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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 5 — May. 1, 2013
  • pp: 1232–1233
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Guided modes in chiral fibers: erratum

Yusheng Cao, Junqing Li, and Qiyao Su  »View Author Affiliations


JOSA B, Vol. 30, Issue 5, pp. 1232-1233 (2013)
http://dx.doi.org/10.1364/JOSAB.30.001232


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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

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

] 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 S3 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 βk+1 and βk1 are almost purely RCP and LCP, respectively. Among these modes, M11 and M11 are of interest for they are the two fundamental modes. Figure 4 gives the S3 distribution for M11 and M11 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 S3. Figure 3 shows the radial distribution version of Fig. 1, in which the z-component of the average energy flux Sz (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.

Fig. 1. S3 distribution of the modes M±11, M±12, and M±13 at ka=60. Dashed line indicates boundary of the demonstration region r<2a.
Fig. 2. S3 distribution of the fundamental modes M±11 at ka=10. Dashed line indicates boundary of the demonstration region r<4a.
Fig. 3. Radial distribution of S3 and Sz of the six modes in Fig. 1.
Fig. 4. Dispersion curves for the first three pairs of guided modes of n=1 and n=1 in a fiber with a chiral cladding.

Additionally, there are several typing errors in [1

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

]. “the time derivation /t therein implies multiplying by iωt or iωt” in the second paragraph of the second section should read as “the time derivation /t therein implies multiplying by iω or iω”. “p1” in Eq. (11), and Eq. (14a) should read as “p”. “(n<1)” in Eq. (14b) and “(n>1)” in Eq. (15a) should read as “(n1)” and “(n1)”, respectively. “k2” in Eq. (15b) should read as “k2”. Equation (16) should be
(1κ)k22an±1+κPn(a,p±,k±12+k22,±2k±1k2)Pn(a,p,k12+k22,2k1k2)=0.
(1)

Due to the mistyped Eq. (14b), cutoffs of the modes in Fig. 5 of [1

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

] 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

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

], 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 M11, which is determined by the cutoff of mode M11, is an important parameter in practical considerations. For different chiral parameters of the cladding, the dependence of cutoff value of a/λ on the relative permittivity of the core is given in Fig. 5. The guided modes have a propagation constant located in the interval (k1,k+2). We could define the index contrast of the core and cladding for guided modes as Δ=ϵ1/ϵ0k+2/k. A smaller Δ promises a larger cutoff value of a/λ as expected. When ϵ1/ϵ0=1.02 and ξ2=105mho (correspondingly Δ=6×103), the maximum design value of the core radius (3.6 μm for λ=1.2μm), which could be further enlarged by choosing a lager ξ2, has been close to the single-mode condition of the practical achiral fibers [2

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

]. 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 M11 and M11. It is known that the fundamental modes in achiral fibers, which are the counterparts of modes M11 and M11 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.

Fig. 5. Dependence of the cutoff value of a/λ on the relative permittivity of the core for different chiral parameters of the cladding.

The inset of Fig. 4 shows the mode transition between modes M12 and M13. The dashed circle roughly indicates the wavelength region of mode transition, which is difficult to be quantitatively determined. The dispersion curves of M12 and M13 are judged to have an avoided crossing rather than a real crossing near ka=13, because the difference between the effective indices of M12 and M13 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. M12 and M13 exchange their field patterns after the mode transition. As a result, we must consider modes (M12,M13) and (M13,M12) ((M12,M12) and (M13,M13)) 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. M11 and M11 always constitute a mode pair since neither of them involves a mode transition. Thus, the degenerate cutoff of M12 and M12 in Fig. 4 does not mean that two coupled modes have a same cutoff.

The “modal degeneracy” indicated in Fig. 2 of [1

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

] should be a mode transition. The dispersion curves of M11 and M12 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).

OCIS Codes
(060.2290) Fiber optics and optical communications : Fiber materials
(060.2400) Fiber optics and optical communications : Fiber properties
(260.2110) Physical optics : Electromagnetic optics
(350.5500) Other areas of optics : Propagation

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 19, 2013
Published: April 18, 2013

Citation
Yusheng Cao, Junqing Li, and Qiyao Su, "Guided modes in chiral fibers: erratum," J. Opt. Soc. Am. B 30, 1232-1233 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-5-1232

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