1. Introduction

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 [1–3] for recent reviews. So called Lugiato-Lefever (LL) model has become a paradigm in this research area [3–6]. Its soliton solutions have some pre- and post- Lugiato-Lefever history in and outside the optics context, see, e.g., [7–14]. 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 [15–19]. Bi-directionally pumped and, related dual-ring, microresonators have also been recently studied for symmetry breaking [20–24] and gyroscope [25–30] related effects, including idealised PT-symmetric cases [31–33].

A variety of models has been reported in the context of experiments dealing with a single mode operation in each direction [20–24,34]. 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., [35–37]. 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.

2. Hierarchies of mode expansions and envelope functions

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

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

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

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

${{\cal E}}$ is measured in V/m, hence normalising linear modes as $\max _{r,z} |F_{ j}|=1$ makes units of $b_jB^{\pm }_j$ 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 $F_j^{\prime }=F_j(r,z)\vert _{y=0}$. $n_j$ 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

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

$D_1$ 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 $S_{j_p}\simeq 30\mu$m$^{2}$, which gives $b_j^{2}\simeq 4\times 10^{12}$V$^{2}$W$^{-1}$m$^{-2}$. 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:

In order to take control of $D_1$ in our future calculations, we define yet another set of slow amplitudes

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.

3. Coupled mode equations

3.1 Separating equations for CW and CCW amplitudes

Substituting $t'=t-\tau$ in Eqs. (1), (3) we then assume that material response is fast so that $C_j(t-\tau )\simeq C_j(t)-\tau \partial _t C_j+\dots$. Neglecting all the 2nd and higher order time derivatives of $B^{\pm }_\mu$ we find that Eq. (1) transforms to

We now multiply the left and right hand-sides of Eq. (17) by $b_{j_p}F_{j_p}\exp ^{-ij_{\mu '}\theta }$, integrate in $r,z$ and $\theta$, and approximate $\omega _\mu \simeq \omega _0=\omega _{j_p}$, $n_\mu \simeq n_0$ inside all the pre-factors, but not in the powers of the exponents. The resulting model, see Eqs. (22), makes use of the two scattering matrices having dimensions of angular frequencies. One characterises scattering induced coupling between the co-propagating modes

Equation (22) can now be split, as per rotating wave approximation, into the parts proportional to $e^{\pm i\omega _{j_\mu }t}$ exponents, so that we have two equations defined on the slow, $D_2$ related, time scales:

In order to be used to describe laboratory experiments with microresonators, Eqs. (23) have to be amended with the single mode pump term and losses accounting for the finite linewidth. We take, for the laser frequency at the exact cavity resonance $\Omega =\omega _{\mu =0}=\omega _{j=j_p}$ and for the low pump levels, i.e., linear regime, the intracavity powers of CW and CCW waves to be $|{{\cal H}}_\pm |^{2}=|B^{\pm }_\mu |^{2}$. This is achieved via a phenomenological substitution

If pump is absent, then the field power would decay with the rate $\kappa$ (full width of the resonance). An expression linking ${{\cal H}}_\pm$ with the laser powers ${{\cal W}}_\pm$ is

$\widehat {{\cal R}}_{\mu \mu }\sim 2\pi \times 4$ kHz in Ref. [15]. In this regime, it is safe to assume that $\kappa$ dominates over $\widehat \Gamma$ and $\widehat {{\cal R}}$ terms. Using this we disregard $\widehat \Gamma _{\mu '\mu }$ in what follows, and retain only the dominant diagonal terms in $\widehat {{\cal R}}_{\mu '\mu }$, i.e., $\widehat {{\cal R}}_{\mu '\mu \ne \mu '}\approx 0$. Dispersion of the diagonal terms is also disregarded, $\widehat {{\cal R}}_{\mu \mu }\simeq \widehat {{\cal R}}_{00}= R$. Accounting for all of the above and complex conjugating second of Eqs. (23) we conclude this subsection with

3.2 Opening up nonlinearity

Using Eqs. (4), (7) we have

Using Eqs. (11) to express amplitudes $B_\mu ^{\pm }$ via $Q_\mu ^{\pm }$ we find that all the time dependent exponents cancel out and the resulting coupled mode equations for $Q_\mu ^{\pm }$ amplitudes are

4. Washout of the repetition rate timescales from the coupled mode equations

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),

Using $A_{\mu }^{\pm }$ amplitudes and detunings $\delta '_\mu$, which are both $D_1$ free, see Eqs. (14), allows to hide $e^{iD_2\mu ^{2}t/2}$ exponents in Eqs. (35). Adding the CCW equation, we have

5. Envelope models

Connection of the coupled mode equations to the wave dynamics becomes more intuitive, if one now derives the envelope, Lugiato-Lefever like, equations. First, we take the $Q_\mu ^{\pm }$ model, see Eqs. (34), and multiply Eq. (34a) with $e^{i\mu \theta }$ and (34b) with $e^{-i\mu \theta }$. We then sum up each of the equations in $\mu$ and use Eqs. (12), (13) connecting the envelopes $Q_{\pm }$ and the reflected envelopes $Q_{\pm }^{(r)}$ to their mode amplitudes. This procedure is free from approximations and it leads to a system of partial differential equations for $Q_\pm$ and $Q_\pm ^{(r)}$,

Starting from the equations for $A_\mu ^{\pm }$, see Eqs. (39), we follow a modified procedure. Namely, we multiply both CW and CCW equations by the same exponent $e^{i\mu \theta }$, use the envelope definitions in Eqs. (16), observe that $\int _0^{2\pi }|A_-|^{2}A_-e^{i\mu \theta }d\theta =\int ^{0}_{-2\pi }|A_-^{(r)}|^{2}A_-^{(r)}e^{-i\mu \theta }d\theta$ and, due to periodicity, $=\int _0^{2\pi }|A_-^{(r)}|^{2}A_-^{(r)}e^{-i\mu \theta }d\theta$, sum up in $\mu$, and derive the following envelope equations

Eqs. (40) (not Eqs. (41)) could in fact, be written without a rigorous derivation, by simply relying on common knowledge, let aside reflected envelopes in the backscattering terms. These equations include traditional cross-phase modulation, and also repetition rates terms and other odd order dispersion terms with the opposite signs. Contrary, Eqs. (41) have no repetition rate terms, i.e., $D_1$-terms, and the remaining odd dispersions, i.e., $D_3$, $D_5$, etc., come with the same signs. Simultaneously, phase sensitive nonlinear wave mixing effects induced by CW-CCW interaction have been washed out. The only nonlinear cross-interaction left comes from the integrated power, which merely shifts the detuning parameters. Thus, in the absence of backscattering a nonlinear bi-directional resonator operates as a uni-directional one, but with the detuning parameter altered by the total power of the counter-propagating wave.

6. Summary

We have derived coupled mode equations describing nonlinear wave mixing processes in Kerr microresonator with counter-propagating waves. Features of the first two coupled mode formulations given by Eqs. (30) and Eqs. (34) are that they fully account for the repetition rate effects and that nonlinear terms are taken in the real space, and can be evaluated via Fourier transforms, see also [42]. We then proceeded to present simplified multi-mode equations that neglect the repetition rate dynamics driving the phase sensitive terms responsible for nonlinear interaction between the counter-propagating fields (washout effect, Section 4), and again deal with the nonlinearity in the real space, see Eqs. (39).

Finally, we demonstrated that coupled mode equations ( 34) and Eqs. (39) are equivalent to two different, Lugiato-Lefever-like, envelope models. The one that involves the repetition rate dynamics, see Eqs. (40), links two usual envelopes for the CW and CCW fields, with two of their space reflections. While the one with the repetition rate averaged out, see Eqs. (41), makes a closed system already for two envelopes, $A_\pm$, one of which is reflected. We note, that $Q_\pm$ can be used directly to reconstruct total electric field, see Eq. (10c), while $A_\pm$ can not, but their respective mode amplitudes can, see Eqs. (15), (16a).

We have taken care to reveal all mathematical transformations, that allow a reader to verify our derivation steps and apply modifications if required. Opportunities for future theoretical and numerical studies offered by the models presented here are numerous, as well as their potential to guide and interpret experimental work.