Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture: errata

An error was made in converting the units of bend radius within the computer program used to obtain the reported results. This caused the calculated values of the bending induced stress to be significantly smaller than their true value. Corrected results for stress within the fiber and bend losses are reported. Since bending-induced stresses cause a relatively small correction to the refractive index profile, these corrections result in no changes in the main conclusions of this work. ©2012 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.5295) Photonic crystal fibers; (230.7370) Waveguides; (140.3510) Lasers, fiber; (060.2320) Fiber optics amplifiers and oscillators; (060.2280) Fiber design and fabrication. References and links 1. B. G. Ward, “Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture,” Opt. Express 16(12), 8532–8548 (2008). 2. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18(5), 737–743 (2000). 3. B. G. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19(12), 11852–11866 (2011). In evaluating Eq. (2) of the original manuscript [1], the coiling radii rx and ry were expressed in units of cm and the positions x and y were expressed in units of μm requiring an additional factor of 10 4 in the numerator of this expression. In the in-house computer program used to calculate the mechanical and optical properties of the fiber this factor was incorrectly entered as 10 6 as would be the case if converting from μm to m. This error caused the bendinginduced stress to be effectively neglected. This error affected all of the reported results therefore all of the calculations were repeated. Since the publication of the original manuscript, several improvements in the computational method have been made including a higher resolution mesh enabled by increases in available computational power, perfectly matched layer boundary conditions with element-wise arbitrary orientation, hybrid linear tangential quadratic nodal elements [2], and transverse variation of the modal propagation constant [3]. The corrected results reflect these improvements. Furthermore the overall stress field was normalized with respect to that of a pure fused silica fiber which maintains some residual stress. All of the figures and tables in the original paper have been revised to reflect the corrected results (see Figs. 1–13 and Table 1 below). The original discussion regarding the numerical results remains valid with the revised loss values given here in the corrected figures. The main conclusion that a particular photonic crystal fiber design exhibits bend loss mode discrimination slightly superior to a comparable step index fiber stands. The author sincerely regrets this error. #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7966 Fig. 1. Photonic crystal fiber geometry. The stress applying parts (SAP) are indicated by a lighter shade of grey. The air holes are white. The labels “B” indicate the perfectly matched layer boundary regions. Fig. 2. A region of the revised finite element mesh near the core boundary showing both the core meshing scheme (upper right bolded hexagonal boundary) and the capillary meshing scheme (lower left bolded hexagonal boundary). #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7967 Fig. 3. Stress fields in MPa for a coil diameter of 47.6 cm: σxx (a), σyy (b), σzz (c), and σxy (d). The center of the coil is on the positive x axis. Fig. 4. Calculated propagation losses as a function of air hole diameter to lattice pitch ratio (d/Λ) for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the x direction. #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7968 Fig. 5. Calculated modal intensity profiles of the first four modes of the PCF with diameter to lattice pitch ratio (d/Λ = 0.12) with a coil diameter of 47.6 cm and coiling in the x direction. Fig. 6. Calculated propagation losses as a function of air hole diameter to lattice pitch ratio (d/Λ) for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the y direction. Fig. 7. Calculated modal intensity profiles of the first four modes of the PCF with diameter to lattice pitch ratio (d/Λ = 0.12) with a coil diameter of 47.6 cm and coiling in the y direction. #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7969 Fig. 8. Calculated propagation losses as a function of coiling plane angle relative to the plane of the SAP for the first three modes of the PCF with a coil diameter of 47.6 cm and diameter to lattice pitch ratio (d/Λ = 0.12). Fig. 9. Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ = 0.12 and coiling in the x direction. The modal intensity plot insets are logarithmic in scale and show an example of how the SAP influence the guided modes depending on the coiling diameter. Fig. 10. Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ = 0.12 and coiling in the y direction. The modal intensity plot insets are logarithmic in scale and show an example of how the guided modes change with coiling diameter. #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7970 Fig. 11. Calculated propagation losses as a function of coiling diameter for the first three modes of a step-index fiber with 40 μm core diameter and a numerical aperture of 0.06. Fig. 12. Bend-distorted mode field intensity plots of the first three modes of a step-index fiber with 40 μm core diameter and a numerical aperture of 0.06 coiled to a diameter of 6 cm. The circle on each plot indicates the extent of the core. Fig. 13. Calculated propagation losses as a function of coiling diameter for the first three modes of a step-index fiber with 40 μm core diameter and a numerical aperture of 0.035. #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7971 Table 1. A comparison of LP01 and LP11 modal losses for a PCF with d/Λ = 0.12 and 41mm core diameter and a SIF with a numerical aperture of 0.035 and a core diameter of 40mm. The 48 cm coil is in the x direction for the PCF. The optimal coiling diameter for the PCF is 95 cm in the y direction. The optimal coiling diameter for the SIF is 53 cm. All losses are in units of dB/m. 48cm coil LP01 loss 48cm coil LP11 loss optimal coil LP01 loss optimal coil LP11 loss PCF 0.00069 3.5 0.0059 12.0 SIF 0.030 3.9 0.0046 1.5 #165000 $15.00 USD Received 19 Mar 2012; published 21 Mar 2012 (C) 2012 OSA 26 March 2012 / Vol. 20, No. 7 / OPTICS EXPRESS 7972

In evaluating Eq. (2) of the original manuscript [1], the coiling radii r x and r y were expressed in units of cm and the positions x and y were expressed in units of µm requiring an additional factor of 10 4 in the numerator of this expression.In the in-house computer program used to calculate the mechanical and optical properties of the fiber this factor was incorrectly entered as 10 6 as would be the case if converting from µm to m.This error caused the bendinginduced stress to be effectively neglected.This error affected all of the reported results therefore all of the calculations were repeated.
Since the publication of the original manuscript, several improvements in the computational method have been made including a higher resolution mesh enabled by increases in available computational power, perfectly matched layer boundary conditions with element-wise arbitrary orientation, hybrid linear tangential quadratic nodal elements [2], and transverse variation of the modal propagation constant [3].The corrected results reflect these improvements.Furthermore the overall stress field was normalized with respect to that of a pure fused silica fiber which maintains some residual stress.All of the figures and tables in the original paper have been revised to reflect the corrected results (see Figs. 1-13 and Table 1 below).The original discussion regarding the numerical results remains valid with the revised loss values given here in the corrected figures.The main conclusion that a particular photonic crystal fiber design exhibits bend loss mode discrimination slightly superior to a comparable step index fiber stands.The author sincerely regrets this error.

Fig. 1 .
Fig. 1.Photonic crystal fiber geometry.The stress applying parts (SAP) are indicated by a lighter shade of grey.The air holes are white.The labels "B" indicate the perfectly matched layer boundary regions.

Fig. 2 .
Fig. 2. A region of the revised finite element mesh near the core boundary showing both the core meshing scheme (upper right bolded hexagonal boundary) and the capillary meshing scheme (lower left bolded hexagonal boundary).

Fig. 3 .
Fig. 3. Stress fields in MPa for a coil diameter of 47.6 cm: σxx (a), σyy (b), σzz (c), and σxy (d).The center of the coil is on the positive x axis.

Fig. 4 .
Fig. 4. Calculated propagation losses as a function of air hole diameter to lattice pitch ratio (d/Λ) for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the x direction.

Fig. 6 .
Fig. 6.Calculated propagation losses as a function of air hole diameter to lattice pitch ratio (d/Λ) for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the y direction.

Fig. 7 .
Fig. 7. Calculated modal intensity profiles of the first four modes of the PCF with diameter to lattice pitch ratio (d/Λ = 0.12) with a coil diameter of 47.6 cm and coiling in the y direction.

Fig. 9 .
Fig. 9. Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ = 0.12 and coiling in the x direction.The modal intensity plot insets are logarithmic in scale and show an example of how the SAP influence the guided modes depending on the coiling diameter.

Fig. 10 .
Fig. 10.Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ = 0.12 and coiling in the y direction.The modal intensity plot insets are logarithmic in scale and show an example of how the guided modes change with coiling diameter.

Fig. 12 .
Fig. 12. Bend-distorted mode field intensity plots of the first three modes of a step-index fiber with 40 µm core diameter and a numerical aperture of 0.06 coiled to a diameter of 6 cm.The circle on each plot indicates the extent of the core.

Fig. 13 .
Fig. 13.Calculated propagation losses as a function of coiling diameter for the first three modes of a step-index fiber with 40 µm core diameter and a numerical aperture of 0.035.