## Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture

Optics Express, Vol. 16, Issue 12, pp. 8532-8548 (2008)

http://dx.doi.org/10.1364/OE.16.008532

Acrobat PDF (968 KB)

### Abstract

Numerical calculations predict that particular birefringent photonic crystal fiber designs exhibit slightly better performance in a coiled configuration than non-birefringent step-index fiber designs with respect to higher order mode suppression for the realization of large mode area effectively single transverse mode fibers. The passive losses of the fundamental and first few higher order modes of a birefringent photonic crystal fiber design with a 41*µ*m diameter core incorporating stress applying parts (SAP) were calculated using an integrated electromechanical finite element method. Minimum higher order mode losses of up to 5.5 dB/m were predicted for fundamentalmode losses of only 0.0014 dB/m. The bend performance of this PCF design was predicted to be relatively insensitive to manufacturing tolerances with respect to air hole size and device assembly tolerances with respect to coiling diameter based on the calculated dependence of the mode losses on these parameters. The positions and refractive index of the SAP render the numerical aperture of the core anisotropic allowing further tailoring of the bend performance by adjusting the angle between the coiling plane of the fiber and the orientation of the SAP within the cladding. Fundamental and higher-order mode losses are calculated for step-index fiber (SIF) designs with a 40*µ*m diameter core for comparison. The step-index fiber designs were predicted to exhibit slightly inferior bend loss mode discrimination and higher sensitivity to packaging configuration compared to the photonic crystal fiber designs presented.

© 2008 Optical Society of America

## 1. Introduction

1. Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. **30**, 459–461 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-5-459. [CrossRef] [PubMed]

2. M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express **14**, 11071–11076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071. [CrossRef] [PubMed]

3. A. Liem, J. Limpert, H. Zellmer, and A. Tnnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. **28**, 1537–1539 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-17-1537. [CrossRef] [PubMed]

4. S. Hfer, A. Liem, J. Limpert, H. Zellmer, A. Tnnermann, S. Unger, S. Jetschke, H. -R. Mller, and I. Freitag, “Single-frequency master-oscillator fiber power amplifier system emitting 20 Wof power,” Opt. Lett. **26**, 1326–1328 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=ol-26-17-1326. [CrossRef]

5. I. Zawischa, K. Plamann, C. Fallnich, H. Welling, H. Zellmer, and A. Tnnermann, “All-solid-state neodymiumbased single-frequency master-oscillator fiber power-amplifier system emitting 5.5 W of radiation at 1064 nm,” Opt. Lett. **24**, 469–471 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-7-469. [CrossRef]

6. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. **25**, 442–444 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-7-442. [CrossRef]

7. J. Limpert, O. Schmidt, J. Rothhardt, F. Rser, T. Schreiber, A. Tnnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express **14**, 2715–2720 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2715. [CrossRef] [PubMed]

8. T. -w Wu, L. Dong, and H. Winful, “Bend performance of leakage channel fibers,” Opt. Express **16**, 4278–4285 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-4278. [CrossRef] [PubMed]

9. W. S. Wong, X. Peng, J. M. McLaughlin, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**, 2855–2857 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-21-2855. [CrossRef] [PubMed]

10. J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express **13**, 3477–3490 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3477. [CrossRef] [PubMed]

11. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. **43**, 899–909 (2007). [CrossRef]

11. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. **43**, 899–909 (2007). [CrossRef]

*µ*m or more has a large torsional stiffness and robustly maintains its orientation.

19. T. Schreiber, F. Rser, O. Schmidt, J. Limpert, R. Iliew, F. Lederer, A. Petersson, C. Jacobsen, K. Hansen, J. Broeng, and A. Tnnermann, “Stress-induced single-polarization single-transverse mode photonic crystal fiber with low nonlinearity,” Opt. Express **13**, 7621–7630 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7621. [CrossRef] [PubMed]

*C*

_{6v}symmetry. Although the finite element method for structural problems is well documented [12], a brief discussion is included here to facilitate the explanation of the details of its application to coiled optical fibers.

## 2. Structural model of the fiber

*within the fiber. The SAP in the cladding cause transverse stress loading within the fiber as it cools from the softening temperature during the drawing process. Bending of the fiber introduces a longitudinal strain distribution throughout the fiber cross-section in order to satisfy displacement boundary conditions for the fiber (e. g. the cladding boundary on the outside of a bend is longer than that on the inside.) The standard prescription for straight fibers is to employ the plane strain approximation so that deformations only occur in the plane normal to the electromagnetic propagation axis of the fiber with no longitudinal strain along this axis. If the bend is caused by a uniform constant-diameter coiling of the fiber, symmetry dictates that the longitudinal strain assumes a known distribution. The appropriate structuralmodel therefore incorporates fixed longitudinal strain variations which serve as an input to the determination of the strain in the plane perpendicular to the fiber axis. In order to determine the plane strain the elastic energy functional*

_{ij}**r**is the two dimensional position within the fiber cross-section Ω,

**x**(

*r*) is the three-component displacement field,

*ε*(

*r*) is the strain field,

**C**is the elasticity matrix, and

**R**

*is the local thermal load vector [12].*

_{th}*x,y*coordinate system in the plane perpendicular to the fiber axis, the longitudinal strain in the

*z*direction may be written

*r*and

_{x}*r*are the bend radii in the

_{y}*x*and

*y*directions respectively. The terms in (1) which depend on

*ε*may then be separated from the rest to yield

_{z}*t*indicates the transverse component of the strain field. The term incorporating

**C**is now linear in the transverse displacements and therefore takes the form of a force in the elastic energy functional

_{zi}*N*which interpolate the values of the displacements at the nodes on the boundary of the element throughout the interior of that element. Summing the elastic energy over all of the elements and minimizing the global elastic energy results a linear equation that may be solved to determine the global transverse displacement vector

**x**

*.*

_{t}**x**is used to calculate the transverse strain field. This is combined with the longitudinal strain field to yield the full three-dimensional strain field within the fiber. The strain field is then combined with the stiffness matrix to yield the full three-dimensional stress-tensor.

_{t}13. B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. **44**, 150–156 (2008). [CrossRef]

## 3. Optical model of the fiber

13. B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. **44**, 150–156 (2008). [CrossRef]

11. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. **43**, 899–909 (2007). [CrossRef]

17. F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guid. Wave Lett. **8**, 223–225 (1998). [CrossRef]

18. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, pp. 185–200, (1994). [CrossRef]

20. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

*i*is:

*is the extent of the element within the fiber cross-section,*

_{i}*H⃗*is the magnetic field which is interpolated within the element from values at the nodes bounding the element, and

*ω*is the temporal frequency of the wave.

*is constant on each element and on the interior of the computational area is obtained by calculating the photoelastic correction to the bulk permeability of fused silica based on the stress distribution imposed by the SAP and bending [13*ε ˜

_{i}13. B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. **44**, 150–156 (2008). [CrossRef]

*T*is the rotation transformation from the local principal stress axes in element

_{i}*i*to the global coordinate axes

*x,y,r⃗*is the location of the center of element

_{i}*i*relative to the center of the fiber cross-section,

*R⃗*is the vector from the center of the fiber cross-section to the center of the fiber coil,

*ε*

_{0}is the permeability of free space,

*n*

_{0}is the bulk refractive index of the fiber material,

*C*are the photoelastic constants, and

_{i}*σ*are the principle stresses.

_{i}17. F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guid. Wave Lett. **8**, 223–225 (1998). [CrossRef]

*s*depend on the position and orientation of the layer relative to the center of the computational region. For layers oriented along the global coordinate

_{i}*x*, the PML transformation is represented by a diagonal matrix with the values

*s*=1-

_{x}*αd*

^{2}

_{i}

*j*,

*s*,

_{y}*s*=1 where

_{z}*d*is the depth within the layer of element

_{i}*i*and

*α*is the stretching parameter which is effectively the strength of the absorbing boundary condition and is proportional to the local conductivity of the medium. The parameter

*α*is unphysical in the sense that its value is not set by a property of the fiber, therefore the results of the calculations should be independent of its value. In order to preserve the symmetry of the computational mesh, 6 PML regions arranged as shown in Figure 1 are employed. The transformation matrices for these regions are obtained from this single region through a rotation transformation

*S*is the transformationmatrix for the element in the primary PML region with the same relative position within the layer as element

_{ix}*i*. Thus, the final forms used for the refractive index tensor and magnetic permittivity in the finite element calculations are

*µ⃡*=

_{i}*µ*

_{0}

*I⃡*and all transformations

*S*are the identity transformation for elements that are not contained within the PML regions. Implementing the details of the finite element scheme as described in [13

**44**, 150–156 (2008). [CrossRef]

*K*and

*M*are square sparse matrices of dimension 3

*N*where

*N*is the number of nodes in the mesh,

*ϕ*is the vector defining the magnetic field vector

*H*on the nodes, and

*β*is the propagation constant of the fields along the fiber axis. This eigenproblem is then solved to yield the vector field distributions and complex propagation constants of the modes. Appendix A briefly describes the software implementation of this method. The imaginary part of the propagation constant determines the propagation loss through the relationship

## 4. Numerical results

*µ*m yielding a flat-to-flat core diameter of 41

*µ*m. The Young’s modulus of borosilicate glass comprising the SAP is typically 60–64 GPa whereas that of fused silica is 73 GPa or approximately 20% higher. Thus the fiber naturally coils perpendicular to the axis along which the SAP are located or along the

*x*axis defined here. Figure 4 shows the propagation loss of the first four guided modes of the fiber as a function of the hole diamter to lattice pitch ratio

*d*/Λ for a coil diameter of 47.6 cm in the

*x*direction. As this is a birefringent fiber, the two LP

_{01}modes have different propagation constants. The one with the greater (lesser) propagation constant is the “slow” (“fast”) mode. The fast mode has a lesser effective index and experiences higher propagation losses. Also plotted are the losses for the LP

_{11o}and LP

_{11e}modes.

*μ*m diameter pump cladding yielding device lengths of approximately 4–5 meters of active fiber for a single-end-pumped configuration. This means that the fundamental mode loss should be kept below approximately 8×10

^{-3}dB/m. This requirement may be relaxed somewhat depending on variations in signal power along the length of the fiber. Figure 4 reveals a fairly slow variation in propagation losses with air hole size. The optimal hole size appears to be in the range of

*d*/Λ=0.11-0.12 which yields a minimum higher-order mode loss of 1 dB/m.

*y*direction, the modal losses are much higher for a given air hole size as shown in Figure 6. This is due to the SAP which assist the air hole lattice in confining the mode when the fiber is coiled in the

*x*direction. In these calculations, the SAP have a refractive index that is 0.006 below that of pure fused silica, a value that is currently employed in manufactured birefringent photonic crystal fibers. This effect is so pronounced that even for

*d*/Λ=0.17, the fundamentalmode loss is 8.2×10

^{-2}, a factor of 10 larger than the target design value. If the air hole size is further increased in order to reduce fundamental mode losses, the higher order mode losses when the fiber is coiled in the

*x*direction become negligible leading to poor mode discrimination. The best option appears to be to maintain the smaller air hole size and take care to coil the fiber in its natural orientation in the plane parallel to the SAP. A previously reported fiber design similar to the one considered here [19

19. T. Schreiber, F. Rser, O. Schmidt, J. Limpert, R. Iliew, F. Lederer, A. Petersson, C. Jacobsen, K. Hansen, J. Broeng, and A. Tnnermann, “Stress-induced single-polarization single-transverse mode photonic crystal fiber with low nonlinearity,” Opt. Express **13**, 7621–7630 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7621. [CrossRef] [PubMed]

*µm*and

*d*/Λ=0.12 which are very close to the optimum values determined here for coiling in the

*x*direction. In the proposed scenario, the sensitivity to slight misalignment of the coiling direction becomes important. Figure 8 shows the variation in mode losses as the coiling plane is changed relative to the SAP plane. For

*d*/Λ=0.12, the coiling plane may be misaligned by as much as 11° while keeping fundamental mode loss below 1×10

^{-2}dB/m.

*x*and

*y*directions on the modal losses. Perhaps the most important quality of the fiber in this regard is the relatively gentle variance of bend losses with coiling diameter. As the diameter is varied within the range of 20–28 cm, the higher-order mode loss remains above 1 dB/m while the fundamental mode loss drops from 0.007 dB/m to 0.0006 dB/m. This means that practically speaking, a coiling diameter variation of a few cm in this range is unlikely to have a drastic impact on performance. This relaxes the requirement for precise design and assembly tolerances for devices incorporating fibers such as these.

*d*/Λ=0.12 when coiled gently in the

*y*direction. A higher order mode loss of 5.5 dB/m is achievable at a diameter of 95 cm with only a 0.0014 dB/m loss in the fundamental mode. If a 1 meter form factor is tolerable, this configuration should be able to achieve superior mode discrimination and efficiency. Alternatively, a larger diameter of curvature in the

*y*direction may be incorporated into a tighter primary coil in the

*x*direction.

**43**, 899–909 (2007). [CrossRef]

20. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

*µ*m and a numerical aperture of 0.06 which is typical for large mode area step index fibers. The results are shown in Figure 11. No SAP are incorporated into this fiber design so only one of the polarizations of the fundamental mode is considered although there is some slight stress-induced birefringence due to bending. At a coiling diameter of 12 cm, the calculated stress-induced birefringence of this fiber is on the order of 10

^{-7}supporting the necessity of SAP for ensuring sufficient birefringence for single-polarization operation. The calculations indicate that the optimal coiling diameter for this fiber is between 8 cm and 12 cm. A diameter of 8.4 cm yielded a predicted fundamental mode loss of 0.5 dB/m and a higher-order mode (LP

_{11o})loss of 2 dB/m. At a somewhat larger diameter of 12.0 cm the fundamental and LP

_{11o}mode losses were predicted to be 0.0016 dB/m and 0.019 dB/m respectively. Interpolating the higher-order mode loss to the

*x*-coiled PCF value of 1 dB/m yields a fundamentalmode loss of 0.2 dB/m which for a device of several meters would impose a significant penalty on device efficiency. The much tighter optimum coiling diameter for the step-index fiber results in a severe distortion of the fundamental mode and reduction in the mode field area as shown in Figure 12.

*µ*m core diameter but a smaller numerical aperture (0.035). This represents and index step of 5×10

^{-4}. The reduced NA enables the SIF to behave more like the PCF with respect to bend-loss mode discrimination. This fiber appears to have an optimal coil diameter of 56cm at which the fundamental mode loss is around 0.006 dB/m with a LP

_{11o}mode loss of about 2 dB/m. If the fiber is coiled down to the design target of a 19″ (48 cm) diameter, the fundamental mode loss increases to about 0.1 dB/m. The results are summarized in Table 1. The change in fundamental mode loss as coiling diameter is varied between 67cm and 37cm provides a measure of the sensitivity of the fiber to irregularities in the coil configuration. For the low NA SIF the loss decreases from about 2 dB/m to about 3×10

^{-4}dB/m or about 5 orders of magnitude. The loss for the PCF coiled in the

*y*direction decreases from about 60 dB/m to about 0.06 dB/m or about 3 orders of magnitude. When coiled in the

*x*direction, the PCF loss decreases from about 4×10

^{-2}dB/m to 2×10

^{-5}dB/m, also a change of 3 orders of magnitude. These results show that the PCF is less sensitive to coiling diameter than the SIF and exhibits lower losses when coiled in the plane of the stress rods.

15. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

*d*/Λ=0.5-0.8. The multipole method predicted losses of 46±9% of those predicted by the FEM over 7 different hole sizes. This represents differences in the complex part of the effective index of 1.1±1.0×10

^{-8}. The real part of the effective indices differed by 5.9±1.4×10

^{-7}. While the agreement could definitely be improved upon, the same numerical scheme was applied here to both PCF and SIF so the conclusions derived from comparing their bend losses should be valid.

*α*characterizing the boundary region, and the approximate effective index of the modes to be calculated. All other inputs are set by the physical properties of the fiber. In order to assess the effect of the running parameters on the results, the results shown in Figure 10 which displayed the most oscillatory behavior, were calculated with boundary thicknesses of 35 and 55

*µm*, cladding to boundary distances of 10 and 20

*µm*and values of

*α*of 0.02 and 0.1. The two results for propagation losses in dB/m were indistinguishable when plotted and agreed to within 5%. For the remainder of the calculations, the value α=0.1 was used, a thickness of 35

*µm*, and a boundary distance of 10

*µm*as depicted in Figure 1.

21. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express **13**, 6015–6022 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

**43**, 899–909 (2007). [CrossRef]

18. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, pp. 185–200, (1994). [CrossRef]

**43**, 899–909 (2007). [CrossRef]

_{01}modes), the fast fundamentalmode is unique in that it is the only mode polarized along the fast axis. The LP

_{11}modes are both slow modes. Stress-induced birefringence can account for modes with different polarizations exhibiting oscillatory loss behavior with peaks and valleys at different points in the parameter space as the stress-induced birefringence causes the guided modes to couple to cladding modes with different effective indices. The simultaneous presence of all of these factors indicates that oscillatory behavior is not in itself cause to doubt the validity of the calculations. Rather, they should be checked with other methods applicable to bend-loss analysis of large mode area fibers as well as experimental results. One possible future avenue of investigation is to use the mode field distributions calculated here to evaluate Equation 13 of reference [11

**43**, 899–909 (2007). [CrossRef]

## 5. Discussion

**43**, 899–909 (2007). [CrossRef]

## Appendix A: Computational details

24. Vicente Hernandez, Jose E. Roman, and Vicente Vidal, “SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigenvalue Problems,” ACM Transactions on Mathematical Software, 31, 351–362 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. |

2. | M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express |

3. | A. Liem, J. Limpert, H. Zellmer, and A. Tnnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. |

4. | S. Hfer, A. Liem, J. Limpert, H. Zellmer, A. Tnnermann, S. Unger, S. Jetschke, H. -R. Mller, and I. Freitag, “Single-frequency master-oscillator fiber power amplifier system emitting 20 Wof power,” Opt. Lett. |

5. | I. Zawischa, K. Plamann, C. Fallnich, H. Welling, H. Zellmer, and A. Tnnermann, “All-solid-state neodymiumbased single-frequency master-oscillator fiber power-amplifier system emitting 5.5 W of radiation at 1064 nm,” Opt. Lett. |

6. | J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. |

7. | J. Limpert, O. Schmidt, J. Rothhardt, F. Rser, T. Schreiber, A. Tnnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express |

8. | T. -w Wu, L. Dong, and H. Winful, “Bend performance of leakage channel fibers,” Opt. Express |

9. | W. S. Wong, X. Peng, J. M. McLaughlin, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. |

10. | J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express |

11. | R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. |

12. | Robert D. Cook, |

13. | B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. |

14. | Kazuaki Sakoda, |

15. | T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

16. | B. T. Kuhlmey, University of Sydney Australian Research Counsel Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems Microstructured Optical Fibre Utilties Software 2004. |

17. | F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guid. Wave Lett. |

18. | J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

19. | T. Schreiber, F. Rser, O. Schmidt, J. Limpert, R. Iliew, F. Lederer, A. Petersson, C. Jacobsen, K. Hansen, J. Broeng, and A. Tnnermann, “Stress-induced single-polarization single-transverse mode photonic crystal fiber with low nonlinearity,” Opt. Express |

20. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. |

21. | J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express |

22. | Jeremy Kepner, “Parallel Programming with MatlabMPI,” in |

23. | Satish Balay, Kris Buschelman, Victor Eijkhout, William D. Gropp, Dinesh Kaushik, Matthew G. Knepley, Lois Curfman McInnes, Barry F. Smith, and Hong Zhang, |

24. | Vicente Hernandez, Jose E. Roman, and Vicente Vidal, “SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigenvalue Problems,” ACM Transactions on Mathematical Software, 31, 351–362 (2005). [CrossRef] |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(140.3510) Lasers and laser optics : Lasers, fiber

(230.7370) Optical devices : Waveguides

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: April 10, 2008

Revised Manuscript: May 21, 2008

Manuscript Accepted: May 23, 2008

Published: May 27, 2008

**Citation**

Benjamin G. Ward, "Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture," Opt. Express **16**, 8532-8548 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8532

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### References

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- M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, "Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power," Opt. Express 14, 11071-11076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071. [CrossRef] [PubMed]
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