Author Correction: Ladder of Eckhaus instabilities and parametric conversion in chi(2) microresonators

Correction to: Communications Physics https://doi.org/10.1038/s42005-022-00907-1, published online 01 June 2022.

It has come to the authors’ attention that equation [22] and [23] on page 6 of the published manuscript contained some notation mistakes.

The original equation [22] was written as

$${\lambda }_{v,u} =-\frac{1}{2}{\kappa }_{a}+\frac{1}{2}{\kappa }_{a}\sqrt{1+4\{{|{\Delta }_{va}|}^{2}-{|{\Delta }_{\mu a}|}^{2}\}}\\ =-\frac{1}{2}{\kappa }_{a}+\frac{1}{2}\sqrt{{\kappa }_{a}^{2}+4\{{\delta }_{va}^{2}-{\delta }_{\mu a}^{2}\}}.$$

(22)

And has now been changed to

$${\lambda }_{v,u} =-\frac{1}{2}{\kappa }_{a}+{\kappa }_{a}\sqrt{{|{\varDelta }_{va}|}^{2}-{|{\varDelta }_{\mu a}|}^{2}}\\ =-\frac{1}{2}{\kappa }_{a}+\sqrt{{\delta }_{va}^{2}-{\delta }_{\mu a}^{2}}.$$

(22)

The original equation [23] was written as

$$-\frac{1}{2}{D}_{2a}[{\nu}^{2}+{(\nu -1)}^{2}]-{\varepsilon }_{0} \; \leqslant \; {\delta }_{0b} \; \leqslant \; -\frac{1}{2}{D}_{2a}[{\nu}^{2}+{(\nu +1)}^{2}]-{\varepsilon }_{0}.$$

(23)

and has now been corrected to

$$-\frac{1}{2}{D}_{2a}[{\nu}^{2}+{(\nu -1)}^{2}]-{\varepsilon }_{-} \; \leqslant \; {\delta }_{0b} \; \leqslant \; -\frac{1}{2}{D}_{2a}[{\nu}^{2}+{(\nu +1)}^{2}]-{\varepsilon }_{+},$$

(23)

As a result of these changes the text has been modified as follow: (i) the second sentence of the paragraph below equation [22] has been changed from

“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 ${\delta }_{va}^{2} \; \leqslant \; {\delta }_{\mu a}^{2}.$”to

“The instability threshold is reached when λ_ν,μ becomes zero, while the generation of the ±ν sideband pair is stable if the pump frequency is tuned to provide δ_0b such that ${\delta }_{va}^{2} \; \leqslant \; {\delta }_{\mu a}^{2}+\frac{1}{4}{\kappa }_{a}^{2}$”.

(ii) the following text has been added immediately following equation [23] where ${\varepsilon}_{\!\pm}={\varepsilon }_{0}+{\kappa }_{a}^{2}/2{D}_{2a}(1\pm 2v).$

In addition to the above, some omissions to the authorship of bibliographic references [13], [14], [15], were corrected. These references have been published with just the first author name appearing and were not reported in standard Nature referencing style as

13. Billat, A. et al. Large second harmonic generation enhancement in Si₃N₄ waveguides by all-optically induced quasi-phase-matching. Nat. Commun. 8, 1016 (2017).

14. Hickstein, D.D. et al. Self-organized nonlinear gratings for ultrafast nanophotonics. Nat. Photonics 13, 494–499 (2019).

15. Lu, X. et al. Efficient photoinduced second-harmonic generation in silicon nitride photonics. Nat. Photonics 15, 131–136 (2021).

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