A hierarchy of coupled mode and envelope models for bi-directional microresonators with Kerr nonlinearity

Microresonator frequency combs have been attracting a significant recent attention with their numerous practical applications and as an experimental setting to study fundamental physics of dissipative optical solitons, see [1, 2, 3] for recent reviews. So called Lugiato-Lefever (LL) model has become a paradigm in this research area [3, 4, 5, 6]. Its soliton solutions have some pre- and post- Lugiato-Lefever history in and outside the optics context, see, e.g., [7, 8, 9, 10, 11, 12, 13, 14]. However, the area has exploded after a breakthrough experimental demonstration of Ref. [5]. In terms of the first principle approach to the Kerr microresonator model development, the decade old work [6] has remained a main reference. However, together with experimental progress in the area of Kerr microresonators, the underpinning theory deserves a refreshed outlook. One of the recent challenges has emerged after a series of experiments with birectionally pumped microresonators, where combs and solitons have been observed in counter-propagating waves [15, 16, 17, 18, 19]. Bi-directionally pumped and, related dual-ring, microresonators have also been recently studied for symmetry breaking [30, 31, 32, 33, 34] and gyroscope [20, 21, 22, 23, 24, 25] related effects, including idealised PT-symmetric cases [26, 27, 28].

A variety of models has been reported in the context of experiments dealing with a single mode operation in each direction [30, 31, 32, 33, 34, 29]. We note, here that studies into single mode bidirectional lasers, laser gyroscopes and symmetry breaking in them have history going back to 1980’s, see, e.g., [35, 36, 37]. To interpret recent soliton experiments, [15, 18] have used models without nonlinear cross-coupling, while [16] has accounted for it. As we will see below, neglecting by the nonlinear cross-coupling was probably a better approach to analyse the experimental measurements under the circumstances, when modelling in neither of [15, 16, 18] included the effect of opposing group velocities, i.e., opposite signs of the resonator repetition rates for counter-propagating waves.

Due to complexity of the problem and diversity of equations both met in literature and the ones that are encountered during first principle analysis of the problem, it appears beneficial to have a detailed reference derivation that can be followed and tailored by a reader. Such mathematically transparent and physically motivated derivation that can be readily mapped onto a variety of experimental setups is present below. Focus of our work is to identify a hierarchy of the mode expansions and envelope functions evolving on different time scales set by the cavity linewidth, nonlinearity and 2nd order dispersion (slow time scales), and by the repetition rate (fast time scale), which can be used to derive a hierarchy of the coupled mode and envelope equations. We pay particular attention to comprehensive explanations of our derivation steps and interpretation of the results.

This Section introduces physical system and discusses a hierarchy of mode amplitudes and envelope functions accounting for different time-scales. It also outlines plan of work for the rest of the paper.

Maxwell equations written for the electric field components ${\cal E}_{\alpha}$ using Einstein’s notations read as

Here $\widehat{\varepsilon}$ is the dielectric response function varying in space and time. It is assumed to be scalar for the sake of brevity. $\theta\in[0,2\pi)$ is the azimuthal coordinate varying along the ring circumference. $z$ axis is perpendicular to the ring, while $r=\sqrt{x^{2}+y^{2}}$ measures distance from the ring centre. ${\cal N}_{\alpha}$ is the nonlinear part of the material polarization and $c$ is the vacuum speed of light. We assume 3rd order nonlinearity, so that

where $\alpha_{1,2,3}$ and $\alpha$ represent either of the three Cartesian projections, $x,y$ or $z$, of a physical quantity they are used with. An implicit summation is assumed over any repeated $\alpha^{\prime}$s. $\chi^{(3)}_{\alpha\alpha_{1}\alpha_{2}\alpha_{3}}$ is a 4th rank tensor of the third order nonlinear susceptibility, which is taken to be nondispersive (Kleinman condition, i.e., interchangeability of all four indices).

Electric field vector ${\cal E}_{\alpha}$ inside a ring resonator is expressed as a superposition of its linear modes $F_{\alpha j}(r,z)e^{ij\theta\pm i\omega_{j}t}$, which are solutions of Eq. (1) with ${\cal N}_{\alpha}=0$:

Here $j>0$ is an azimuthal mode number, or angular momentum, and $\omega_{j}>0$ is the corresponding mode frequency. $B^{\pm}_{j}$ are the amplitudes of the clockwise (CW) and counter-clockwise (CCW) modes. For typical microresonators geometries, either bulk crystalline or chip integrated, the transverse mode profiles $F_{\alpha j}$ can be divided into quasi-TE quasi-radial modes ($|F_{xj,yj}|\gg|F_{zj}|$) and quasi-TM ($|F_{zj}|\gg|F_{rj,\theta j}|$). For many practical purposes, which is in our case calculation of the overlap integrals in the nonlinear terms, it often suffices to neglect the smaller components of $F_{\alpha j}$. We also assume that the dominant components of $F_{\alpha j}$ and $\omega_{j}$ are real, so that $F_{\alpha j}=F_{\alpha j}^{*}$, $\omega_{j}=\omega_{j}^{*}$. Thus for TE modes $F_{xj}\approx\cos\theta F_{r}$, $F_{yj}\approx\sin\theta F_{r}$, $F_{zj}\approx 0$ and for TM modes $F_{xj,yj}\approx 0$.

In order to cut notational complexity and drop the $\alpha$ index, we consider TM family, so that from now on $F_{zj}\to F_{j}$, and

Results of our derivations would be the same for TE modes, $F_{rj}\to F_{j}$, and therefore, what we are loosing is only formal consideration of the nonlinear coupling between the TE and TM families.

We assume that inhomogeneities of the resonator surfaces result in scattering in general and in backscattering, in particular, and hence lead to the linear coupling between the modes. We account for these effects assuming

Here $\varepsilon_{id}$ is the dispersive dielectric function of the ideal (no backscattering) geometry, that does not depend on $\theta$, while relatively small $\varepsilon_{in}(\theta)$ accounts for inhomogeneities along the ring. Mode profiles $F_{j}(r,z)$ are calculated for $\varepsilon_{in}=0$.

${\cal E}$ is measured in V/m, hence normalising linear modes as $\max_{r,z}|F_{j}|=1$ makes units of $b_{j}B^{\pm}_{j}$ to be V/m. Real field amplitude of a CCW mode is $2b_{j}|B^{+}_{j}|$, so that its intensity is $I^{+}_{j}=2c\epsilon_{vac}n_{j}b_{j}^{2}|B^{+}_{j}|^{2}$ and power is $I^{+}_{j}/S_{j}$. $S_{j}$ is the effective transverse mode area, $S_{j}=\big{(}\iint|F_{j}^{\prime}|^{2}dxdz\big{)}^{2}/\iint|F_{j}^{\prime}|^{4}dxdz$ and $F_{j}^{\prime}=F_{j}(r,z)|_{y=0}$. $n_{j}$ is the linear refractive index, $n_{j}^{2}=\int_{-\infty}^{\infty}\varepsilon_{id}(\tau,r=r_{0},z=0)e^{i\omega_{j}\tau}d\tau$, where $r_{0}$ is the distance between the $z$ axis and a point of maximum of $|F_{j}|$. We define scaling factors $b_{j}$ as

so that the $|B^{\pm}_{j}|^{2}$ are measured in Watts. $\epsilon_{vac}$ is the vacuum susceptibility.

We assume that the resonator is pumped into its $j_{p}$ mode and introduce the mode index offset $\mu=j-j_{p}$. The real field expression, Eq. (3), is then

where $j_{\mu}=j_{p}+\mu=j$ and $\mu=-|j_{p}-j_{min}|,\dots,0,\dots,|j_{max}-j_{p}|$. Here ${\cal E}(t,\theta)$ is the real electric field measured in $W^{1/2}$, which dependence on the transverse coordinates has been factored out. Introducing pump laser frequency $\Omega$, we define mode detunings

where $\delta_{0}$ is the detuning for mode $j_{p}$, and

where a desired number of the dispersion orders can be included to approximate $\omega_{\mu}$ over a required spectral range.

$D_{1}$ is the resonator repetition rate (or free spectral range (FSR)) and $D_{2}$ is its group velocity dispersion. $D_{2}>0$ implies anomalous and $D_{2}<0$ normal dispersion. For example, the work [15] deals with a bi-directionally pumped silica ring with radius $1.5$mm and it has $D_{1}=2\pi\times 22$GHz, $D_{2}=2\pi\times 16$kHz. The linewidth of this resonator is $\kappa=2\pi\times 1.5$MHz, and hence the corresponding finesse ${\cal F}=D_{1}/\kappa\simeq 13000$. The mode area estimate is $S_{j_{p}}\simeq 30\mu$m², which gives $b_{j}^{2}\simeq 4\times 10^{12}$V²W^-1m^-2. Pump laser wavelength was $\simeq 1550$nm ($\omega_{0}\simeq 2\pi\times 193$THz) and the comb spectra observed there were relatively narrow and span over $\sim 20$nm bandwidth, corresponding to about $300$ modes, and the momentum of a mode nearest to the pump is estimated as $j_{p}=8700$.

In order to introduce a new set of mode amplitudes important in what follows, we transform Eq. (7) further:

Newly introduced mode amplitudes $Q_{\mu}^{\pm}$ are defined as

and as we can see they absorb frequency scales associated with both $D_{1}$ and $D_{2}$. The corresponding CW and CCW envelope functions are

Inclusion of the backscattering effects to the envelope equations, see Section 5, requires introducing of the envelope functions with reflections of their spatial coordinate,

The above definition of the space reflected functions follows a text-book list of properties of Fourier transforms, where an equivalent transformation is typically introduced in time domain and could be called as either time reflection or time inversion transformation. Differential equations involving functions with reflections of their arguments also attracted some attention from a more general mathematics prospective, see, e.g., [38], while our system reveals their role in nonlinear photonics.

In order to take control of $D_{1}$ in our future calculations, we define yet another set of slow amplitudes

Here $D_{1}$ is moved away from the exponential factors defining our third and final set of amplitudes $A_{\mu}^{\pm}$. Instead, exponents with $D_{1}$ appear explicitly in the total field equation that uses $A_{\mu}^{\pm}$,

We also use $A_{\mu}^{\pm}$ to define the corresponding envelope functions and their reflections

Though the envelopes $A_{\pm}$ can not be used themselves to define the electric field ${\cal E}$ (only their mode amplitudes can), cf. Eq. (15) and (16), they play a pivotal role in the transition from the coupled mode to the partial differential equations, see Section 5.

To summarize this section: $B_{\mu}^{\pm}$ amplitudes absorb only the slowest time scales associated with the nonlinear effects and resonator losses. $A_{\mu}^{\pm}$ absorb time scales associated with the second and higher order dispersions, in addition to the ones already inside $B_{\mu}^{\pm}$. $Q_{\mu}^{\pm}$ amplitudes evolve with the highest in our hierarchy frequency determined by the resonator repetition rate. To see how these different time scales and mode amplitudes are used to express the total field, ${\cal E}$, one should compare Eqs. (10a), (10b) and (15).

The rest of this work is structured as follows: In section 3, we first derive a system of equations for $B^{\pm}_{\mu}$ and perform its exact reduction to the equations for $Q^{\pm}_{\mu}$. In Section 4, we come back to the equations for $B^{\pm}_{\mu}$, make the $D_{1}$ role explicit, eliminate the associated fast oscillations and derive a simpler system for $A^{\pm}_{\mu}$. Corresponding mean-field equations for the envelope functions $Q_{\pm}$ and $A_{\pm}$ and their counter parts with the reflected spatial coordinates are derived in Section 5.

Systems of Eqs. (30), (11), (12) on one side, and Eqs. (34), (12) on the other, are mathematically and physically equivalent. However, there are important observations to be made here. If one could assume that $|Q_{+}|^{2}+2|Q_{-}|^{2}$ under the integrals in the right hand sides of Eqs. (30) and Eqs. (34) is a slow function of time, then these integrals would be approximately equal to $Q_{\mu}^{\pm}e^{-i\delta_{\mu}t}$, see Eqs. (11). Balancing these with the $e^{i\delta_{\mu}t}$ exponents before the integrals in Eqs. (30), one would end up with equations involving time scales determined only by the linewidths, pump detuning and nonlinear resonance shifts, which are all order of MHz. MHz frequencies would be far simpler to resolve numerically, compare to GHz-THz frequencies associated with $D_{1}$, that are directly implicated inside $\delta_{\mu}$ in the linear parts of Eqs. (34).

In this Section, we demonstrate that there are both slow and fast time scales inside the nonlinear terms in Eqs. (30), and that the latter can be eliminated resulting in a simpler and better balanced system of equations for the $A_{\mu}^{\pm}$ amplitudes, see Eqs. (14), (39).

We proceed by taking Eqs. (30a), express $Q_{\pm}$ via $B_{\mu}^{\pm}$, see Eqs. (11), (12), calculate integrals in the nonlinear terms, see Eq. (35a), and perform the two step transformation, see Eqs. (35b), (35c),

The four-wave mixing momentum matching conditions are reflected in the Kronecker delta’s in front of the nonlinear terms in the second line of the above and directly follow from taking the integrals in $\theta$. Swapping of $\mu_{2}$ and $\mu_{3}$ inside the nonlinear cross-coupling is a critical step that a reader should pay attention to, see Eq. (35b). This operation equals the Kronecker delta’s, but it re-orders the amplitudes and respective frequency detunings in the second nonlinear term. After inserting explicit expressions for $\delta_{\mu}$, see Eqs. (8), (9), and using the momentum matching condition,

we find that $D_{1}$ frequencies cancel out inside the nonlinear self-action terms, but remain in the cross-action ones providing $\mu_{2}\neq\mu_{3}$, see Eq. (35c). Thus if $D_{1}$ oscillations are much faster than dynamics associated with the other time scales left in the equations, i.e., $\kappa$, $R$ and nonlinear frequency shifts, then the fast oscillating components can be disregarded [19, 41]. This leaves us only with $\mu_{2}=\mu_{3}$ components in the cross-action terms, so that

Nonlinear terms in Eqs. (37) are now grouped into the phase insensitive pure cross-Kerr term, that contains nonlinear shift of the CW resonance frequencies due to CCW wave, and into the term that mixes both phase sensitive and phase insensitive four-wave mixing CW-CW nonlinearities. The phase sensitive effects come only from the CW-CW interaction, because all the phase sensitive CW-CCW dynamics develops with the $2D_{1}$ frequencies and is washed out by the high repetition rates. This can be called the washout effect of high repetition rates on nonlinear frequency mixing of the counter-propagating waves in a ring resonator.

Using $A_{\mu}^{\pm}$ amplitudes and detunings $\delta^{\prime}_{\mu}$, which are both $D_{1}$ free, see Eqs. (14), allows to hide $e^{iD_{2}\mu^{2}t/2}$ exponents in Eqs. (37). Adding the CCW equation, we have