Ladder of Eckhaus instabilities and parametric conversion in chi(2) microresonators
Model and classification of OPO regimes
Frequencies of the modes in highfinesse resonators are rigidly linked to the quantised set of the wavenumbers. Therefore, the step in frequency also implies the abrupt change of the spatial period, as it happens in the Eckhaus instability scenario. Accounting for the quantisation of the spectrum calls for the model formulation and methodology different from the continual models of transverse nonlinear optics^{27,31,32,33,34,35}, fibreloop^{28,29,36} and bowtie^{37,38,39} cavities, cf. with the soliton and frequency conversion theories in the highfinesse microresonators^{40,41,42,43,44,45,46,47,48}.
We now set the model in a transparent and rigorous manner. We assume that the pump laser frequency, ω_{p}, is tuned around the frequency, ω_{0b}, of the resonator mode with the number 2M. 2M equals the number of wavelengths fitting along the ring circumference. We also express the multimode intraresonator electric field centred around ω_{p} and its halfharmonic, ω_{p}/2, as
$$\begin{array}{ll}&{e}^{iM\vartheta i\frac{1}{2}{\omega }_{p}t}\mathop{\sum}\limits_{\mu }{a}_{\mu }{e}^{i\mu \theta }+c.c.,\\ &{e}^{i2M\vartheta i{\omega }_{p}t}\mathop{\sum}\limits_{\mu }{b}_{\mu }{e}^{i\mu \theta }+c.c.,\theta =\vartheta {D}_{1}t,\end{array}$$
(1)
respectively. Here, \(\vartheta =\left(0,2\pi \right]\) is the angular coordinate, and θ is its transformation to the reference frame rotating with the rate D_{1}/2π. ‘a’ and ‘b’ mark the halfharmonic, i.e., signal, and pump fields, respectively. μ = 0, ±1, ±2, … is the relative mode number, and the resonator frequencies are
$${\omega }_{\mu \zeta }={\omega }_{0\zeta }+\mu {D}_{1\zeta }+\frac{1}{2}{\mu }^{2}{D}_{2\zeta },\,\zeta =a,b,$$
(2)
where, D_{1ζ}/2π are the free spectral ranges, FSRs, and D_{2ζ} are dispersions. In what follows, we choose D_{1} = D_{1a}. The frequency, i.e., phase, matching parameter for the nondegenerate parametric process is defined as
$${\varepsilon }_{\mu } ={\omega }_{\mu a}+{\omega }_{\mu a}{\omega }_{0b}\\ =\frac{c}{R}\left[\frac{M+\mu }{{n}_{M+\mu }}+\frac{M\mu }{{n}_{M\mu }}\frac{2M}{{n}_{2M}}\right].$$
(3)
Here, n_{m} is the effective refractive index taken for the frequencies of the modes with the absolute numbers m = M ± μ (signal and idler) and 2M (pump), c is the vacuum speed of light and R is the resonator radius. For example, ε_{0} = 2ω_{0a} − ω_{0b} = 0 corresponds to the exact matching for the degenerate parametric conversion, n_{M} = n_{2M}.
Coupledmode equations governing the evolution of a_{μ}(t), b_{μ}(t) are^{8,42,44}
$$i{\partial }_{t}{a}_{\mu }= \; {\delta }_{\mu a}{a}_{\mu }\frac{i{\kappa }_{a}}{2}{a}_{\mu }\\ {\gamma }_{a}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}{\mu }_{2}}{b}_{{\mu }_{1}}{a}_{{\mu }_{2}}^{* },\\ i{\partial }_{t}{b}_{\mu }= \; {\delta }_{\mu b}{b}_{\mu }\frac{i{\kappa }_{b}}{2}({b}_{\mu }{\widehat{\delta }}_{\mu ,0}{{{{{{{\mathcal{H}}}}}}}})\\ {\gamma }_{b}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}+{\mu }_{2}}{a}_{{\mu }_{1}}{a}_{{\mu }_{2}},$$
(4)
where \({\widehat{\delta }}_{\mu ,\mu ^{\prime} }=1\) for \(\mu =\mu ^{\prime}\) and is zero otherwise. \({{{{{{{\mathcal{H}}}}}}}}\) is the pump parameter, \({{{{{{{{\mathcal{H}}}}}}}}}^{2}={{{{{{{\mathcal{F}}}}}}}}{{{{{{{\mathcal{W}}}}}}}}/2\pi\), where \({{{{{{{\mathcal{W}}}}}}}}\) is the laser power, and \({{{{{{{\mathcal{F}}}}}}}}={D}_{1b}/{\kappa }_{b}\) is finesse^{44}. δ_{μζ} are the detuning parameters in the rotating reference frame, \({\delta }_{\mu a}=({\omega }_{\mu a}\frac{1}{2}{\omega }_{p})\mu {D}_{1a}\) and δ_{μb} = (ω_{μb} − ω_{p}) − μD_{1a}. δ_{0b} is the pump detuning that is the main control parameter^{19,20,22}. The parameter values used to scale our results to physical units are listed in the caption of Fig. 1.
Simple algebra reveals that all modal detunings in the halfharmonic signal can be expressed via δ_{0b} and the respective phasematching parameters,
$${\delta }_{\mu a}={\omega }_{0a}\frac{1}{2}{\omega }_{p}+\frac{1}{2}{\mu }^{2}{D}_{2a}=\frac{1}{2}{\delta }_{0b}+\frac{1}{2}{\varepsilon }_{\mu }.$$
(5)
While δ_{μa} and ε_{μ} do not depend on the repetitionrate difference, δ_{μb} does,
$${\delta }_{\mu b}={\delta }_{0b}+\mu ({D}_{1b}{D}_{1a})+\frac{1}{2}{\mu }^{2}{D}_{2b}.$$
(6)
Depending on the pump power, the classical halfharmonic signal can be either zero or not. This is reflected in the structure of Eq. (4) and its solutions. Three types of solutions we should highlight are

(i)
noOPO state:
$${a}_{\mu }=0,{b}_{0}=\frac{i{\kappa }_{b}{{{{{{{\mathcal{H}}}}}}}}}{2{\delta }_{0b}i{\kappa }_{b}},{b}_{\mu \ne 0}=0;$$
(7)

(ii)
degenerate OPO state:
$${a}_{0}\;\ne\; 0,{b}_{0}\;\ne\; 0,{a}_{\mu \ne 0}=0,{b}_{\mu \ne 0}=0;$$
(8)
and a family of

(iii)
nondegenerate OPO states:
$${a}_{\pm \nu }\;\ne\; 0,{b}_{0}\;\ne\; 0,{a}_{ \mu  \ne  \nu  }\approx 0,{b}_{\mu \ne 0}\approx 0.$$
(9)
Though Eq. (4) does not have a closed analytical solution for the nondegenerate OPO, the experimental data demonstrate that the states with the ∣b_{0}∣^{2} and ∣a_{±ν}∣^{2} powers strongly dominating across the whole spectrum both exist and can be tuned to change from one ν to the other, and therefore, they represent the practically desirable regimes of the microresonator operation^{18,19,20,21}. The explicit expressions for the nonzero modes in Eqs. (8) and (9) are introduced later.
In the fluid dynamics context^{23,24,25}, the transition from the noOPO to an OPO state would correspond to the Benjamin–Feir instability (emergence of the signal), while moving from the OPO state operating in the ±ν pair of modes to the ±(ν + 1) pair would be the Eckhaus instability, see, e.g., Ref. ^{23} that elucidates the difference between the two. The general form of Eq. (4) is not tractable analytically regarding the issue of the interplay between the different OPO states, and, therefore, in the next section, we derive the reduced model that allows such analysis to be carried out in a transparent form.
Apart from the OPO regimes listed above, Eq. (4) allows for the multimode frequency comb solutions that could be either stationary or timedependent. The left column of Fig. 1 schematically illustrates a solution of Eq. (4) corresponding to the generic frequency comb with the spatial period 2π/ν. The structure of Eq. (4) also admits a family of the spectrally staggered combs, see the right column in Fig. 1, and we will show below that the nondegenerate OPO states are, in fact, approximations of the staggered combs.
Method of slowly varying amplitudes
Here we assume a condition that is quite common for the frequency conversion experiments in χ^{(2)} microresonators. If a resonator is made to operate close to the μ = 0 phasematching, i.e., ∣ε_{0}∣ ~ κ_{ζ}, then the simultaneous control of the repetitionrate difference between the pump and signal, D_{1b} − D_{1a}, is hard to achieve. Therefore, μ(D_{1b} − D_{1a}) easily becomes the dominant frequency scale in Eq. (4), i.e., μ∣D_{1a} − D_{1b}∣ ≫ ∣δ_{0b}∣, κ_{ζ}, ∣ε_{0}∣, γ_{a}b_{0}. For example, for a bulkcut (R ~ 1 mm)^{22} and integrated (R ~ 100 μm)^{19} LiNbO_{3} resonators, (D_{1b} − D_{1a})/κ_{ζ} ~ 10^{3} and ~10, respectively. Thus, the above conditions work very well for the former starting from ∣μ∣ = 1 and for the latter from ∣μ∣ ~ 10.
Now, the natural methodological step is to separate the fast and slow time scales in the pump sidebands,
$${b}_{\mu }={B}_{\mu }{e}^{i\mu ({D}_{1b}{D}_{1a})t},\,\mu \;\ne\; 0,$$
(10)
where B_{μ} are the slowly varying amplitudes. Then, the μ = 0 part of Eq. (4) becomes
$$i{\partial }_{t}{a}_{0}= \; {\kappa }_{a}{\Delta }_{0a}{a}_{0}{\gamma }_{a}{b}_{0}{a}_{0}^{* }\\ {\gamma }_{a}\mathop{\sum}\limits_{{\mu }_{1}\ne 0}{B}_{{\mu }_{1}}{a}_{{\mu }_{1}}^{* }{e}^{i{\mu }_{1}({D}_{1b}{D}_{1a})t},\\ i{\partial }_{t}{b}_{0}= \; {\kappa }_{b}{\Delta }_{0b}{b}_{0}+\frac{i{\kappa }_{b}}{2}{{{{{{{\mathcal{H}}}}}}}}{\gamma }_{b}{a}_{0}^{2}{\gamma }_{b}\mathop{\sum}\limits_{{\mu }_{1}\ne 0}{a}_{{\mu }_{1}}{a}_{{\mu }_{1}},$$
(11)
and the μ ≠ 0 part is
$$i{\partial }_{t}{a}_{\mu }= \; {\kappa }_{a}{\Delta }_{\mu a}{a}_{\mu }{\gamma }_{a}{b}_{0}{a}_{\mu }^{* }{B}_{\mu }{a}_{0}^{* }{e}^{i\mu ({D}_{1b}{D}_{1a})t}\\ {\gamma }_{a}\mathop{\sum}\limits_{{\mu }_{1},{\mu }_{2}\ne 0}{\widehat{\delta }}_{\mu ,{\mu }_{1}{\mu }_{2}}{B}_{{\mu }_{1}}{a}_{{\mu }_{2}}^{* }{e}^{i{\mu }_{1}({D}_{1b}{D}_{1a})t},\\ i{\partial }_{t}{B}_{\mu }= \; {\kappa }_{b}{\Delta }_{\mu b}{B}_{\mu }{\gamma }_{b}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}+{\mu }_{2}}{a}_{{\mu }_{1}}{a}_{{\mu }_{2}}{e}^{i\mu ({D}_{1b}{D}_{1a})t}.$$
(12)
Here, Δ_{μζ} are the auxiliary dimensionless detuning parameters,
$${\Delta }_{\mu \zeta }=\left({\delta }_{0\zeta }+\frac{1}{2}{\mu }^{2}{D}_{2\zeta }i\frac{1}{2}{\kappa }_{\zeta }\right)\frac{1}{{\kappa }_{\zeta }},$$
(13)
which include the losses and, hence, are complexvalued. We note that Δ_{μζ} are free from D_{1b} − D_{1a} which has been absorbed by the fast oscillating exponents, see Eq. (10).
Integrating the ∂_{t}B_{μ} equation, while assuming that a_{μ} is a slow function of time, we express the pump sidebands via the signal ones,
$${B}_{\mu }\approx {\gamma }_{b}\frac{{e}^{i\mu ({D}_{1b}{D}_{1a})t}}{\mu ({D}_{1b}{D}_{1a})}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}+{\mu }_{2}}{a}_{{\mu }_{1}}{a}_{{\mu }_{2}}.$$
(14)
Substituting Eq. (14) into Eqs. (11) and (12) would make up the Kerrlike nonlinear terms. These terms represent the socalled cascaded Kerr nonlinearity^{49}, which is, however, negligible in the leading order, because it scales inversely with μ(D_{1b} − D_{1a}). Hence, Eqs. (11) and (12), and the whole of the master system, Eq. (4), simplify to
$$i{\partial }_{t}{a}_{0} ={\kappa }_{a}{\Delta }_{0a}{a}_{0}{\gamma }_{a}{b}_{0}{a}_{0}^{* },\\ i{\partial }_{t}{a}_{\mu } ={\kappa }_{a}{\Delta }_{\mu a}{a}_{\mu }{\gamma }_{a}{b}_{0}{a}_{\mu }^{* },\,\mu \;\ne\; 0,\\ i{\partial }_{t}{a}_{\mu } ={\kappa }_{a}{\Delta }_{\mu a}{a}_{\mu }{\gamma }_{a}{b}_{0}{a}_{\mu }^{* },\\ i{\partial }_{t}{b}_{0} ={\kappa }_{b}{\Delta }_{0b}{b}_{0}+\frac{i{\kappa }_{b}}{2}{{{{{{{\mathcal{H}}}}}}}}{\gamma }_{b}{a}_{0}^{2}2{\gamma }_{b}\mathop{\sum}\limits_{{\mu }_{1} > 0}{a}_{{\mu }_{1}}{a}_{{\mu }_{1}}.$$
(15)
Thus, the pump sidebands, b_{μ≠0}, play no significant role in the frequency conversion when the repetitionrate difference, μ(D_{1b} − D_{1a}), is large. The latter simply is not featured in Eq. (15). The pumped mode, b_{0}, is driven by the sumfrequency processes of the ±μ signal sidebands, which feeds back to the equations for the signal sidebands a_{±μ} via the b_{0}terms. The absence, in the leading order, of the sumfrequency interaction between the sidebands with ∣μ_{1}∣ ≠ ∣μ_{2}∣ hints that the generation of the isolated sideband pairs should be a preferential regime over the broadband frequency combs.
Our approach is different from, e.g., the method when the whole of the highfrequency field is adiabatically eliminated by one way or the other so that the lowfrequency field becomes driven by the cascaded Kerr effect, see, e.g., ^{28,38,49}. The transition from Eq. (4) to Eq. (15) reduces the phasespace dimensionality of the pump field to one, but does not eliminate it entirely, and retains the leading order quadratic nonlinearity.
Solutions and thresholds
The reduced model, Eq. (15), allows examining in detail the properties of the OPO states (this section) and studying their instabilities with respect to each other (next section). In what follows we keep using μ as the running sideband index and ν designates a specific OPO state. Fixing ∂_{t}a_{±ν} = ∂_{t}b_{0} = 0, and after some engaging algebra with Eq. (15), the explicit solutions for the nondegenerate OPO states are found in the following form,
$${a}_{\nu } = {a}_{\nu } {e}^{i{\phi }_{\nu }},\,{a}_{\nu }= {a}_{\nu } ,\,{b}_{0}=\frac{{e}^{i{\phi }_{\nu }}{\kappa }_{a}{\Delta }_{\nu a}}{{\gamma }_{a}},\\ {e}^{i{\phi }_{\nu }} =\frac{iH}{{\Delta }_{\nu a}{\Delta }_{0b}{q}_{\nu }\frac{{\gamma }_{a}{\gamma }_{b}}{{\kappa }_{a}{\kappa }_{b}} {a}_{\nu }{ }^{2}},\,H=\frac{{\gamma }_{a}{{{{{{{\mathcal{H}}}}}}}}}{2{\kappa }_{a}},$$
(16)
where, H is the dimensionless pump parameter, and q_{ν} = 2 for ν ≠ 0. Eq. (16) with ν = 0 also covers for the degenerate case, but q_{0} = 1. Two possible solutions for the sideband amplitudes are found by taking modulus squared of the equation for \({e}^{i{\phi }_{\nu }}\),
$$ {a}_{\nu }^{\pm }{ }^{2} =\frac{\,{{\mbox{Re}}}({\Delta }_{\nu a}{\Delta }_{0b})\pm \sqrt{{H}^{2}{H}_{\nu }^{2}+{{{\mbox{Re}}}}^{2}({\Delta }_{\nu a}{\Delta }_{0b})}}{{q}_{\nu }{\gamma }_{a}{\gamma }_{b}/{\kappa }_{a}{\kappa }_{b}},\\ {H}_{\nu }^{2} ={{{\mbox{Im}}}}^{2}({\Delta }_{\nu a}{\Delta }_{0b})+{{{\mbox{Re}}}}^{2}({\Delta }_{\nu a}{\Delta }_{0b}).$$
(17)
The recent microresonator theories^{19,45,50} reported the doublevalued solutions for the degenerate case. The nondegenerate case was also considered in^{19}, but only for the zero detunings and exact phasematching, while Eq. (16) and stability analysis that follow depend critically on the full account and ability to vary both of those flexibly.
The bifurcation points from the noOPO to OPO regimes, i.e., Benjamin–Feir instabilities, are conditioned by the zeros of the sideband powers,
$$ {a}_{\nu }^{+}{ }^{2} =0,\,\,{{\mbox{i.e.}}}\,,\,{H}^{2}={H}_{\nu }^{2},\,\,{{\mbox{Re}}}\,({\Delta }_{\nu a}{\Delta }_{0b}) \; < \; 0,\,\,{{\mbox{or}}}\,\\  {a}_{\nu }^{}{ }^{2} =0,\,\,{{\mbox{i.e.}}}\,,\,{H}^{2}={H}_{\nu }^{2},\,\,{{\mbox{Re}}}\,({\Delta }_{\nu a}{\Delta }_{0b}) \; > \; 0.$$
(18)
The laser power \({{{{{{{{\mathcal{W}}}}}}}}}_{\nu }\) at the OPO threshold is then calculated from \({{{{{{{{\mathcal{W}}}}}}}}}_{\nu }=8\pi {\kappa }_{a}^{2}{H}_{\nu }^{2}/{{{{{{{\mathcal{F}}}}}}}}{\gamma }_{a}^{2}\). The left column in Fig. 2 shows \({{{{{{{\mathcal{W}}}}}}}}={{{{{{{{\mathcal{W}}}}}}}}}_{\nu }\) vs δ_{0b}, for the negative, zero, and positive phasemismatch ε_{0}. The minimum of \({{{{{{{\mathcal{W}}}}}}}}\) near δ_{0b} = 0 exists in all three cases. For ε_{0} ≤ 0, this minimum is provided by the ν = 0 mode, i.e., the noOPO state losses its stability to the degenerate OPO first. The second minimum of \({{{{{{{\mathcal{W}}}}}}}}\) at δ_{0b} = − ε_{0}, i.e., δ_{0a} = 0 (see Eq. (6)), also happens for ν = 0.
The top row of Fig. 3 shows the results of numerical simulations of Eq. (4) across the regions of the unstable noOPO state for the relatively small input powers marked with the black dashed lines in the first column of Fig. 2. The ε_{0} = 0 case in Fig. 3a demonstrates how the degenerate OPO is first excited from the noOPO state and then switches, in a cascaded manner, to the nondegenerate OPOs. Figure 3b shows how this scenario can happen twice for ε_{0} < 0. We note that the two cascades in Fig. 3b occur in the reverse order. The righttoleft cascade involves oscillations of the μ = 0 and other modes (breathing comb), while all the lefttoright cascades in Fig. 3a–c appear as the direct transitions between the neighbouring nondegenerate OPO states, as it is expected to happen in the Eckhaus instability scenario. We recall that we consider the nearphasematching between the μ = 0 modes in the pump and signal fields. It leads to the Rabilike oscillations between the a_{0} and b_{0} modes, which acquire small gain in the narrow interval of detunings and give birth to the breather states, see Refs. ^{46,47} for details of the theory of the χ^{(2)} Rabi oscillations.
For ε_{0} > 0, the minimum of \({{{{{{{\mathcal{W}}}}}}}}\) is also found at δ_{0b} ≈ 0, but now it happens for ν ≠ 0, see Fig. 2b. Thus, here, the noOPO state transits directly to the nondegenerate regime. Fixing δ_{0b} = 0 in Eq. (17) gives \(16{H}_{\nu }^{2}={\varepsilon }_{\nu }^{2}/{\kappa }_{a}^{2}+1\), where ε_{ν} = ε_{0} + ν^{2}D_{2a}. Hence, the minimum threshold power, \({H}_{\widehat{\nu }}^{2}=1/16\), for the Benjamin–Feir instabilities is achieved at the exact phasematching, \({\varepsilon }_{\widehat{\nu }}=0\), where \(\widehat{\nu }\approx \sqrt{{\varepsilon }_{0}/{D}_{2a}}\), ε_{0}D_{2a} < 0. Figure 3c demonstrates how the below threshold OPO switches directly to the nondegenerate state and passes through the cascade of νs.
Solving \({H}^{2}={H}_{\nu }^{2}\), Eqs. (18), one could find either two or four real values of δ_{0b} where the ±ν sidebands bifurcate from zero, cf., Fig. 3d, f with Fig. 3e. In the left column of Fig. 2, the different colours mark the parts of the \({H}^{2}={H}_{\nu }^{2}\) thresholds where either \( {a}_{\nu }^{+}{ }^{2}=0\) (red) or \( {a}_{\nu }^{}{ }^{2}=0\) (blue). The points of transition between the two colours are found by setting
$$\,{{\mbox{Re}}}\,({\Delta }_{\nu a}{\Delta }_{0b})=0.$$
(19)
Tuning the pump laser across the red boundaries and moving into the instability interval leads to \( {a}_{\nu }^{+}{ }^{2}\) gradually increasing from zero, which corresponds to the softexcitation regime (supercritical bifurcation), see Fig. 3d. Entering the instability tongue across the blue boundary leads to \( {a}_{\nu }^{+}{ }^{2}\) popping out stepwise and \( {a}_{\nu }^{}{ }^{2}\) bifurcating from zero subcritically (hard excitation), see Fig. 3e, f.
The right column in Fig. 2 shows the existence and stability ranges of the degenerate, ν = 0, OPO states. They bifurcate from the noOPO state supercritically along the red line and subcritically from the blue line. The parameter range where the \( {a}_{0}^{+}{ }^{2}\) and \( {a}_{0}^{}{ }^{2}\) solutions coexist is located between the blue and dashedgrey lines. Generalising for arbitrary ν, the dashedgrey boundary is found from
$$ {a}_{\nu }^{+}{ }^{2}= {a}_{\nu }^{}{ }^{2},\,{{\mbox{i.e.}}},\,{H}^{2}={{{\mbox{Im}}}}^{2}({\Delta }_{\nu a}{\Delta }_{0b}).$$
(20)
Eckhaus instabilities and OPO tuning
While the top row in Fig. 3 shows the sequential switching between the modes with different numbers and intervals of the pump detuning that select a particular sideband pair, the bottom row shows the changes of the sideband amplitudes computed from Eq. (4) and maps them on the analytical solutions for the OPO states. Apart from the oscillatory instabilities around δ_{0b}/κ_{b} ≈ 0.5 in Fig. 3b, e, all the insatiabilities of a given OPO state converge to the nearby stable one. In other words, these instabilities lead to the ν → ν + 1 swaps and, hence, to the change of the spatial period and frequency of the waveform in the resonator, i.e., these are the discrete spectrum Eckhaus instabilities.
By taking the OPO state with an arbitrary ν ⩾ 0, adding small perturbations \({\hat{a}}_{\mu }(t)\), where ∣μ∣ ≠ ν, and linearising the slowly varying amplitude model (15) we find
$$\begin{array}{ll}&i{\partial }_{t}{\hat{a}}_{\mu }={\kappa }_{a}{\Delta }_{\mu a}{\hat{a}}_{\mu }{\gamma }_{a}{b}_{0}{\hat{a}}_{\mu }^{* },\\ &i{\partial }_{t}{\hat{a}}_{\mu }={\kappa }_{a}{\Delta }_{\mu a}{\hat{a}}_{\mu }{\gamma }_{a}{b}_{0}{\hat{a}}_{\mu }^{* },\end{array}$$
(21)
where b_{0} is a function of ν in accordance with Eq. (16). Solving Eq. (21) with \({\hat{a}}_{\mu }(t)={\tilde{a}}_{\mu }\exp \{t{\lambda }_{\nu ,\mu }\}\) and \({\hat{a}}_{\mu }^{* }(t)={\tilde{a}}_{\mu }\exp \{t{\lambda }_{\nu ,\mu }\}\) yields a set of the sidebandpair growth rates,
$${\lambda }_{\nu ,\mu } = \frac{1}{2}{\kappa }_{a}+\frac{1}{2}{\kappa }_{a}\sqrt{1+4\left\{ {\Delta }_{\nu a}{ }^{2} {\Delta }_{\mu a}{ }^{2}\right\}}\\ = \frac{1}{2}{\kappa }_{a}+\frac{1}{2}\sqrt{{\kappa }_{a}^{2}+4\left\{{\delta }_{\nu a}^{2}{\delta }_{\mu a}^{2}\right\}}.$$
(22)
Thus, Eq. (22) describe the growth rates of the Eckhaus instabilities in the microresonator OPO, i.e., destabilization of the nondegenerate OPO state corresponding to the ±ν sideband pair through the excitation of any other pair ±μ. The instability threshold is reached when the curly brackets become zero, while the generation of the ±ν sideband pair is stable if the pump frequency is tuned to provide δ_{0b} such that \({\delta }_{\nu a}^{2}\;\leqslant \;{\delta }_{\mu a}^{2}\). The oscillatory bifurcations, i.e., the birth of breathers highlighted by the white lines in Fig. 2, correspond to ν = μ, and, therefore, are exempt from the above theory.
To find conditions of the switching from one sideband pair to the next, we set μ = ν ± 1. For D_{2a} < 0 (normal dispersion), the interval of stable generation of the ±ν sideband pair is found as
$$\begin{array}{l}\frac{1}{2}{D}_{2a}[{\nu }^{2}+{(\nu 1)}^{2}]{\varepsilon }_{0}\leqslant \\ {\delta }_{0b}\leqslant \frac{1}{2}{D}_{2a}[{\nu }^{2}+{(\nu +1)}^{2}]{\varepsilon }_{0}.\end{array}$$
(23)
The instability boundaries, λ_{ν,ν±1} = 0, as given by the above condition, are shown in Figs. 3 and 4a by the white dashed vertical lines and they match perfectly with the transitions found from the numerical modelling of Eq. (4).
The detuning corresponding to the midpoint of every step on the Eckhaus ladder, i.e., the left plus right limits divided by two, is \({\delta }_{0b}={\varepsilon }_{0}{D}_{2a}({\nu }^{2}+\frac{1}{2})\). Thus, the latter reproduces the characteristic parabolic shape of δ_{0b} vs ν in Fig. 4a. Recalling Eqs. (2) and (3), one can show that the same parabola is described by
$${\delta }_{0b}={\varepsilon }_{\nu }\frac{1}{2}{D}_{2a},$$
(24)
Thus, the analysis done so far has demonstrated that a sequence of the OPO regimes achieved by the scan of the pump frequency follows the steps of the Eckhaus instabilities ladder and the phasematching condition, cf., Eqs. (3) and (24). It also reveals that the size of the Eckhaus ladder’s steps and the tuning curve’s discreetness are controlled by the secondorder dispersion. We shell note that the applicability of Eq. (22) extends beyond the secondorder expansion applied for ω_{μ}, Eq. (2), while Eq. (24) relies on it. Therefore, the microresonators with dispersion dominated by the higher orders could be an interesting case to consider in future.
Figure 4a shows a sequence of the OPO transitions for the power which is much higher than in Fig. 3 (cf. black and white lines in Fig. 2b). The Eckhaus ladder in Fig. 4a is shown up to ν = 30, but extends up to ν = ν_{max} = 120. The data in the top row of Fig. 3 also demonstrate that, for ε_{0} > 0, the parabolic tuning curve is also cut at its minimum. The power reduction, from the levels in Fig. 4a to the ones in Fig. 3, does not change the shape of the parabola but reduces its extent in ν and δ_{0b}. Since the value of ν_{max} sets the practical limit for the OPO tunability, it is now mandatory to apply our theory to elaborate this point further.
First, we should recall that the data in Fig. 3 have confirmed that Eqs. (16) and (17) provide an excellent approximation for the OPO states. Figure 4d includes the much weaker, practically negligible, part of the spectrum and, thereby, explicitly reveals the degree of accuracy of Eqs. (16) and (17). It also uncovers that the nondegenerate OPO states belong to the family of staggered combs. It follows from Eq. (16) that the power of the signal sidebands, a_{±ν}, depends on the laser power, \({H}^{2} \sim {{{{{{{\mathcal{W}}}}}}}}\), and the power of b_{0} does not, and only its phase does. Indeed, \({b}_{0}^{\pm }={e}^{i{\phi }_{\nu }^{\pm }}{\kappa }_{a}{\Delta }_{\nu a}/{\gamma }_{a}\), where \({\phi }_{\nu }^{\pm }={\phi }_{\nu }( {a}_{\nu }^{\pm }{ }^{2})\). This is a reason why H^{2} does not explicitly enter the Eckhaus instability rate, see Eq. (22). However, the limits of existence of \( {a}_{\nu }^{\pm }{ }^{2}\;\geqslant \;0\), and, hence, of \({b}_{0}^{\pm }\) in Eq. (21) do critically depend on H^{2} as is given by Eq. (20).
Substituting the detunings δ_{0b} = −ε_{ν} (corresponding to the numbered tips in the left column of Fig. 2) in Eq. (20) yields \(4{\kappa }_{b}^{2}{H}^{2}={({\varepsilon }_{0}+{\nu }^{2}{D}_{2a})}^{2}\). From here one can work out the power dependencies of the minimal, \({\nu }_{\min }\) vs \({{{{{{{\mathcal{W}}}}}}}}\), and maximal, \({\nu }_{\max }\) vs \({{{{{{{\mathcal{W}}}}}}}}\), mode numbers corresponding to the ±ν OPO states. Recalling that the dispersion is normal, D_{2a} < 0, the maximal number is
$${\nu }_{{{\mbox{max}}}}^{2}\approx \frac{{\varepsilon }_{0}2{\kappa }_{b}H}{{D}_{2a}}=\frac{{\varepsilon }_{0}}{{D}_{2a}}\frac{{\kappa }_{b}{\gamma }_{a}}{{\kappa }_{a}{D}_{2a}}\sqrt{\frac{{{{{{{{\mathcal{W}}}}}}}}{{{{{{{\mathcal{F}}}}}}}}}{2\pi }}.$$
(25)
The \({\nu }_{\min }\) is conditioned by the sign of ε_{0}. It is zero for ε_{0} ⩽ 0 and \({\nu }_{{{{\min}}}}^{2}\approx ({\varepsilon }_{0}+2{\kappa }_{b}H)/{D}_{2a}\) if ε_{0} > 0, see Fig. 3. The plots of \({\nu }_{\max }\), \({\nu }_{\min }\) vs the laser power are shown in Fig. 5.
For the whispering gallery and integrated LiNbO_{3} resonators, the mode number change by one corresponds to the wavelength step ~0.1 nm^{22} and 2 nm^{20}, respectively. Hence ν_{max} = 50 corresponds to the wavelength difference between the signal and idler 10 and 200 nm, respectively. The pumpfrequency tuning data in Ref. ^{20} report up to 200nm signalidler separations. The experimental reports of the parabolic pumpfrequency tuning curves in the microresonator^{18,19,20}, fibreloop^{36}, and bowtie^{38} OPOs do not contain the data sufficient for making a comparison with the reported here powerdependent parabola cutoff at \({\nu }_{\max }\) and \({\nu }_{\min }\). Therefore, further research into this problem is necessary.