## Abstract

We correct the errors that were made in “Guided modes in chiral fibers” [J. Opt. Soc. Am. B **28**, 319–324(2011)].

© 2013 Optical Society of America

It was indicated by Figs. 3 and 4 in our previously published paper [1] that the evanescent field of the demonstrated modes experiences a inversion of the handedness of circular polarization from some location far from the core-cladding interface. Actually, these modes should be either RCP or LCP in the whole demonstration region except where the field vanishes. The two figures should be corrected respectively as in Figs. 1 and 2 below. Accordingly, the describing paragraph should have been:

The ${S}_{3}$ distribution of the modes (those shown in Fig. 2) at $ka=60$ is demonstrated in Fig. 3. In the demonstration region $r<2a$, the modes with $\beta \to {k}_{+1}$ and $\beta \to {k}_{-1}$ are almost purely RCP and LCP, respectively. Among these modes, ${M}_{-11}$ and ${M}_{11}$ are of interest for they are the two fundamental modes. Figure 4 gives the ${S}_{3}$ distribution for ${M}_{-11}$ and ${M}_{11}$ at $ka=10$; although $ka$ is near the cutoff value, these two modes still are LCP and RCP, respectively, in the region $r<4a$.

The detail of the circular polarization state of the modes can be demonstrated more clearly by the radial distribution of ${S}_{3}$. Figure 3 shows the radial distribution version of Fig. 1, in which the $z$-component of the average energy flux ${S}_{z}$ (normalized) is also presented for reference. The normalized third Stokes parameter of the modes is nearly plus unity or minus unity (consequently, the first and second Stokes parameters nearly vanish) in the region where the fields concentrate.

Additionally, there are several typing errors in [1]. “the time derivation $\partial /\partial t$ therein implies multiplying by $i\omega t$ or $-i\omega t$” in the second paragraph of the second section should read as “the time derivation $\partial /\partial t$ therein implies multiplying by $i\omega $ or $-i\omega $”. “${p}_{-1}$” in Eq. (11), and Eq. (14a) should read as “${p}_{-}$”. “$(n<-1)$” in Eq. (14b) and “$(n>1)$” in Eq. (15a) should read as “$(n\le 1)$” and “$(n\ge -1)$”, respectively. “${k}_{2-}$” in Eq. (15b) should read as “${k}_{-2}$”. Equation (16) should be

Due to the mistyped Eq. (14b), cutoffs of the modes in Fig. 5 of [1] were incorrectly evaluated. The correct version should be Fig. 4 given here. Chirality in the cladding splits cutoffs of the originally degenerate modes in achiral fibers. It is of particular interest that the LCP fundamental mode has a nonzero cutoff value of the normalized frequency $ka$ when the cladding has a positive chiral parameter. Thus in [1], we missed out on a significant feature of the chiral fibers that adding chirality into the cladding of fibers could lead to single-mode operation of RCP or LCP fundamental mode. The size of the single-mode operation window of mode ${M}_{11}$, which is determined by the cutoff of mode ${M}_{-11}$, is an important parameter in practical considerations. For different chiral parameters of the cladding, the dependence of cutoff value of $a/\lambda $ on the relative permittivity of the core is given in Fig. 5. The guided modes have a propagation constant located in the interval $({k}_{1},{k}_{+2})$. We could define the index contrast of the core and cladding for guided modes as $\mathrm{\Delta}=\sqrt{{\u03f5}_{1}/{\u03f5}_{0}}-{k}_{+2}/k$. A smaller $\mathrm{\Delta}$ promises a larger cutoff value of $a/\lambda $ as expected. When ${\u03f5}_{1}/{\u03f5}_{0}=1.02$ and ${\xi}_{2}={10}^{-5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mho}$ (correspondingly $\mathrm{\Delta}=6\times {10}^{-3}$), the maximum design value of the core radius (3.6 μm for $\lambda =1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$), which could be further enlarged by choosing a lager ${\xi}_{2}$, has been close to the single-mode condition of the practical achiral fibers [2]. To operate a single LCP mode guidance, we just need to change the sign of the chiral parameter in the cladding, which leads to a exchange of waveguide dispersion between modes ${M}_{11}$ and ${M}_{-11}$. It is known that the fundamental modes in achiral fibers, which are the counterparts of modes ${M}_{11}$ and ${M}_{-11}$ here, are two-fold degenerate (approximately RCP and LCP, respectively, under weakly guiding conditions) and both have no cutoff. The significant effect of cladding chirality is to bifurcate the cutoff conditions of these two fundamental modes, which brings a property of single polarization mode guidance.

The inset of Fig. 4 shows the mode transition between modes ${M}_{12}$ and ${M}_{13}$. The dashed circle roughly indicates the wavelength region of mode transition, which is difficult to be quantitatively determined. The dispersion curves of ${M}_{12}$ and ${M}_{13}$ are judged to have an avoided crossing rather than a real crossing near $ka=13$, because the difference between the effective indices of ${M}_{12}$ and ${M}_{13}$ is observed to converge to a nonzero value. Mode transition makes the mode labeling to be a complex issue. We simply give one certain mode label to a continued dispersion curve. ${M}_{12}$ and ${M}_{13}$ exchange their field patterns after the mode transition. As a result, we must consider modes $({M}_{-12},{M}_{13})$ and $({M}_{-13},{M}_{12})$ ($({M}_{-12},{M}_{12})$ and $({M}_{-13},{M}_{13})$) as two mode pairs below (above) the frequencies of mode transition. The two modes of each pair have the similar field pattern, and are degenerate in waveguide dispersion when chirality is removed. ${M}_{11}$ and ${M}_{-11}$ always constitute a mode pair since neither of them involves a mode transition. Thus, the degenerate cutoff of ${M}_{12}$ and ${M}_{-12}$ in Fig. 4 does not mean that two coupled modes have a same cutoff.

The “modal degeneracy” indicated in Fig. 2 of [1] should be a mode transition. The dispersion curves of ${M}_{-11}$ and ${M}_{-12}$ should experience an avoided crossing instead of a real crossing near $ka=33$.

The authors are indebted to the reviewer, who suggested a more detailed discussion on the single RCP/LCP mode operation in the fibers with a chiral cladding and pointed out that the crossing type of dispersion curves should be carefully determined.

## REFERENCES

**1. **Y. Cao, J. Li, and Q. Su, “Guided modes in chiral fibers,” J. Opt. Soc. Am. B **28**, 319–324 (2011).

**2. **G. Agrawal, *Fiber-Optic Communication Systems* (Wiley, 2002).