# Blog

## Ladder of Eckhaus instabilities and parametric conversion in chi(2) microresonators

11:32 11 junio in Artículos por Website

### Model and classification of OPO regimes

Frequencies of the modes in high-finesse 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 optics27,31,32,33,34,35, fibre-loop28,29,36 and bow-tie37,38,39 cavities, cf. with the soliton and frequency conversion theories in the high-finesse microresonators40,41,42,43,44,45,46,47,48.

We now set the model in a transparent and rigorous manner. We assume that the pump laser frequency, ωp, is tuned around the frequency, ω0b, of the resonator mode with the number 2M. 2M equals the number of wavelengths fitting along the ring circumference. We also express the multi-mode intra-resonator electric field centred around ωp and its half-harmonic, ωp/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 D1/2π. ‘a’ and ‘b’ mark the half-harmonic, 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, D1ζ/2π are the free spectral ranges, FSRs, and D2ζ are dispersions. In what follows, we choose D1 = D1a. The frequency, i.e., phase, matching parameter for the non-degenerate 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, nm 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, ε0 = 2ω0a − ω0b = 0 corresponds to the exact matching for the degenerate parametric conversion, nM = n2M.

Coupled-mode equations governing the evolution of aμ(t), bμ(t) are8,42,44

$$i{\partial }_{t}{a}_{\mu }= \; {\delta }_{\mu a}{a}_{\mu }-\frac{i{\kappa }_{a}}{2}{a}_{\mu }\\ -{\gamma }_{a}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}-{\mu }_{2}}{b}_{{\mu }_{1}}{a}_{{\mu }_{2}}^{* },\\ i{\partial }_{t}{b}_{\mu }= \; {\delta }_{\mu b}{b}_{\mu }-\frac{i{\kappa }_{b}}{2}({b}_{\mu }-{\widehat{\delta }}_{\mu ,0}{{{{{{{\mathcal{H}}}}}}}})\\ -{\gamma }_{b}\mathop{\sum}\limits_{{\mu }_{1}{\mu }_{2}}{\widehat{\delta }}_{\mu ,{\mu }_{1}+{\mu }_{2}}{a}_{{\mu }_{1}}{a}_{{\mu }_{2}},$$

(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 finesse44. δμζ are the detuning parameters in the rotating reference frame, $${\delta }_{\mu a}=({\omega }_{\mu a}-\frac{1}{2}{\omega }_{p})-\mu {D}_{1a}$$ and δμb = (ωμb − ωp) − μD1a. δ0b is the pump detuning that is the main control parameter19,20,22. 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 half-harmonic signal can be expressed via δ0b and the respective phase-matching 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 δμa and εμ do not depend on the repetition-rate difference, δμb 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 half-harmonic 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

1. (i)

no-OPO state:

$${a}_{\mu }=0,{b}_{0}=\frac{-i{\kappa }_{b}{{{{{{{\mathcal{H}}}}}}}}}{2{\delta }_{0b}-i{\kappa }_{b}},{b}_{\mu \ne 0}=0;$$

(7)

2. (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

3. (iii)

non-degenerate 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 non-degenerate OPO, the experimental data demonstrate that the states with the b02 and a±ν2 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 operation18,19,20,21. The explicit expressions for the non-zero modes in Eqs. (8) and (9) are introduced later.

In the fluid dynamics context23,24,25, the transition from the no-OPO 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 multi-mode frequency comb solutions that could be either stationary or time-dependent. 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 non-degenerate 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 phase-matching, i.e., ε0 ~ κζ, then the simultaneous control of the repetition-rate difference between the pump and signal, D1b − D1a, is hard to achieve. Therefore, μ(D1b − D1a) easily becomes the dominant frequency scale in Eq. (4), i.e., μD1a − D1bδ0b, κζ, ε0, γab0. For example, for a bulk-cut (R ~ 1 mm)22 and integrated (R ~ 100 μm)19 LiNbO3 resonators, (D1b − D1a)/κζ ~ 103 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 Bμ 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, Δμζ 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 complex-valued. We note that Δμζ are free from D1b − D1a which has been absorbed by the fast oscillating exponents, see Eq. (10).

Integrating the ∂tBμ equation, while assuming that aμ 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 Kerr-like nonlinear terms. These terms represent the so-called cascaded Kerr nonlinearity49, which is, however, negligible in the leading order, because it scales inversely with μ(D1b − D1a). 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, bμ≠0, play no significant role in the frequency conversion when the repetition-rate difference, μ(D1b − D1a), is large. The latter simply is not featured in Eq. (15). The pumped mode, b0, is driven by the sum-frequency processes of the ±μ signal sidebands, which feeds back to the equations for the signal sidebands a±μ via the b0-terms. The absence, in the leading order, of the sum-frequency interaction between the sidebands with μ1 ≠ μ2 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 high-frequency field is adiabatically eliminated by one way or the other so that the low-frequency field becomes driven by the cascaded Kerr effect, see, e.g., 28,38,49. The transition from Eq. (4) to Eq. (15) reduces the phase-space 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 ∂ta±ν = ∂tb0 = 0, and after some engaging algebra with Eq. (15), the explicit solutions for the non-degenerate 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 qν = 2 for ν ≠ 0. Eq. (16) with ν = 0 also covers for the degenerate case, but q0 = 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 theories19,45,50 reported the double-valued solutions for the degenerate case. The non-degenerate case was also considered in19, but only for the zero detunings and exact phase-matching, 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 no-OPO 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 δ0b, for the negative, zero, and positive phase-mismatch ε0. The minimum of $${{{{{{{\mathcal{W}}}}}}}}$$ near δ0b = 0 exists in all three cases. For ε0 ≤ 0, this minimum is provided by the ν = 0 mode, i.e., the no-OPO state losses its stability to the degenerate OPO first. The second minimum of $${{{{{{{\mathcal{W}}}}}}}}$$ at δ0b = − ε0, i.e., δ0a = 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 no-OPO state for the relatively small input powers marked with the black dashed lines in the first column of Fig. 2. The ε0 = 0 case in Fig. 3a demonstrates how the degenerate OPO is first excited from the no-OPO state and then switches, in a cascaded manner, to the non-degenerate OPOs. Figure 3b shows how this scenario can happen twice for ε0 < 0. We note that the two cascades in Fig. 3b occur in the reverse order. The right-to-left cascade involves oscillations of the μ = 0 and other modes (breathing comb), while all the left-to-right cascades in Fig. 3a–c appear as the direct transitions between the neighbouring non-degenerate OPO states, as it is expected to happen in the Eckhaus instability scenario. We recall that we consider the near-phase-matching between the μ = 0 modes in the pump and signal fields. It leads to the Rabi-like oscillations between the a0 and b0 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 ε0 > 0, the minimum of $${{{{{{{\mathcal{W}}}}}}}}$$ is also found at δ0b ≈ 0, but now it happens for ν ≠ 0, see Fig. 2b. Thus, here, the no-OPO state transits directly to the non-degenerate regime. Fixing δ0b = 0 in Eq. (17) gives $$16{H}_{\nu }^{2}={\varepsilon }_{\nu }^{2}/{\kappa }_{a}^{2}+1$$, where εν = ε0 + ν2D2a. Hence, the minimum threshold power, $${H}_{\widehat{\nu }}^{2}=1/16$$, for the Benjamin–Feir instabilities is achieved at the exact phase-matching, $${\varepsilon }_{\widehat{\nu }}=0$$, where $$\widehat{\nu }\approx \sqrt{-{\varepsilon }_{0}/{D}_{2a}}$$, ε0D2a < 0. Figure 3c demonstrates how the below threshold OPO switches directly to the non-degenerate 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 δ0b 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 soft-excitation 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 sub-critically (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 no-OPO state super-critically along the red line and sub-critically from the blue line. The parameter range where the $$| {a}_{0}^{+}{| }^{2}$$ and $$| {a}_{0}^{-}{| }^{2}$$ solutions coexist is located between the blue and dashed-grey lines. Generalising for arbitrary ν, the dashed-grey 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 δ0b/κb ≈ 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 b0 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 sideband-pair 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 non-degenerate 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 δ0b 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 D2a < 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, λν,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 δ0b 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 phase-matching 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 second-order dispersion. We shell note that the applicability of Eq. (22) extends beyond the second-order expansion applied for ωμ, 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 = νmax = 120. The data in the top row of Fig. 3 also demonstrate that, for ε0 > 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 δ0b. Since the value of νmax 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 non-degenerate 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 b0 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 H2 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 H2 as is given by Eq. (20).

Substituting the detunings δ0b = −εν (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, D2a < 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 ε0. It is zero for ε0 0 and $${\nu }_{{{{\min}}}}^{2}\approx (-{\varepsilon }_{0}+2{\kappa }_{b}H)/{D}_{2a}$$ if ε0 > 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 LiNbO3 resonators, the mode number change by one corresponds to the wavelength step ~0.1 nm22 and 2 nm20, respectively. Hence νmax = 50 corresponds to the wavelength difference between the signal and idler 10 and 200 nm, respectively. The pump-frequency tuning data in Ref. 20 report up to 200-nm signal-idler separations. The experimental reports of the parabolic pump-frequency tuning curves in the microresonator18,19,20, fibre-loop36, and bow-tie38 OPOs do not contain the data sufficient for making a comparison with the reported here power-dependent parabola cut-off at $${\nu }_{\max }$$ and $${\nu }_{\min }$$. Therefore, further research into this problem is necessary.