Takeshi Fujisawa, Takanori Sato, Kunimasa Saitoh
Journal of Lightwave Technology 36 (18) 4211 - 4212 0733-8724 2018/09/15
[Refereed][Not invited] © 2018 IEEE. We reported the development of novel beam propagation method (BPM) for helicoidal waveguides and its application to twisted photonic crystal fibers (PCFs) in [1]. Although the formulation presented in [1] is correct, we found a careless mistake in our numerical code. In this erratum, we would like to describe a corrected point in the code and demonstrate corrected results. As shown here, corrected results are very much closer to the experimental results [2]. The corrections do not affect the claim of the original paper, namely, the development of novel BPM and the BPM is more useful for analyzing twisted PCF than a guided mode analysis. II. CORRECTED POINTS Following equations are finite element matrices [Mtz ] and [Mz t ] (the equation numbers are the same as [1]). (Equation presented) In our numerical code, minus 1 is multiplied to these matrices by mistake. Here, we correct this point, and the results obtained by the corrected code are presented. Again, mathematical formulation presented in [1] is correct. III. CORRECTED RESULTS Here, the corrected major numerical results are presented. The condition of the calculation is completely the same as in [1]. For easy comparison with [1], the same Figure numbers are given in this erratum. Figure 3(a) and (b) showtransmission spectra of twisted PCF fora = 0.004, 0.006, 0.008 and 0.0108 rad/μm and α = 0.0136 rad/μm. Dots in Figs. 3(a) and (b) indicate reported measured spectra [2] for α = 0.0108 and 0.0136 rad/μm, respectively. In [1], if the value of a for the (Figure Presented) Fig. 3. Transmission spectra of twisted PCF for (a) α = 0.004, 0.006, 0.008, 0.0108 rad/μm and (b) α = 0.0136 rad/μm. Dots are reported measured data in [2]. calculation is the same with the experiment, the loss is overestimated and we had to fit the value. In [1], α = 0.008 and 0.01 rad/μm were used to compare the calculated results with the measured results for α = 0.0108 and 0.0136 rad/μm. In the corrected results, however, calculated results with α = 0.0108 and 0.0136 rad/μm are in very good agreement with the measured results. Therefore, we do not have to fit the value of a in the corrected code, which is a very reasonable result. The positions of the dips are very close to the experiment with slight difference (calculated dips are on shorter wavelength side) and the background loss at off resonance wavelength can be taken into account (the background loss calculated by the guided mode analysis is almost zero [2]). (Figure Presented) Fig. 4. Polarization power ratio as a function of z for (a) λ = 0.66 μm and (b) λ = 0.8 μm. (Figure Presented) Fig. 7. Transmission spectra of twisted PCF for different values of d. Figure 4(a) and (b) show the polarization power ratio as a function of z for λ = 0.66 and 0.8 μm. The twisting rate is α = 0.0136 rad/μm. Compared with Fig. 4 of [1], we changed the value of a from 0.01 to 0.0136 rad/μm, since we do not have to fit the value any more as shown in Fig. 3. Compared with [1], in the results obtained by the corrected code, the polarization rotation occurrs in the helicoidal system (xyz coordinate). This seems to be reasonable if we consider that the modeling space (where the numerical discretization is made) is related to the xyz coordinate system [3]. (Figure Presented) Fig. 9. Transmission spectra of twisted PCF for (a) α = 0.0108 rad/μm and (b) α = 0.0136 rad/μm. Dots are reported measured data in [2]. Figure 7 shows the transmission spectra of twisted PCF for different values of d. The twisting rate is α = 0.0136 rad/μm (which is changed from α = 0.01 rad/μm in [1]). Qualitatively, similar results are obtained compared with [1] (smaller loss for larger air-hole size). Figure 9 (a) and (b) show the transmission spectra of twisted PCF for α = 0.0108 and 0.0136 rad/μm, respectively, calculated by using the finite-element mesh shown in Fig. 8 of [1]. The positions of the calculated dips are shifted to longerwavelength side and are in excellent agreement with the measured results. From these results, the corrected BPM gives more reasonable and accurate results. At the same time, the claims and conclusion in [1] (mathematical formulation of helicoidal BPM and the claim that the BPMis useful for the analysis of twisted PCF comparedwith the guided mode analysis) are not affected by the corrected results.