Intra-cavity field dynamics near avoided mode crossing in concentric silicon nitride ring resonator

Optical resonator supports multiple mode families depending on the number of total internal reflection a light ray suffers within a single round trip. These are called resonant modes $(M)$ of the system, which can be estimated from the relation $2\pi Rn_{\text{clad}}/\lambda_{p}\leq M\leq 2\pi Rn_{\text{core}}/\lambda_{p}$, (more specifically can be defined as, $M=2\pi Rn_{eff}/\lambda_{p}$ ) where, $n$ is the refractive index of the material and $R$ is the radius of the cavity Lam . A stable single mode-locked pulse dynamics can be interpreted as a double balance between Kerr nonlinearity, dispersion, the external CW pump, and the total loss in the dissipative systems Haldar ; Roy . Moreover, anomalous dispersion is desirable to obtain a bright soliton pulse and broad FC. Resonators with normal dispersion, in general, hinders the possibility of CS, where coupling between two normal dispersion modes modifies the local dispersion of one mode to the anomalous region and helps to produce a stable, bright pulse in the same resonator system Fujii ; Liu . Researchers have studied the dynamics of the resonator field in the presence of various mode coupling phenomena such as coupling between clockwise-anticlockwise modes Fujii2 ; Yoshiki , orthogonally polarized modes Saha ; Suzuki ; Hansson2 or modes which encounters avoided mode crossing (AMC) Fujii ; Liu ; Aguanno ; Lucas .
In the present work, our sole interest is to study the temporal and spectral dynamics of the resonator field in presence of mode coupling near the AMC region in a concentric microring resonator. When two modes of a resonator system have an identical resonant frequency, a new pair of hybrid modes may appear due to mode coupling between them Wiersig ; Soltani . We find out the intra-cavity field dynamics in two regions i.e. strongly coupled hybrid mode region (SCR) and weakly coupled hybrid mode region (WCR) based on the eigenfrequency separation of the two-hybrid modes Soltani . We observe oscillations of peak power as well as transfer of energy between the hybrid modes in the temporal domain and periodic variation of FC width in the spectral domain. We also witness non-identical and identical phase variations of the mode field in SCR and WCR respectively. To distinguish the pump wavelength region within which hybrid modes will exist, we calculate the overlap integral (OI) between the two hybrid mode fields. Depending on the OI values, we obtain the boundary region for strongly coupled and weakly coupled hybrid modes. Outside the boundary region for weakly coupled hybrid modes, we can safely say that hybrid modes do not exist as their resonance frequencies are too far to have sufficient amount of spatial overlap. We further verified the field dynamics theoretically with the help of Rayleigh’s dissipation function (RDF) in the framework of variational principle RoyJLT ; Saha2 ; Sahoo . We also study the polarization state of the intra-cavity field in these two separate regions with Stokes parameters and Jones vector.

In this section, we model and characterize concentric microresonators exhibiting avoided mode crossing Soltani ; Kim . The concentric microresonators is made of Si₃N₄ with SiO₂ cladding, shown in Fig.1(a). The structural parameters that we consider for this waveguide are: thickness (h) of the individual ring is $300$ nm, ring radius (R) is $50$ $\mu$m, widths (W${}_{\text{in}}$ and W${}_{\text{out}}$) are $2600$ nm and $1085$ nm for inner and outer rings, respectively, the gap ($g$) between the two rings is $900$ nm. The design geometry is shown in Fig.1(b). It is observed that in the absence of any mode-coupling, the effective refractive index ($n_{\text{eff}}$) of the fundamental transverse electric modes (TE${}_{0}^{\text{inner}}$ and TE${}_{0}^{\text{outer}}$ ) for two resonators crosses each other at a particular wavelength. In Fig.1(c), we plot $n_{\text{eff}}$ as a function of $\lambda$ where the cross-over takes place around $\lambda\approx 1.57$ $\mu$m (solid lines). This AMC position can be tuned with structural parameters like $g$ or W${}_{\text{in}}$ and W${}_{\text{out}}$. The hybrid mode, which moves from outer resonator to inner resonator with increasing wavelength, is known as symmetric mode and the mode which undergoes a transition from inner resonator to outer one, known as asymmetric mode, respectively Kim . We demonstrate the vectorial field profile of the electric field confined in the two hybrid modes (indicated by arrow lines). It is evident that, for the symmetric mode, the electric field orientation in two coupled resonators are in the same direction whereas, for anti-symmetric modes, the orientation of the field is in opposite direction and the field also reverses its direction in either side of the AMC point. From the coupled-mode analysis for a loss-less resonator, one can readily write down the eigenfrequencies $(\omega^{\text{as,s}})$ of these two hybrid modes as Fujii :

Exploiting the ($n_{\text{eff}}$-$\lambda$) plot obtained from the mode analysis in COMSOL, we calculate the eigenfrequencies of the modes both in the uncoupled $(\omega_{1},\omega_{2})$ and coupled conditions using the following relation $\omega=\frac{2\pi c}{n_{\text{eff}}\lambda}$. Thus, we readily obtain the mode coupling value $(\kappa)$ from Eq. (1). The resonant modes have slightly different coupling factor i.e., $\kappa_{1}\neq\kappa_{2}$ because the concentric ring waveguides are slightly different in size. We further use coupling coefficient $(\kappa)$ : $\kappa=\sqrt{\kappa_{1}\kappa_{2}}$ as defined in Okamoto . In Fig.2(a), we demonstrate the resonance splitting $\Delta\omega_{R}=(\omega_{as}-\omega_{s})$ of two hybrid modes which is normalized by the free spectral range ($FSR_{\text{outer}}$) of the outer ring resonator. We mark two regions for hybrid modes namely, (i) strongly coupled hybrid mode region (SCR) (1.53 $\mu$m $\leq\lambda\leq$ 1.61 $\mu$m) and (ii) weakly coupled hybrid mode region (WCR)(1.49 $\mu$m $\leq\lambda<$ 1.53 $\mu$m and 1.61 $\mu$m $<\lambda\leq$ 1.67 $\mu$m). In SCR, the two hybrid modes are correlated with the eigenmodes of each microring and can exist only when $\Delta\omega_{R}/FSR_{\text{outer}}\leq 1$. For WCR, $1<\Delta\omega_{R}/FSR_{\text{outer}}<2$, indicating reduced coupling effect between the hybrid modes. In Fig.2(b), we demonstrate the variation of effective mode area ($A_{\text{eff}}$) for both hybrid modes as a function of pump wavelength. Note that, $A_{\text{eff}}$ attains a peak value near AMC wavelength. In Fig.2(c), we plot the variation of group velocity inverse for two modes, $\beta_{\text{1(as,s)}}=v_{\text{g(as,s)}}^{-1}$, where $v_{\text{g(as,s)}}$ represent the group velocities of corresponding hybrid modes. It is observed that at AMC wavelength, both the modes have nearly identical $\beta_{1}$ values. In Fig.2(d), we demonstrate the group velocity dispersion (GVD) profile for symmetric and anti-symmetric modes. It is interesting to note that under mode coupling condition, symmetric mode always remains in normal GVD regime. Conversely, in the SCR the anti-symmetric mode undergoes anomalous GVD regime. Vertical dotted lines mark the lower and upper cutoff values of the SCR.

In order to study the intra-cavity field dynamics, we obtain the fundamental mode $(\omega_{l0})$ family whose eigenfrequency is nearest to the pump frequency $(\omega_{p})$. For two hybrid modes with different $n_{\text{eff}}$, the fundamental pump mode number $(l_{0\text{(as,s)}})$ and their eigenfrequency $(\omega_{l0})$ will be slightly different $(l_{0\text{(as,s)}}=\frac{n_{\text{eff(as,s)}}\omega_{p}R}{c})$ for a single $\omega_{p}$. One can expand the eigenfrequencies of the fundamental mode family $(l)$ using a Taylor series expansion to obtain for first N elements Grelu :

where $\omega_{l0\text{(as,s)}}$ is the eigenfrequency of the individual pump mode (nearest to the external pump frequency) and $\zeta_{k}=\frac{d^{k}\omega}{dl^{k}}|_{l=l_{0,\text{(as,s)}}}$. Now, $\zeta_{1}=\frac{d\omega}{dl}|_{l=l_{0,\text{(as,s)}}}=\Delta{\omega_{\text{FSR}}}$ is the FSR of the corresponding resonator supporting the hybrid mode at pump frequency. The resonator being dispersive, the eigenmodes are not exactly equidistant, $\zeta_{2}=\frac{d^{2}\omega}{dl^{2}}|_{l=l_{0,\text{(as,s)}}}$ is the second order dispersion coefficient for the hybrid modes. $\zeta_{2}$ can be related to the well-known notation for GVD ($\beta_{2}$) as, $\zeta_{2}=-\frac{c}{n_{\text{eff}}}\zeta_{1}^{2}\beta_{2}$ Grelu . We study field dynamics at different pump frequencies near the AMC region. At each time, resonant mode number changes with external pump frequency and correspondingly $\omega_{l0(\text{as,s})}$, $\Delta{\omega_{\text{FSR}}}$, dispersion coefficients are calculated from the Taylor expansion coefficients.

The field dynamics inside a dissipative system like Kerr microresonator can be modelled by the well-known Lugiato-Lefever equation (LLE) Lugiato , which incorporates the Kerr nonlinearity, dispersion, external pump source and the total loss of the system. Apart from all these parameters, this system being dissipative, the field dynamics is highly sensitive to the external parameters like frequency detuning, i.e., the difference between pump frequency and the nearest cavity mode resonant frequency ($\omega_{p}-\omega_{l0(\text{as,s})}$). Here, we use coupled LLE to find the dynamics of hybrid modes experiencing the AMC. The coupled LLE’s are as follows:

In the construction of coupled LLE, Eq.(4) and Eq.(5) ${u_{as}}$ and ${u_{s}}$ represent the field amplitudes of anti-symmetric and symmetric modes, respectively. In this scenario, either one of the hybrid modes (anti-symmetric or symmetric) is pumped at any particular pump frequency depending on, which mode confines in the outer microring resonator. At the lower pump frequency, anti-symmetric mode is excited $(F_{as}\neq 0,F_{s}=0)$ whereas for higher pump frequency, symmetric mode is excited $(F_{as}=0,F_{s}\neq 0)$. The coupling coefficient ($\kappa$) arises due to the degeneracy in eigenfrequency of two different resonator eigenmodes. Here, $\kappa$ is equivalent to the separation between the two hybrid modes, which varies with pumping wavelength near the AMC region suggesting the linear coupling effect between these two hybrid modes. Due to the sufficient amount of overlapping field between these modes near AMC, we also include nonlinear cross-phase modulation (XPM) effect (sixth term on the right hand side of coupled equations (Eqs.4-5)). Though, for the choice of the external parameters in this case, we have seen the effect of XPM is negligible compare to the effect of linear coupling between the two modes (see, Appendix 11). We further neglect any coherent coupling terms. It is possible when light is confined within the resonator for a long time covering multiple round trips, such that the traverse path length is much greater than the corresponding beat length Agrawal . Here, $t$ is the slow time, $\theta$ is the azimuthal coordinate of the resonator, $\Delta\Gamma_{\text{as,s}}$ is the resonance linewidth of a mode which can be determined from the relation $\Delta\Gamma_{\text{as,s}}=\frac{\omega_{l0\text{(as,s)}}}{Q}$. The Q-factor of a cavity can be calculated as $Q=\frac{\omega_{p}T_{\text{rt}}}{l_{\text{rt}}}$ where $T_{\text{rt}}=\frac{2\pi}{\zeta_{1}}\simeq 1.98$ ps is the round trip time and $l_{\text{rt}}$ is the round trip loss. Round trip number can be estimated as, ‘total time confined/ round trip time’=$t/T_{rt}$. The nonlinear coefficient $g_{0}$ is defined as, $g_{0\text{(as,s)}}=\frac{\hbar\omega_{l0\text{(as,s)}}^{2}n_{2}\zeta_{1\text{(as,s)}}}{2\pi n^{2}A_{\text{\text{eff(as,s)}}}}$, $\hbar$, $n_{2}$, $n$, $A_{eff(as,s)}$ are the reduced Planck constant, Kerr coefficient, the linear refractive index of the medium and effective area of the mode, respectively. $\kappa$ is the mode coupling constant, $F_{\text{as,s}}$ is the external pump field defined as $(1/2)\Delta\Gamma_{\text{as,s}}F_{\text{as,s}}=\sqrt{\Delta\Gamma_{\text{as,s}}}\sqrt{\frac{P_{\text{in}}}{\hbar\omega_{l0,\text{(as,s)}}}}$, where $P_{\text{in}}$ is the input power. We observe linewidth of the hybrid modes are of the same order, i.e. $\frac{\Delta\Gamma_{as}}{\Delta\Gamma_{s}}\approx 1$. So, from now onward we designate $\Delta\Gamma_{as}\approx\Delta\Gamma_{s}=\Delta\Gamma$ $\approx 50\text{ GHz}$. $\sigma_{\text{as,s}}=\omega_{p}-\omega_{l0\text{(as,s)}}$ is the detuning between the pump and resonance frequencies of the resonator. $\Delta\zeta_{1}=\zeta_{1,as}-\zeta_{1,s}$ is the difference between cavity FSR for the two hybrid modes. The spectral variation of FSR for the hybrid modes follow the similar pattern of $\beta_{1}$ (Fig.2(c)), mentioned in Aguanno . For numerical simulations, it is useful to normalize the coupled LLE which takes the form,

where the parameters are rescaled as : $\psi_{as}=\sqrt{\frac{2g_{0as}}{\Delta\Gamma}}u_{as}$, $\psi_{s}=\sqrt{\frac{2g_{0s}}{\Delta\Gamma}}u_{s}$, $\alpha_{\text{as,s}}=-\frac{2\sigma_{\text{as,s}}}{\Delta\Gamma}$, $D_{2\text{(as,s)}}=\frac{\zeta_{2\text{(as,s)}}}{\Delta\Gamma}$, $D_{1}=\frac{2\Delta\zeta_{1}}{\Delta\Gamma}$, $\chi=\frac{\kappa}{\Delta\Gamma}$, $S_{\text{as,s}}=\sqrt{\frac{4g_{0\text{(as,s)}}P_{\text{in}}}{\Delta\Gamma^{2}\hbar\omega_{p}}}$ and $\tau=\frac{\Delta\Gamma}{2}t$. For simplicity, in simulations, we have excluded all higher-order dispersion and nonlinearity terms. We choose our initial wave-function of the form,

where, $\eta_{as,s}=\sqrt{2|\alpha_{as,s}|}$ and $\alpha_{as,s}$ is the normalized detuning parameter for the two hybrid modes. For a particular pump wavelength, the difference between the eigenfrequencies of the hybrid modes are very small $(\omega_{l0\text{(as)}}\approx\omega_{l0\text{(s)}})$, such that the detuning value for both modes are nearly same. Here, the normalized detuning value $\alpha_{as,s}$ is considered as unity for both cases. Further, we analyze two regions of strongly and weakly coupled hybrid modes on the basis of their eigenfrequency separation. We observe damped oscillatory motion of intra-cavity field due to mode coupling for a particular choice of detuning frequency. Phase and the polarization state of the cavity field evolve significantly in a different way in these two regions.

In this section, we introduce the Lagrangian perturbative approach based on variational principle Anderson ; RoyJLT to analyze the field dynamics semi-analytically. The treatment requires a suitable ansatz whose parameters may evolve with slow time. Note, a major approximation of this treatment is the intactness of the ansatz function. We proceed with standard Kerr soliton ansatz having the mathematical form:

where the amplitudes $(\eta_{as},\eta_{s})$ and the phases $(\phi_{as},\phi_{s})$ of two CSs are allowed to vary over slow time, $\tau$. $D_{1}$ is proportional to the group-velocity mismatch between the two hybrid modes defined as, $D_{1}\approx\delta/L_{r}$, where $L_{r}$ is the resonator length and $\delta$ is the group-velocity mismatch between the two hybrid modes. As a standard procedure we define the Lagrangian in such a way that the following Euler-Lagrangian (EL) equation retrieve the governing Eq.(6) and Eq.(7). Note, for mathematical simplicity, we ignore the negligible effect of nonlinear coupling terms in Lagrangian formalism.

where L and R consist of:

Here ($L_{as}$, $R_{as}$) and ($L_{s}$, $R_{s}$) represent the Lagrangian function and RDF of the anti-symmetric and symmetric modes, respectively. Next, we reduce the Lagrangian and RDF as, $L_{g}=\int_{-\infty}^{\infty}L\hskip 2.84526ptd\theta$ and $R_{g}=\int_{-\infty}^{\infty}R\hskip 2.84526ptd\theta$ (details are given in Appendix B). Now we use the reduced form of $L_{g}$ and $R_{g}$ in EL equation:

where $p_{j}=\eta_{as},\eta_{s},\phi_{as},\phi_{s}$ and $\dot{p_{j}}=\frac{\partial\eta_{as}}{\partial\tau},\frac{\partial\eta_{s}}{\partial\tau},\frac{\partial\phi_{as}}{\partial\tau},\frac{\partial\phi_{s}}{\partial\tau}$, we obtain the coupled ODEs for two hybrid mode field parameters. The coupled differential equations governing the amplitude and phase of these fields are as follows: