## Abstract

We present an erratum to our Letter [Opt. Lett. **46**, 1486 (2021) [CrossRef] ]. This
erratum corrects an inadvertent error in defining the electric field
and a typographical error in Eq. (5). The corrections have no
influence on the results and conclusions of the original Letter.

© 2021 Optical Society of
America

There was an inadvertent error in writing the electric field in our Letter
[1] that we intend to correct in this
erratum. In the numerical simulation, we had already taken the correct
definition of the electric field, and the error occurred while writing the
equations in our Letter. Therefore, the correction does not affect our results
and conclusions.

The electric field at frequency ${\omega _l}$ in the Letter [1] should be defined as

$${E_l}(x,y,z,t) = \sqrt {2c{\mu
_0}P_{{\omega _p}}^0/{n_l}} {A_l}(y){\psi _l}(x,z){e^{i({\omega _l}t -
{\beta _l}y)}} + \text{c.c.},$$

where $P_{{\omega _p}}^0$ is the input power of fundamental frequency ${\omega _p}$ and the optical power ${P_{{\omega _l}}}$ carried out by the frequencies at ${\omega _l}$ is given by $P_{{\omega
_p}}^0|{A_l}{|^2}$. This correction does not change the form of
coupled Eqs. (1)–(4), except for a slight modification in defining ${\alpha _1}$, which should be read in the corrected form
as ${\alpha _1} =
(2/\pi){d_{\text{eff}}}\sqrt {2c{\mu _0}P_{{\omega
_p}}^0/n_p^2{n_{2p}}}$.There was also an error in writing the equation for the dark mode given by
Eq. (5), where the square root should only be in the denominator. The
corrected equation must be read as

$$A_p^0 = \frac{{{\kappa
_2}}}{{\sqrt {8\kappa _1^2 + \kappa _2^2}}};A_{2p}^0 = 0;A_s^0 = - A_i^0 =
\frac{{2{\kappa _1}}}{{\sqrt {8\kappa _1^2 + \kappa _2^2}}}.$$

This gives the dark mode at $y = L$ as $A_p^0 \approx 0,{A_{2p}} =
0,A_s^0 \approx 1/\sqrt 2 ,A_i^0 \approx - 1/\sqrt 2$.

## REFERENCE

**1. **P. Aashna and K. Thyagarajan, Opt. Lett. **46**, 1486 (2021). [CrossRef]

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### Equations (2)

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(1)
$${E}_{l}(x,y,z,t)=\sqrt{2c{\mu}_{0}{P}_{{\omega}_{p}}^{0}/{n}_{l}}{A}_{l}(y){\psi}_{l}(x,z){e}^{i({\omega}_{l}t-{\beta}_{l}y)}+\text{c.c.},$$
(2)
$${A}_{p}^{0}=\frac{{\kappa}_{2}}{\sqrt{8{\kappa}_{1}^{2}+{\kappa}_{2}^{2}}};{A}_{2p}^{0}=0;{A}_{s}^{0}=-{A}_{i}^{0}=\frac{2{\kappa}_{1}}{\sqrt{8{\kappa}_{1}^{2}+{\kappa}_{2}^{2}}}.$$