## Long wavelength behavior of the fundamental mode in microstructured optical fibers

Optics Express, Vol. 13, Issue 6, pp. 1978-1984 (2005)

http://dx.doi.org/10.1364/OPEX.13.001978

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### Abstract

Using a novel computational method, the fundamental mode in index-guided microstructured optical fibers with genuinely infinite cladding is studied. It is shown that this mode has no cut-off, although its area grows rapidly when the wavelength crosses a transition region. The results are compared with those for w-fibers, for which qualitatively similar results are obtained.

© 2005 Optical Society of America

## 1. Introduction

*fundamental*mode, possibly degenerate, remains. Since almost all optical fibers operate in this single mode regime, the fundamental mode is the most important bound mode. In most conventional optical fibers the fundamental mode itself is not cut off, i.e., it remains bound as λ approaches infinity [1].

2. A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic crystal fibers* (Kluwer, Boston, 2003). [CrossRef]

2. A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic crystal fibers* (Kluwer, Boston, 2003). [CrossRef]

3. B.T. Kuhlmey, R.C. McPhedran, and C.M. de Sterke, “Modal ‘cutoff’ in microstructured optical fibers,” Opt. Lett. **27**, 1684–1686 (2002). [CrossRef]

3. B.T. Kuhlmey, R.C. McPhedran, and C.M. de Sterke, “Modal ‘cutoff’ in microstructured optical fibers,” Opt. Lett. **27**, 1684–1686 (2002). [CrossRef]

4. B.T. Kuhlmey, R.C. McPhedran, C.M. de Sterke, P.A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where is the edge?,” Opt. Express10, 1285–1291 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285. [PubMed]

*fictitious source superposition*(FSS) method. We do not detail it here but outline the three concepts underpinning it [5]. The first is the use of fictitious sources which, when placed within any cylinder, can compensate exactly the reflected field generated by an incident field. It can thus generate an exterior field identical to that in the cylinder’s absence, as in a defect. While such a calculation is simple for a single cylinder, the interrelationships between the infinity of scatterers makes the solution of the problem for a non-periodic structure very difficult. Instead, we construct the defect mode from a superposition of quasiperiodic solutions, each corresponding to fictitious multipole sources

*=*

**q**_{p}*exp(*

**q***i*), where

**k**_{0}·**r**_{p}

**k**_{0}is the Bloch vector, embedded in each cylinder (

*p*) of the lattice at

*. The superposition is then formed by Brillouin zone (BZ) integration, thus generating a solution that satisfies the wave equations and boundary conditions, and which is associated with the fictitious source distribution*

**r=r**_{p}**∫**

*q*_{BZ}exp(

*i*)

**k**_{0}·**r**_{p}*d*at each cylinder. The BZ integration thus eliminates all sources except the one within the cylinder at

**k**_{0}

*r=r*_{0}=

**0**, which remains available to modify the response field, and thus to form the defect mode. As described, the BZ integration is two-dimensional and thus time consuming. The third key idea reduces the problem to a single integration: we model the structure as a diffraction grating, whose cylinders contain a phased line of sources, sandwiched between two semi-infinite photonic crystals, the actions of which are modelled by the Fresnel reflection matrix

*∞ [6*

**R**6. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E **64**, 046603:1-20 (2001). [CrossRef]

*∞. The grating is characterized by plane wave reflection and transmission scattering matrices, calculated using a multipole method [6*

**R**6. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E **64**, 046603:1-20 (2001). [CrossRef]

*n*

_{eff}, and the modal fields.

7. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” J. Quantum Electron. **10**, 879–887 (1974). [CrossRef]

## 2. Results

*d*=0.24Λ, with a single defect, constituting the core. The refractive index of the background glass is

*n*

_{b}=1.45. Following the literature for conventional fibers the normalized frequency

*V*and propagation constant U can be defined as [1, 8

8. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. **29**, 1739–1741 (2004). [CrossRef] [PubMed]

8. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. **29**, 1739–1741 (2004). [CrossRef] [PubMed]

*ρ*is the core diameter, which is taken to be

*ρ*=Λ/√3 [8

8. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. **29**, 1739–1741 (2004). [CrossRef] [PubMed]

*n*

_{fsm}is the effective index of the fundamental space filling mode [9

9. T.A. Birks, J.C. Knight, and P.St .J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

*n*

_{fsm}<

*n*

_{eff},

*U*<

*V*. Definitions (1) differ from those of Mortensen

*et al*. [10

10. N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, and K.P. Hansen, ”Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. **28**, 1879–1881 (2003). [CrossRef] [PubMed]

*n*

_{fsm}we obtain the solid curve in Fig. 1; because of definitions (1), the wavelength increases in the direction indicated by the large diagonal arrow. Note first the behavior at long wavelengths, where

*U*≈

*V*. From (1) this implies that

*n*

_{eff}≈

*n*

_{fsm}, and so the modal effective index is determined by the effective index of the cladding. Indeed, at these long wavelengths the mode extends well into the cladding. The longest wavelength for which we can obtain a result is λ=1.6Λ, corresponding to

*V*≈0.6. Here

*n*

_{fsm}and

*n*

_{eff}are very close, with

*n*

_{eff}-

*n*

_{fsm}≈10

^{-6}. Therefore, the (average) transverse wavenumber

*k*

_{⊥}=(

^{1/2}is very small; in fact at

*V*=0.6,

*k*

_{⊥}Λ≈7×10

^{-3}. The evaluation of the lattice sums that are used in the FSS method, for the multipole calculation of the grating scattering matrices, now becomes increasingly difficult, as their magnitudes scale as powers of 1/

*k*

_{⊥}[11].

*U*<

*V*, so that, by Eqs (1),

*n*

_{fsm}<

*n*

_{eff}, implying that a substantial fraction of the mode’s energy is in the core. The solid circles refer to λ/Λ=0.25,0.15,0.05 from left to right. These are approximately equally spaced in

*V*, implying that

*U*and

*V*remain finite, even as λ→0. This is because in this limit,

*n*

_{fsm},

*n*

_{eff}→

*n*

_{b}, which, consistent with the argument of Birks

*et al*., leads to the MOFs’ well-known endlessly single-mode behavior [9

9. T.A. Birks, J.C. Knight, and P.St .J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

4. B.T. Kuhlmey, R.C. McPhedran, C.M. de Sterke, P.A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where is the edge?,” Opt. Express10, 1285–1291 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285. [PubMed]

*n*

_{eff}. This can be understood as follows: the finite cladding reduces the degree of confinement and the effective index is thus smaller than in structures with an infinite cladding. As mentioned previously, based on the properties of MOFs of finite cross section, some of us previously concluded that the fundamental MOF mode does have a cutoff [4

4. B.T. Kuhlmey, R.C. McPhedran, C.M. de Sterke, P.A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where is the edge?,” Opt. Express10, 1285–1291 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285. [PubMed]

*transition region*, the extent of which was deduced from extrapolation [4

*V*=1.55 (first column), λ/Λ=0.50 (

*V*=1.23; second column), λ/Λ=1.1 (

*V*=0.79; third column) and λ/Λ=1.6 (

*V*=0.58, fourth column), are also indicated by the vertical lines in Fig. 1(a). As a comparison, we show in the bottom row, the modal field at the same frequencies in a MOF with infinite cross section. At the shortest wavelength λ/Λ=0.133 the mode is well confined to the core region and the fields in the MOFs with finite and infinite cross sections are very similar. We thus expect

*n*

_{eff}to be very similar as well, as is confirmed in Fig. 1(a). In contrast, at the longest wavelength λ/Λ=1.6 the mode is poorly confined. In the finite MOF it fills essentially the entire MOF cross section, with the mode size limited by antiguiding at the interface at the outer cladding [12

12. M. Yan and P. Shum, “Antiguiding in microstructured optical fibers,” Opt. Express12, 104–116 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104. [CrossRef] [PubMed]

*U*

^{f}>

*V*, indicating that

*n*

_{fsm}. This means that the field is not evanescent in the cladding region. This is consistent with the mode’s poor confinement and indicates strong confinement losses in the finite structure. At the two remaining wavelengths the mode exhibits intermediate behavior: at λ/Λ=1.1 we also find that

*n*

_{fsm}, with the mode no longer confined to the core, but antiguided by the outer cladding [12

12. M. Yan and P. Shum, “Antiguiding in microstructured optical fibers,” Opt. Express12, 104–116 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104. [CrossRef] [PubMed]

*λ*/Λ=0.50 the mode is just starting to lose confinement and is somewhat more extended than at λ/Λ=0.133.

*U*and

*V*on the axes are very similar to those in Fig 1(a) indicating that the choice

*ρ*=Λ/√3 [8

**29**, 1739–1741 (2004). [CrossRef] [PubMed]

*a*=1

*µ*m,

*n*

_{core}=1.45 and

*n*

_{clad}=1.43. It is similar to that in Fig. 1(a), except that now

*V*→∞ as λ→ 0. The dashed curves again apply to fibers with a finite cladding, here of radius 5

*µ*m (long-dashed curve) and 3

*µ*m (short-dashed curve). For these fibers the part of the cross section outside these radii was taken to have a refractive index

*n*

_{core}, as in MOFs. Note that the curves in Figs 1 are qualitatively similar, confirming that the cut-off behavior for MOFs and for conventional fibers is comparable. Note also the difference at short wavelengths, where

*V*→∞. Finally, Figs 3 are the equivalent of Figs 2, but for conventional fibers. They show the Poynting vector for a w-fiber with outer cladding radius of 3

*µ*m (top row) and a step index fiber with infinite cladding (bottom row) for the same V-values as in Fig. 2. As for the MOF, at the longest wavelength where

*V*=0.58, the field is not confined to the core, but does not extend as far in the w-fiber as in the fiber with infinite cladding [12

12. M. Yan and P. Shum, “Antiguiding in microstructured optical fibers,” Opt. Express12, 104–116 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104. [CrossRef] [PubMed]

*V*=1.55 the field is well confined to the core in both fibers and the finite extent of the cladding is irrelevant. At

*V*=0.79 the field is still confined to the core, while at

*V*=1.23 the wavelength is sufficiently large to extend well beyond the core, approaching the cladding’s outer edge.

## 3. Discussion and conclusions

*n*

_{b}

*-n*

_{fsm}≈0.023, and we can therefore understand the modes’ dominant field component using the scalar approximation [1]. As mentioned, at this wavelength

*k*

_{⊥}Λ≈7×10

^{-3}, and so, on average, we expect the field of the fundamental mode to vary in the cladding as K

_{0}(

*k*

_{⊥}

*r*), where K

_{0}is a modified Bessel function. We have confirmed this numerically. As a corollary, we may use this approximation to estimate the effective mode size. Solving

*r*

_{1/2}where the average field intensity decays to half the value at the core edge

*ρ*=Λ/√3 (see Eqs (1)). At λ/Λ=1.6, we find

*r*

_{1/2}>200Λ. This means that the mode field extends over hundreds of periods; to our knowledge, no conventional calculational method exists that can be used to calculate such large modes.

**29**, 1739–1741 (2004). [CrossRef] [PubMed]

*any*wavelength to determine the number of rings of holes required for finite cladding effect in MOFs to be negligible. Note that this method, based upon the analogy between MOFs and conventional fibers, requires accurate knowledge of the effective cladding index

*n*

_{fsm}.

**29**, 1739–1741 (2004). [CrossRef] [PubMed]

10. N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, and K.P. Hansen, ”Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. **28**, 1879–1881 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | A.W. Snyder and J.D. Love, |

2. | A. Bjarklev, J. Broeng, and A.S. Bjarklev, |

3. | B.T. Kuhlmey, R.C. McPhedran, and C.M. de Sterke, “Modal ‘cutoff’ in microstructured optical fibers,” Opt. Lett. |

4. | B.T. Kuhlmey, R.C. McPhedran, C.M. de Sterke, P.A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where is the edge?,” Opt. Express10, 1285–1291 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285. [PubMed] |

5. | S. Wilcox, L.C. Botten, R.C. McPhedran, C.G. Poulton, and C.M. de Sterke, “Exact modelling of defect modes in photonic crystals,” in press, Phys. Rev. E. |

6. | L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E |

7. | S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” J. Quantum Electron. |

8. | M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. |

9. | T.A. Birks, J.C. Knight, and P.St .J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

10. | N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, and K.P. Hansen, ”Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. |

11. | R.C. McPhedran, L.C. Botten, A.A. Asatryan, N.A. Nicorovici, C.M. de Sterke, and P.A. Robinson, “Ordered and disordered photonic bandgap materials,” Aust. J. Phys. |

12. | M. Yan and P. Shum, “Antiguiding in microstructured optical fibers,” Opt. Express12, 104–116 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104. [CrossRef] [PubMed] |

**OCIS Codes**

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

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2005

Revised Manuscript: March 6, 2005

Published: March 21, 2005

**Citation**

S. Wilcox, L. Botten, C. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, "Long wavelength behavior of the fundamental mode in microstructured optical fibers," Opt. Express **13**, 1978-1984 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1978

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### References

- A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
- A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic crystal fibers (Kluwer, Boston, 2003). [CrossRef]
- B.T. Kuhlmey, R.C. McPhedran, and C.M. de Sterke, �??Modal �??cutoff�?? in microstructured optical fibers,�?? Opt. Lett. 27, 1684-1686 (2002). [CrossRef]
- B.T. Kuhlmey, R.C. McPhedran, C.M. de Sterke, P.A. Robinson, G. Renversez, and D. Maystre, �??Microstructured optical fibers: where is the edge?,�?? Opt. Express 10, 1285-1291 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a>. [PubMed]
- S. Wilcox, L.C. Botten, R.C. McPhedran, C.G. Poulton, and C.M. de Sterke, �??Exact modelling of defect modes in photonic crystals,�?? in press, Phys. Rev. E.
- L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, �??Photonic band structure calculations using scattering matrices,�?? Phys. Rev. E 64, 046603:1�??20 (2001). [CrossRef]
- S. Kawakami and S. Nishida, �??Characteristics of a doubly clad optical fiber with a low-index inner cladding,�?? J. Quantum Electron. 10, 879-887 (1974). [CrossRef]
- M. Koshiba and K. Saitoh, �??Applicability of classical optical fiber theories to holey fibers,�?? Opt. Lett. 29, 1739-1741 (2004). [CrossRef] [PubMed]
- T.A. Birks, J.C. Knight, and P.St .J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, and K.P. Hansen, �??Modal cutoff and the V parameter in photonic crystal fibers,�?? Opt. Lett. 28, 1879-1881 (2003). [CrossRef] [PubMed]
- R.C. McPhedran, L.C. Botten, A.A. Asatryan, N.A. Nicorovici, C.M. de Sterke, and P.A. Robinson, �??Ordered and disordered photonic bandgap materials,�?? Aust. J. Phys. 52, 791-809 (1999).
- M. Yan and P. Shum, �??Antiguiding in microstructured optical fibers,�?? Opt. Express 12, 104-116 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-104</a.> [CrossRef] [PubMed]

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