Correction to: Scientific Reports https://doi.org/10.1038/s41598-022-18145-4, published online 02 September 2022


The original version of this Article contained errors in Equations 22, 29, and 30 where extra brackets were incorrectly added to Equations 22 and 29, and the brackets in Equation 30 were captured in the wrong font size. Additionally, value "r" in Equations 22, 29 and 30 was wrongly presented as "r".


Equation 22

$$\begin{aligned} S_\rho ({{ {r}},t})&= -\mathrm{i}\frac{\omega _0}{8} \bigg \{|G(t)|^2 \bigg [ U_1^*({ {r}}) \partial _\rho U_1({ {r}})-U_1({ {r}}) \partial _\rho U_1^*({ {r}})\bigg ] +\bigg |G(t-\tau _\text {CP})\bigg |^2 \bigg [ U_2^*({ {r}}) \partial _\rho U_2({ {r}})]-U_2({ {r}}) \partial _\rho U_2^*({ {r}})\bigg ] \nonumber \\&\quad + G(t) G^*(t-\tau _\text {CP}) [ U_2^*({ {r}}) \partial _\rho U_1({ {r}})-U_1({ {r}}) \partial _\rho U_2^*({ {r}})] + G^*(t) G(t-\tau _\text {CP}) [ U_1^*({ {r}}) \partial _\rho U_2({ {r}})-U_2({ {r}}) \partial _\rho U_1^*({ {r}})] \bigg \} \nonumber \\&\quad + \text {c.c.}, \end{aligned}$$

now reads:

$$\begin{aligned} S_\rho ({{\textbf {r}},t})&= -\mathrm{i}\frac{\omega _0}{8} \bigg \{|G(t)|^2 \bigg [ U_1^*({\textbf {r}}) \partial _\rho U_1({\textbf {r}})-U_1({\textbf {r}}) \partial _\rho U_1^*({\textbf {r}})\bigg ] +\bigg |G(t-\tau _\text {CP})\bigg |^2 \bigg [ U_2^*({\textbf {r}}) \partial _\rho U_2({\textbf {r}})-U_2({\textbf {r}}) \partial _\rho U_2^*({\textbf {r}})\bigg ] \nonumber \\&\quad + G(t) G^*(t-\tau _\text {CP}) [ U_2^*({\textbf {r}}) \partial _\rho U_1({\textbf {r}})-U_1({\textbf {r}}) \partial _\rho U_2^*({\textbf {r}})] + G^*(t) G(t-\tau _\text {CP}) [ U_1^*({\textbf {r}}) \partial _\rho U_2({\textbf {r}})-U_2({\textbf {r}}) \partial _\rho U_1^*({\textbf {r}})] \bigg \} \nonumber \\&\quad + \text {c.c.}, \end{aligned}$$

Equation 29

$$\begin{aligned} S_\rho ({{{r}},t})&= -\mathrm{i}\frac{\omega _0}{8} \bigg \{g(t)^2 \bigg [ u_1^*({ {r}}) \partial _\rho u_1({ {r}})-u_1({ {r}}) \partial _\rho u_1^*({ {r}})\bigg ] +g(t-\tau _\text {CP})^2 \bigg [ u_2^*({ {r}}) \partial _\rho u_2({ {r}})]-u_2({ {r}}) \partial _\rho u_2^*({ {r}})\bigg ]\nonumber \\&\quad + \exp \bigg \{\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_2^*({ {r}}) \partial _\rho u_1({ {r}})-u_1({ {r}}) \partial _\rho u_2^*({ {r}})\bigg ] \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_1^*({ {r}}) \partial _\rho u_2({ {r}})-u_2({ {r}}) \partial _\rho u_1^*({ {r}})\bigg ] \bigg \} \nonumber \\&\quad +\text {c.c.}, \end{aligned}$$

now reads:

$$\begin{aligned} S_\rho ({{\textbf {r}},t})&= -\mathrm{i}\frac{\omega _0}{8} \bigg \{g(t)^2 \bigg [ u_1^*({\textbf {r}}) \partial _\rho u_1({\textbf {r}})-u_1({\textbf {r}}) \partial _\rho u_1^*({\textbf {r}})\bigg ] +g(t-\tau _\text {CP})^2 \bigg [ u_2^*({\textbf {r}}) \partial _\rho u_2({\textbf {r}})-u_2({\textbf {r}}) \partial _\rho u_2^*({\textbf {r}})\bigg ]\nonumber \\&\quad + \exp \bigg \{\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_2^*({\textbf {r}}) \partial _\rho u_1({\textbf {r}})-u_1({\textbf {r}}) \partial _\rho u_2^*({\textbf {r}})\bigg ] \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_1^*({\textbf {r}}) \partial _\rho u_2({\textbf {r}})-u_2({\textbf {r}}) \partial _\rho u_1^*({\textbf {r}})\bigg ] \bigg \} \nonumber \\&\quad +\text {c.c.}, \end{aligned}$$

Equation 30

$$\begin{aligned} S_\phi ({{ {r}},t})&= -\frac{\sigma \omega _0}{4} \bigg \{g(t) \partial _\rho |u_1({ {r}})|^2 +g(t-\tau _\text {CP})^2 \partial _\rho |u_2({ {r}})|^2 \nonumber \\&\quad + \exp \bigg \{\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_1({ {r}}) \partial _\rho u_2^*({ {r}})+u_2^*({ {r}}) \partial _\rho u_1({ {r}})\bigg ] \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\} g(t) g(t-\tau _\text {CP}) [ u_2({ {r}}) \partial _\rho u_1^*({ {r}})+u_1^*({ {r}}) \partial _\rho u_2({ {r}})] \bigg \} \nonumber \\&\quad +\frac{\omega _0}{4\rho } \bigg \{2 \ell _1 g(t)^2 |u_1({ {r}})|^2 +2\ell _2 g(t-\tau _\text {CP})^2 |u_2({ {r}})|^2 \nonumber \\&\quad + \exp \bigg \{\mathrm{i}\bigg [(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2\bigg ]\bigg \} (\ell _1+\ell _2) g(t) g(t-\tau _\text {CP}) u_1({ {r}}) u_2^*({ {r}}) \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} (\ell _1+\ell _2) g(t) g(t-\tau _\text {CP}) u_1^*({ {r}}) u_2({ {r}}) \bigg \},\end{aligned}$$

now reads:

$$\begin{aligned} S_\phi ({{\textbf {r}},t})&= -\frac{\sigma \omega _0}{4} \bigg \{g(t) \partial _\rho |u_1({\textbf {r}})|^2 +g(t-\tau _\text {CP})^2 \partial _\rho |u_2({\textbf {r}})|^2 \nonumber \\&\quad + \exp \bigg \{\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} g(t) g(t-\tau _\text {CP}) \bigg [ u_1({\textbf {r}}) \partial _\rho u_2^*({\textbf {r}})+u_2^*({\textbf {r}}) \partial _\rho u_1({\textbf {r}})\bigg ] \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg\} g(t) g(t-\tau _\text {CP}) \bigg[ u_2({\textbf {r}}) \partial _\rho u_1^*({\textbf {r}})+u_1^*({\textbf {r}}) \partial _\rho u_2({\textbf {r}})\bigg] \bigg \} \nonumber \\&\quad +\frac{\omega _0}{4\rho } \bigg \{2 \ell _1 g(t)^2 |u_1({\textbf {r}})|^2 +2\ell _2 g(t-\tau _\text {CP})^2 |u_2({\textbf {r}})|^2 \nonumber \\&\quad + \exp \bigg \{\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} (\ell _1+\ell _2) g(t) g(t-\tau _\text {CP}) u_1({\textbf {r}}) u_2^*({\textbf {r}}) \nonumber \\&\quad + \exp \bigg \{-\mathrm{i}[(\ell _1-\ell _2)\phi -C\tau _\text {CP} t - \omega _0\tau _\text {CP} +C\tau _\text {CP}^2/2]\bigg \} (\ell _1+\ell _2) g(t) g(t-\tau _\text {CP}) u_1^*({\textbf {r}}) u_2({\textbf {r}}) \bigg \},\end{aligned}$$

The original Article has been corrected.