We present an erratum and generalization to our Letter [Opt. Lett. 45, 3115 (2020) [CrossRef]  ]. This erratum corrects an error in Eq. (12), and the generalization converts ${R_h}$ to ${k_h}$ for more general situations of wavelengths. Neither has any influence on the conclusions of the original Letter.

© 2020 Optical Society of America

For the proposed spatial pattern-shifting method in our Letter [1], $s\!(x)$ should be a quotient instead of a remainder, and thus, Eq. (12) is corrected as follows:

$$s\!(x ) = floor\!\left[{{{\left({x - 1} \right)} / {LCM({\lambda _h},{\lambda _l})}}} \right],1 \le x \le {\lambda _h}{\lambda _l},$$
where $floor[\cdot]$ is the function of rounding towads minus infinity. In addition, ${R_h}$ should be converted to ${k_h}$ according two situations:
  • (i) If $({{\lambda _l} - {\lambda _h}}) = GCD({{\lambda _h},{\lambda _l}})$, Eq. (15) is used for the conversion;
  • (ii) If $({{\lambda _l} - {\lambda _h}}) \gt GCD({{\lambda _h},{\lambda _l}})$, the conversion can be performed by referring a look-up table constructed from ${R_h}$.

Our 4-6 and 30-36 examples used in [1] belong to the first situation. This situation is useful in a defoused system where ${\lambda _h}$ and ${\lambda _l}$ are close to each other as well as to deal with the gamma distortion [1]. For the second situation, we take ${\lambda _h} = 4$ and ${\lambda _l} = 10$ as an example for explanation, where ${R_h}$ is calculated as 0, 6, 2, 8, 4, 1, 7, 3, 9, 5 using Eq. (9) and should be mapped to ${k_h}$ as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Although it is possible to represent this mapping relationship as an equation or algorithm, we suggest to use a simple look-up table because once ${\lambda _h}$ and ${\lambda _l}$ are chosen, this relationship is fixed. Furthermore, for the first situation, a look-up table can also be used to replace Eq. (15) for higher efficiency, although the latter is more elegant.


Jiangsu Provincial Key Research and Development Program (BE2018126); National Natural Science Foundation of China (61727802, 61971227).


The authors are thankful to Xuanna Zhao for independently discovering the error in Eq. (12).


The authors declare no conflicts of interest.


1. C. Lin, D. Zheng, Q. Kemao, J. Han, and L. Bai, Opt. Lett. 45, 3115 (2020). [CrossRef]  


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  1. C. Lin, D. Zheng, Q. Kemao, J. Han, and L. Bai, Opt. Lett. 45, 3115 (2020).

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s ( x ) = f l o o r [ ( x 1 ) / L C M ( λ h , λ l ) ] , 1 x λ h λ l ,