## Design of single-moded holey fibers with large-mode-area and low bending losses: The significance of the ring-core region

Optics Express, Vol. 15, Issue 4, pp. 1794-1803 (2007)

http://dx.doi.org/10.1364/OE.15.001794

Acrobat PDF (604 KB)

### Abstract

One of the major trends in optical fiber science is to be able to obtain fibers with large-mode-area (LMA), optimized for various applications such as high power delivery, fiber amplifiers, and fiber lasers. In order to ensure the high beam quality and the ultimate controllability of the damage threshold in the fiber’s material, it is required to have a LMA property and of course to operate in a single mode fashion. While the conventional fibers have some difficulties in providing simultaneously LMA, single mode operation, as well as low macro-bending loss characteristics, all-silica holey fibers are highly attractive candidates for realizing LMA single-mode fibers with low bending losses. In this paper, we present a novel type of effectively single-mode holey fibers with effective mode area of about 1400 μm^{2}, small allowable bending radius as small as 5 cm, good beam quality factor of 1.15, and high confinement losses exceeding 1 dB/m for the higher-order mode at 1.064-μm wavelength.

© 2007 Optical Society of America

## 1. Introduction

1. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. Russell, and J. P. deSandre, “Large mode area photonic crystal fiber,” Electron. Lett. **34**,1347–1348 (1998). [CrossRef]

2. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express **11**,818–823 (2003). [CrossRef] [PubMed]

3. W. S. Wong, X. Peng, J. M. McLaughlim, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**,2855–2857 (2005). [CrossRef] [PubMed]

4. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, and A. Tünnermann, “Extended single-mode photonic crystal fiber lasers,” Opt. Express **14**,2715–2720 (2006). [CrossRef] [PubMed]

5. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. **31**,1797–1799 (2006). [CrossRef] [PubMed]

7. M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni, “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express **12**,1775–1779 (2004). [CrossRef] [PubMed]

1. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. Russell, and J. P. deSandre, “Large mode area photonic crystal fiber,” Electron. Lett. **34**,1347–1348 (1998). [CrossRef]

^{2}at 1.064-μm wavelength and small bending losses, has been reported [3

3. W. S. Wong, X. Peng, J. M. McLaughlim, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**,2855–2857 (2005). [CrossRef] [PubMed]

3. W. S. Wong, X. Peng, J. M. McLaughlim, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**,2855–2857 (2005). [CrossRef] [PubMed]

**30**,2855–2857 (2005). [CrossRef] [PubMed]

^{2}, small allowable bending radius as small as 5 cm, good beam quality factor of 1.15, and high confinement losses exceeding the 1 dB/m for the HOM at 1.064-μm wavelength. The designed holey fiber has a central core region formed by seven missing neighboring air-holes, and a ring core region between the innermost and the outermost air-hole rings. With an appropriate choice of the design parameters, the HOMs in the central core can be resonantly coupled to the much leaky ring-core mode around a certain wavelength range, while on the other hand the fundamental mode is strongly confined into the central core, leading to low bending losses and effectively single-mode operation around the desired operating wavelength.

## 2. Holey fiber design

*d*. Figure 1(b) shows the cross section of a holey fiber formed by seven missing neighboring air-holes with one air-hole ring (denoted through the paper as HF7-1), where the distance

*l*is a gap between the air-holes. Figure 1(c) shows the cross section of a holey fiber formed by seven missing neighboring air-holes and only one air-hole ring, with two different kinds of air-hole diameters,

*d*

_{1}and

*d*

_{2}(HF7-2). We also consider a novel type of holey fiber formed by seven missing neighboring air holes, and a ring core region around the inner air-hole ring as shown in Fig. 1(d) (HF7-3). The dashed circles in Fig. 1(d) denote the silica-based ring-core region.

*d*, exhibiting various values of effective mode area

*A*, while the confinement losses of the HOM-

_{eff}*L*

^{2nd}, are 0.1 dB/m, 1dB/m, and 10 dB/m, by using a full vectorial modal solver based on the finite element method (FEM) incorporating anisotropic perfectly matched layers as absorbing boundary conditions to evaluate confinement losses described in detail in Ref. [8

8. 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]

*ck*=

*c*2π/λ (where

*c*is velocity of light, and

*k*and λ are the free-space wavenumber and wavelength, respectively) the numerical calculation provides us with a complex propagation constant γ(ω) = β(ω) +

*i*α(ω) where β is the usual propagation constant component along the fiber axis associated with the mode and α is the attenuation constant associated with the exponential decay along the fiber axis. We present the attenuation on a dB-scale by 20 × log

_{10}(

*e*) × α ≅ 8.686 × α. Regarding the effective node area of the fiber core, we use the following definition:

**is the electric field vector and**

*E**S*denotes the whole fiber cross section.

*d*/Λ and Λ, while the solid black curves depicted inside the graph correspond to different values of the confinement losses of the HOM of HF1, shown Fig. 1(a). In this study, we target an effective mode area of 1400 μm

^{2}as in Ref. [3

**30**,2855–2857 (2005). [CrossRef] [PubMed]

^{2}, can significantly reduce the nonlinear effects even for high-power delivery applications. In addition, we define that the holey fiber with

*L*

^{2nd}of 1 dB/m can operate in an effectively single-mode fashion. In order to satisfy all the above mentioned characteristics, we can determine the structural parameters to be: Λ=41.9 μm and

*d*/Λ=0.68 for HF1, and these parameters correspond to the black filled circle in Fig. 2. However, we have numerically confirmed that the HF1 with Λ=41.9 μm and

*d*/Λ=0.68 has relatively large bending losses of 8 dB/m for 20 cm bending radius at 1.064 μm wavelength, so it is difficult to realize LMA single-mode holey fiber with low bending losses, using the conventional HF1 structure. Notice that the holey fiber structure reported in Ref. [3

**30**,2855–2857 (2005). [CrossRef] [PubMed]

^{-2}dB/m at 1.064-μm wavelength.

*d*/Λ and Λ, while the solid black curves (depicted inside the graph) correspond to different values of the confinement losses of the HOM of HF7-1, while the solid white curves correspond to different values of the distance l between air-holes of HF7-1, shown Fig. 1(b). In order to achieve effective mode area of 1400 μm

^{2}and large confinement losses of the HOM,

*L*

^{2nd}, above 1 dB/m, we can determine the structural parameters to be: Λ=18.5 μm and

*d*/Λ=0.53 for HF7-1, and these parameters correspond to the black filled circle in Fig. 3. However, we have numerically confirmed that the HF7-1 with Λ=18.5 μm and

*d*/Λ=0.53 has relatively large bending losses of 2 dB/m for 20 cm bending radius at 1.064 μm wavelength. In order to achieve low bending losses at 1.064-μm wavelength, small value of

*l*of about 5 μm is required. Therefore, for low bending losses, we should choose the following values of the design parameters: of Λ≈20 μm and

*d*/Λ≈0.75, and these parameters correspond to the white filled circle in Fig. 3. However, the confinement losses of the HOM with these structural parameters are lower than 0.5 dB/km at 1.064-μm wavelength. As a result, the HF7-1 with Λ=20 μm and

*d*/Λ=0.75 can not operate in an effectively single-mode fashion at 1.064-μm wavelength because of the low confinement losses of the HOM. So, in the aim of an enhancement of the confinement losses of the HOM we consider the holey fiber structure as shown in Fig. 1(c) (HF7-2). The 12 air-holes with two different diameters of

*d*

_{1}and

*d*

_{2}are arranged alternatively with hole-pitch Λ. The distance

*l*, is a gap between the air-holes, defined as

*l*=Λ-

*d*

_{1}/2-

*d*

_{2}/2.

*d*

_{2}/Λ, and for different values of

*d*

_{1}/Λ, where the normalized operating wavelength is λ/Λ = 0.05 (which corresponds to Λ≈20 μm and λ≈-1.064 μm). The red and blue circles correspond to HF7-1 structures with

*d*

_{1}/Λ=

*d*

_{2}/Λ=0.7 and

*d*

_{1}/Λ=

*d*

_{2}/Λ=0.75, respectively. In addition, the red and blue dashed curves represent the variation of the normalized confinement losses of the HOM

*L*

^{2nd}Λ by varying

*d*

_{1}/Λ and

*d*

_{2}/Λ while keeping the values of

*l*/Λ=0.30 and

*l*/Λ=0.25, respectively. From these results, we can clearly observe that the confinement losses of the HOM can be controlled only by a small fraction by changing the values of

*d*

_{1}/Λ and

*d*

_{2}/Λ in spite of keeping the same value of

*l*/Λ. The value of

*l*/Λ is an important parameter regarding the bending losses of the fundamental mode and it should be as small as possible. As a result, we can say that, to achieve the effective mode area of the fundamental mode

*A*= 1400 μm

_{eff}^{2}as well as the low bending loss properties at λ= 1.064 μm, we should consider the following range for the value of

*l*/Λ<0.27 while Λ≈20 μm (

*l*≈5.4 μm). However, the confinement losses of the HOM within these structural parameters are still lower than 3 dB/km at 1.064-μm operating wavelength. Therefore, it is difficult to realize low bending loss single-mode LMA fibers using HF7-2 structure as in the case of HF1 topology, and we should enhance the confinement losses of the HOM further.

*d*

_{1}and

*d*

_{2}arranged alternatively with hole-pitch Λ, as in the case of HF7-2. To enhance the confinement losses of the HOM in the central core, a ring core of pure-silica region is introduced outside the central core. The enhancement of the confinement losses of the higher-order central-core mode can be achieved through the index-matching mechanism between the central-core and the ring-core modes [9–11]. The air-hole diameter-

*d*in the outermost air-hole ring is determined in such a way as to match the effective index of the LP

_{11}-like HOM in the central core with that in the ring core, at the desired wavelength, which in our case is λ= 1.064 μm. Around this resonant wavelength, the HOMs of the central core are strongly coupled to those of the ring-core, resulting in a sufficient enhancement of the confinement losses of the coupled modes (HOMs). To achieve

*A*=1400 μm

_{eff}^{2}, index-matching around 1.064-μm wavelength, and low bending loss characteristics, we set the structural parameters as Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51 (

*l*/Λ=0.27), and

*d*/Λ=0.451, and these structural parameters correspond to the black filled circle in Fig. 4.

_{11}-like modes) in the central core and the ring core are plotted, respectively. These two modes can be coupled through the resonant coupling effect and Fig. 5(c) shows the corresponding field distribution of the coupled mode at λ=1.064 μm, where Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51, and

*d*/Λ=0.451. There is an obvious coupling of the central-core HOM into the outer ring-core. On the other hand, in Fig. 5(d), we plot the optical field distribution of the fundamental mode (LP

_{01}-like mode) at λ=1.064 μm. It is evident that the mode is confined strongly into the central core. Figure 6 shows the confinement losses of the HOM for HF7-3 as a function of the operating wavelength, where Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51, and

*d*/Λ=0.451. We can clearly see that large confinement losses of the HOM above 1 dB/m can be obtained over 100 nm wavelength range, leading to an effectively single-mode operation in this wavelength range. Due to the large core and the small value of

*l*, the holey fiber proposed here supports several higher-order modes in the central core region, however, the bending losses of other modes higher than 2nd-order modes increase dramatically around 20 cm bending radius. As a result, an effectively single-mode operation could be achieved for practical application. Figure 7 shows the dependence of the numerically calculated bending losses of the fundamental mode on the bending radius at 1.064-μm wavelength through the vector FEM [12

12. Y. Tsuchida, K. Saitoh, and M. Koshiba, “Design and characterization of single-mode holey fibers with low bending losses,” Opt. Express **13**,4770–4779 (2005). [CrossRef] [PubMed]

*d*/Λ=0.68 and Λ=18.5 μm and

*d*/Λ=0.53, respectively. Although the holey fibers of HF1, HF7-1, and HF7-3 have the same effective mode area of 1400 μm

^{2}and the same level of confinement losses of the HOM at 1.064-μm wavelength, the bending losses of HF7-3 are much less than that of HF1, and HF7-3 can be bent down to a bending radius of 5 cm, without a significant increment of the bending losses. In Fig. 8, we plot the optical field distributions in the curved HF7-3, where the operating wavelength is 1.064 μm and the bending radius is (a) 30 cm, (b) 20 cm, and (c) 10 cm. As we can see, the fiber curvature can not affect the coupling phenomenon between the fundamental mode of the central core and the ring-core mode, so low bending losses can be maintained even at a small bending radius.

## 3. Calculation of the beam quality factor

*M*

^{2}of the HF7-3 with the following design parameters: Λ= 20 μm,

*d*

_{1}/Λ = 0.95,

*d*

_{2}/Λ= 0.51, and

*d*/Λ=0.451.

*M*

^{2}

_{x}which is the

*x*-component of

*M*

^{2}can be expressed as follows [13

13. A. Mafi and J. V. Moloney, “Beam quality of Photonic-crystal fibers,” IEEE J. Lightwave Technol. **23**,2267–2270 (2005). [CrossRef]

*z*

_{0}is the location of the beam waist and λ is the operation wavelength. The second-moment-based beam width

*W*is defined as follows:

_{x}*I*(

*x*,

*y*) stands for the beam intensity profile in the cross section.

*M*

^{2}

_{y}which is

*y*-dependency of

*M*

^{2}can be also calculated by changing the subscript

*x*to

*y*in Eqs. (2) and (3). The

*M*

^{2}value that we present in this paper is the average value of

*M*

^{2}

_{x}and

*M*

^{2}

_{y}. The value of

*M*

^{2}can be evaluated by calculating the modal spot size variation when the mode propagates in free space, as suggested in Ref. [13

13. A. Mafi and J. V. Moloney, “Beam quality of Photonic-crystal fibers,” IEEE J. Lightwave Technol. **23**,2267–2270 (2005). [CrossRef]

14. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” IEEE J. Lightwave Technol. **19**,405–413 (2001). [CrossRef]

*M*

^{2}=1.15 for the fundamental mode at 1.064-μm wavelength. The conclusion is that although the HF7-3 structure has a ring core region, we can achieve a relative high beam quality factor, a fact which can be advantageous utilized in applications such as high power beam delivery.

## 4. Sensitivity analysis of the proposed holey fiber

*A*and

_{eff}*L*

^{2nd}to the tolerances of all the structural parameters that are Λ,

*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ. Figure 10(a) shows the dependence of the

*A*(red curve, right axis) and

_{eff}*L*

^{2nd}(blue dashed curve, left axis) on the variation of the lattice constant-Λ, while keeping all the other design parameters fixed in their optimized values. In Fig. 10(b) we examine the variation of the same quantities as in Fig. 10(a) but this time as a function of the tolerance of the normalized design parameter

*d*

_{1}/Λ. The impact of the parameter variation

*d*

_{2}/Λ and

*d*/Λ is plotted in Figs. 10(c) and 10(d), respectively. From these results we can estimate the variation of the quantities

*A*and the

_{eff}*L*

^{2nd}at the operating wavelength of 1.064-μm for the case of possible tolerances from the ideal nominal values: Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51, and d/Λ=0.451. We can clearly conclude that the

*A*of the proposed holey fiber is roughly constant as a function of the parameter variation Λ,

_{eff}*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ. On the other hand, we see that the effectively single-mode operation can be achieved only by setting the structural parameters of

*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ within a certain allowable range. This is because, by varying these parameters from their idealized values, the proposed holey fiber can not meet any more the condition necessary for the index-matching mechanism between the central-core and the ring-core modes at the desired 1.064-μm wavelength range. The crucial tolerance range within which the index-matching mechanism can result in an effectively single mode operation was estimated to be ± 0.5% to keep the level of

*L*

^{2nd}larger than 1 dB/m.

## 5. Conclusion

*d*

_{1}and

*d*

_{2}in the innermost air-hole ring can be tuned to enhance the confinement losses of the HOM and to reduce the bending losses of the fundamental mode. It has been shown through a detailed numerical analysis that it is possible to design a holey fiber with an effective mode area of 1400 μm

^{2}, sufficiently enhanced confinement losses of the HOM above 1dB/m for effectively single-mode operation, and low bending losses below 1 dB/m at 5 cm bending radius at λ=1.064 μm. We have succeeded to enhance the confinement losses of the HOMs up to a level of 1400 dB/km at λ=1.064 μm, a result that is found to be superior when compared to the reported holey fiber in Ref. [3

**30**,2855–2857 (2005). [CrossRef] [PubMed]

^{-2}dB/km.

## Acknowledgments

## References and links

1. | J. C. Knight, T. A. Birks, R. F. Cregan, P. St. Russell, and J. P. deSandre, “Large mode area photonic crystal fiber,” Electron. Lett. |

2. | J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express |

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

4. | J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, and A. Tünnermann, “Extended single-mode photonic crystal fiber lasers,” Opt. Express |

5. | S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. |

6. | P. Weβels and C. Fallnich, “Highly sensitive beam quality measurements on large-mode-area fiber amplifiers,” Opt. Express |

7. | M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni, “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express |

8. | 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. |

9. | T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya, European Conference on Optical Communications, We2.7.3 (2003). |

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

11. | K. Saitoh, N. J. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: Towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express |

12. | Y. Tsuchida, K. Saitoh, and M. Koshiba, “Design and characterization of single-mode holey fibers with low bending losses,” Opt. Express |

13. | A. Mafi and J. V. Moloney, “Beam quality of Photonic-crystal fibers,” IEEE J. Lightwave Technol. |

14. | K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” IEEE J. Lightwave Technol. |

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

(060.2430) Fiber optics and optical communications : Fibers, single-mode

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: October 26, 2006

Revised Manuscript: January 18, 2007

Manuscript Accepted: January 23, 2007

Published: February 19, 2007

**Citation**

Yukihiro Tsuchida, Kunimasa Saitoh, and Masanori Koshiba, "Design of single-moded holey fibers with large-mode-area and low bending losses: the significance of the ring-core region," Opt. Express **15**, 1794-1803 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1794

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### References

- J. C. Knight, T. A. Birks, R. F. Cregan, P. St. Russell, and J. P. deSandre, "Large mode area photonic crystal fiber," Electron. Lett. 34, 1347-1348 (1998). [CrossRef]
- J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, A. Tunnermann, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818-823 (2003). [CrossRef] [PubMed]
- W. S. Wong, X. Peng, J. M. McLaughlim, L. Dong, "Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers," Opt. Lett. 30, 2855-2857 (2005). [CrossRef] [PubMed]
- J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, and A. Tünnermann, "Extended single-mode photonic crystal fiber lasers," Opt. Express 14, 2715-2720 (2006). [CrossRef] [PubMed]
- S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, "Light propagation with ultralarge modal areas in optical fibers," Opt. Lett. 31, 1797-1799 (2006). [CrossRef] [PubMed]
- P. Weßels and C. Fallnich, "Highly sensitive beam quality measurements on large-mode-area fiber amplifiers," Opt. Express 11, 3346-3351 (2003).
- M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni, "Predicting macrobending loss for large-mode area photonic crystal fibers," Opt. Express 12, 1775-1779 (2004). [CrossRef] [PubMed]
- 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]
- T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya, European Conference on Optical Communications, We2.7.3 (2003).
- J. Fini, "Design of solid and microstructure fibers for suppression of higher-order modes," Opt. Express 13, 3477-3490 (2005). [CrossRef] [PubMed]
- K. Saitoh, N. J. Florous, T. Murao, and M. Koshiba, "Design of photonic band gap fibers with suppressed higher-order modes: Towards the development of effectively single mode large hollow-core fiber platforms," Opt. Express 14, 7342-7352 (2006). [CrossRef] [PubMed]
- Y. Tsuchida, K. Saitoh, and M. Koshiba, "Design and characterization of single-mode holey fibers with low bending losses," Opt. Express 13, 4770-4779 (2005). [CrossRef] [PubMed]
- A. Mafi and J. V. Moloney, "Beam quality of Photonic-crystal fibers," IEEE J. Lightwave Technol. 23, 2267-2270 (2005). [CrossRef]
- K. Saitoh and M. Koshiba, "Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides," IEEE J. Lightwave Technol. 19, 405-413 (2001). [CrossRef]

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