, we have obtained a model describing the dynamics of azimuthal modes which coupled by means of Kerr nonlinearity:
Model (2) includes energy dissipation $V_{diss_i}(z)=D\omega _i/(2\beta _i)f(z)/(\sqrt {2\pi }a)$, where $f(z)=\exp {\left (-\frac {1}{2}\left (\frac {z}{a}\right )^2\right )}$ determines the spatial distribution of the input/output microfiber impact along the fiber axis $z$, $a=1 \mu\textrm{m}$ is characteristic width of microfiber, $D=i 0.0158 \mu \rm {m}^{-1}$ is the coefficient describes the additional losses in the system introduced by the microfiber, which typical value is taken from [20]. The model also includes internal losses in the resonator $\Gamma _{i}=\gamma \omega _i^2/(2\pi c)$, where loss coefficient $\gamma = 0.1 \textrm{pm}$ was chosen as in [11]. The Q-factor of the resonator is defined by the losses introduced by the material and surface $\gamma$ and a taper $D$, and it is of the order of $10^5$ for the parameters under consideration.
The source function is expressed as $f_{s_i}(z,t)=E_s/E_0\left (\omega _i\Delta r(0)/r_0+\Delta \omega _{si}\right )f(z)\exp (i\Delta \omega _{s_i}t)$, where $E_s$ is the source field amplitude, $\Delta \omega _{si}=\omega _s-\omega _i$ - detuning of the source frequency $\omega _s$ from the azimuthal mode resonance frequency. The potential $V(z)=\Delta r_\textrm{eff}(z)/r_0$ is determined by the effective radius variation along the fiber axis, where $r_0=62.5\,\mu\textrm{m}$ is fiber radius and defines the structure of axial eigenmodes of the resonator.
A combined numerical scheme was suggested to solve the system of nonlinear equations. The system can be represented in the following operator form: $i\cfrac {\partial \vec {A_{i}}}{\partial t}+\hat H\vec A_i=\vec f_{s_i}(z,t)$, where $\hat H = \hat {D}+\hat V+\hat L+ \hat N$ includes operators of dispersion, potential, losses, and nonlinearity, respectively. The inhomogeneous equation is solved using the modified Euler method and the action of the operator $H$ is derived using the split-step method. The nonlinear step is expressed in terms of a matrix exponential, which is calculated by the Padé approximation.
3. Potential engineering
Equations 2 describe the dynamics of azimuthal modes along the fiber axis $z$ in a cylindrical WGM microcavity with an effective radius variation. The azimuthal modes localize in the region of nonzero effective radius variation along the fiber axis and split into a set of axial modes. The form of the effective radius variation determines the structure of the resonator eigenmodes. To generate a broadband optical comb in the entire range of the studied azimuthal modes, it is necessary to properly choose a potential such that the spectrum of the resonator eigenmodes consists of equidistant axial-azimuthal modes. We consider a parabolic potential of the form $V(z)=\cfrac {\delta r}{r_0}\left (1-\left ({2z}/{l}\right )^2\right )$, since it has equidistant eigenvalues. The structure of the axial-azimuthal eigenmodes is described by the linear stationary Schrödinger equation [19]:
Figure 2 schematically shows the position of the levels in the potential corresponding to the axial modes for three azimuthal numbers $m = 377, 378, 379$.
Fig. 2. Eigen frequency distribution of axial-azimuthal modes numbered $m = 377, 378, 379$ in the resonator with parabolic effective radius variation.
Each of the three azimuthal modes is split into 86 axial modes corresponding to the levels of the parabolic potential. The frequency distance between the axial modes $\textrm{FSR}_{ax}=5.9\,GHz$ is determined solely by the effective radius variations and thus is the same for all azimuthal modes. The spectral height of the potential for the mode $m=378$ is set to be equal to the frequency distance between the adjacent azimuthal modes $\nu _m-\nu _{m-1}=\textrm{FSR}_{az}=508\, GHz$. The spectral width of the axial series for the azimuthal mode $m=378$ is $\nu _{m=378,q=86}-\nu _{m=378,q=1}=502\, \textrm{GHz}$, apparently being smaller than the spectral height of the potential. Because the potential height is determined by $\Delta \nu _m = \nu _m \delta r/r_0$, it is directly dependent on the azimuthal mode number, and the potential height for $m>378$ exceeds $\textrm{FSR}_{az}$, hence the spectral width and the number of axial modes series corresponding to azimuthal modes with $m>378$ increase. So, for 11 modes with $m=373-383$, the spectral width of axial series changes in the range $496.3-508.2~ \textrm{GHz}$, and the number of axial modes in the series varies from 85 to 87. This leads to a violation of the equidistance of the axial-azimuth modes at the edges of the series of axial modes, that is, to the dispersion of the axial-azimuthal modes. Notably, potentials are not overlapped strongly, and there is no overlap between modes of different axial series.
4. Nonlinear dynamics of azimuthal modes
To study the nonlinear mode behavior we numerically solve Eqs. (2) for different input power levels. Thus about 100 simulations of azimuthal modes dynamics were carried out at various pump powers which correspond to different values of $\Delta n$. The pumping is carried out at a frequency corresponding to the central axial-azimuthal mode $\omega _s=191.75\,\textrm{THz}$ ($\lambda _s=1.56\,\mu\textrm{m}$) with quantum numbers $m_0=378$ and $q_0=43$. This mode has a maximum spatial distribution at the point $z = 0$, where the source is located. Initially the system contained uniformly distributed noise over $z$.
As a result we observe mainly two types of nonlinear dynamics shown in Fig. 3 for 11 azimuthal modes with $m=373-383$. The first is when no generation occurs. Figure 3(a) demonstrates that the amplitude of the pumping azimuthal mode reaches a constant value in about 10 ns, while the noise in other azimuthal modes decays. The second behavior is the chaotic generation of the azimuthal modes. Figure 3(b) illustrates that at the moment of time $t=10\, \textrm{ns}$ nonlinear generation of azimuthal modes begins, while their synchronization is not observed.
Fig. 3. The amplitudes of 11 azimuthal modes in $z = 0$ a) in the absence of nonlinear generation, $\Delta n=0.004$ ($E_{max}=4.5\cdot 10^8 \rm ~V/m$) - the maximum field amplitude in the resonator and b) in the regime of chaotic nonlinear generation, $\Delta n=0.035$ ($E_{max}=19.5\cdot 10^8 \rm ~V/m$).
The more rare behavior was also observed, namely the generation of a stable optical frequency comb when the pump power was determined by $\Delta n=0.022$. Figure 4, a presents the nonlinear generation of azimuthal modes starting at the moment of time $t = \textrm{5}\,ns$ and synchronization of all 11 azimuthal modes starting after 15 ns from the initial moment of time. The mode-locking regime is maintained for the next 45 ns and after (Fig. 4, b), while the pulse length was 8 ps.
Fig. 4. The amplitudes of 11 azimuthal modes at $z = 0$ in the mode-locking regime, $\Delta n=0.022$ ($E_{max}=18.7\cdot 10^8 \rm ~V/m$) a) at the beginning of generation, b) after 55 ns from the beginning of generation. Inset shows the pulse duration of 8 ps. c) Spectrum at the moment of $t = 55 \textrm{ns}$, representing a stable comb with a repetition rate of 12 GHz.
As it is shown in Fig. 4, c the stable optical frequency comb is observed. The repetition rate is 12 GHz and it corresponds to the spectral distance between the axial-azimuth modes with the same parity. The source located at the point $z = 0$ excites the mode with the odd axial number $q=43$. Since the overlap integrals for modes with different parities are small, modes with odd axial numbers only participate in the generation. Thus, the possibility of generating a stable optical frequency comb in a cylindrical resonator with the effective radius variation was demonstrated.
The generation threshold can be identified with some precision. The nonlinear dynamics of the system depends on the particular realization of the initial noise. We have found that for pump power corresponding to $\Delta n<0.02$ generation is not observed, while at $\Delta n>0.02$, there could be chaotic generation, mode-locked generation, or absence of generation depending on the noise realisation.
The generation of the optical frequency comb is obtained by varying the pump power, which is not an optimal method for systematically obtaining the OFC. Therefore, we consider a method to obtain generation by tuning the frequency of the source [5]. For this, we chose the pump power $\Delta n=0.004$ at which nonlinear generation of modes has not been observed at the first simulations (Fig. 3(a)) and study if a comb can be obtained in this case.
Under the influence of nonlinearity, the effective potential is distorted $V_{i}^{(\textrm{eff})}(z,t)=V(z)+\frac {3\Delta n}{2nI_{i}}\left (I_{iiii}|A_i(z,t)|^2+2\sum _{j\neq i}I_{jjii}|A_j(z,t)|^2\right )$, which leads to a shift of axial modes and distortion of their spatial distribution. Figure 5, a shows the frequency offset of the source and the nonlinear axial-azimuthal modes of the resonator with quantum numbers $m=378$, $q=41-45$ as a function of time.
Fig. 5. a) The tuning of pump frequency $\Delta \nu _s$ (black line) and corresponding spectral position of nonlinear axial-azimuthal modes with quantum numbers $m=378$, $q~\!=~\!41~\!-~\!45$. b, c) The time dynamics for amplitudes of 11 azimuthal modes in $z = 0$ in the partial mode-locking regime, $\Delta n=0.004$.
Interestingly, nonlinear generation of azimuthal modes begins right after the moment $t = 11.5\,\textrm{ns}$ when the pump frequency coincides with a mode with number q = 43 (Fig. 5(b)). The effective excitation of the microresonator becomes possible as a nonlinear distortion of the effective potential considerably changes the mode distribution. We were tuning the pump frequency observing the chaotic regime of nonlinear generation till the moment $t=23.88\,\textrm{ns}$ when some periodic pulses appeared. We then fixed the pump frequency, setting the stable partial mode-locking regime (Fig. 5(c)). Thus we conclude that tuning of the pump frequency facilitate generation a stable optical comb. We believe that adjusting the tuning rate and proper choice of the final pump frequency allows achieving a fully coherent well mode-locked regime.
5. Discussion and conclusion
In this work, we present the mathematical model describing the nonlinear dynamics of azimuthal whispering gallery modes on the surface of an optical fiber with nanoscale effective radius variations. Using the implemented framework, we studied the onset of the broadband nonlinear generation in both chaotic and mode-locked regimes. The latter regime corresponds to the generation of OFC. We demonstrated that the tuning of pump frequency does support the mode-locking.
The number of azimuthal modes in the current model was limited to 11 as computations in a model with a large number of equations require significant computational power. At the same time, in the experiment, the number of modes is limited by the dispersion of the azimuthal modes. Such dispersion can be taken into account in the model as well. First, our model does comprise some part of dispersion, that appears from the dependence of the effective potential height on the azimuthal mode number. Indeed, according to the Eqs. (2), (4), the effective potential is preceded by coefficients that depend on the frequency of an azimuthal mode $\omega _i$. Second, the model allows taking into account the dispersion right from experimental measurements by introducing proper values of azimuthal resonances $\omega _i$, since they are parameters of the system (2).
Another way to take into account the material and geometric dispersion of azimuthal modes is using analytical estimates. Azimuthal mode frequencies can be approximated using the approach presented in [21] for the WGMs at the perfect cylinder. The relation for the eigenfrequencies includes the refractive index, which in its turn also depends on the frequency. Using the Sellmeier equation for the refractive index, the frequencies of the azimuthal modes were found as solutions to the implicit equation for the resonant frequency. We performed simulations for 11 azimuthal modes pumped into the central mode $m_0=366$ corresponding to the frequency $\omega _{m_0}=194.712\,\textrm{THz}$ ($\lambda _{m_0}=1.544\,\mu\textrm{m}$) with tuning the frequency of the source and with the same parameters as on (Fig. 5). The results showed that the presence of azimuthal mode dispersion disrupts the partial mode-locking regime for a given set of parameters. Thus, a more detailed study is required to achieve the partial mode-locking regime in a system with azimuthal mode dispersion.
Funding
Ministry of Education and Science of the Russian Federation (FSUS-2020-0034); Russian Foundation for Basic Research (20-32-70170).
Acknowledgments
The authors would like to acknowledge Dr. O. A. Gorbunov for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon reasonable request.
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