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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 6 — Mar. 21, 2005
  • pp: 1978–1984
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Long wavelength behavior of the fundamental mode in microstructured optical fibers

S. Wilcox, L. C. Botten, C. Martijn de Sterke, B. T. Kuhlmey, R. C. McPhedran, D. P. Fussell, and S. Tomljenovic-Hanic  »View Author Affiliations


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

Arguably the most important property of an optical fiber is its bound mode spectrum, indicating the fibers’ eigenfrequencies and associated eigenfields versus wavelength. Using calculations that are based on the Maxwell equations, and taking the fiber cladding to be of infinite extent, it is found that at short wavelengths, conventional fibers support many bound modes. However, they are progressively cut off with increasing wavelength λ, so that at sufficiently long wavelengths only a single, 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

1. A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

].

If a MOF with infinite cladding can be compared to a step-index fiber, then the equivalent of a MOF with finite cross section is a w-fiber [7

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]

] in which the cladding is a low-index ring between a high-index core and a high-index region extending to infinity. Here we therefore also analyze w-fibers and compare them to MOFs with a cladding of finite cross section.

2. Results

The results below refer to a MOF with circular air holes located on an infinite hexagonal lattice with period Λ and hole diameter d=0.24Λ, with a single defect, constituting the core. The refractive index of the background glass is nb =1.45. Following the literature for conventional fibers the normalized frequency V and propagation constant U can be defined as [1

1. A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

, 8

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

]

V2πλρnco2ncl2=2πλΛ3nb2nfsm2,
U2πλρnco2neff2=2πλΛ3nb2neff2,
(1)

At shorter wavelengths 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 effnb , 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]

].

The fundamental modes of MOFs with a finite cross section were calculated earlier [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]

] and we find that the real part of the effective mode index of finite structures nefff, where here and below the superscript “f” refers to finite structures, satisfies nefff < 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]

]. This was based on the existence of a transition region, the extent of which was deduced from extrapolation [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]

]. The approximate positions of the edges of this transition region are indicated by the open circles in Fig. 1(a).

Fig. 1. Parameter U versus V for (a) MOF with parameters given in the text. Solid curve refers to a structure with infinite cladding. The three closed circles correspond to wavelengths λ=0.25Λ, 0.15Λ and 0.05Λ from left to right. The open circles correspond to the wavelengths indicated. The long (short)-dashed curve is the equivalent result for a MOF with a finite cross section with five (three) rings of holes. Dotted curve, included for convenience, gives U=V. (b) Same as (a) but for a conventional geometry with parameters given in the text. The points indicated by the circles in (a) have no equivalent here.

Fig. 2. Axial Poynting vector for a MOF with finite cross section with three rings of holes (top row), and an infinite cross section (bottom row), for V=1.55 (λ/Λ=0.133), V=1.23 (λ/Λ=0.50), V=0.79 (λ/Λ=1.1), and V=0.58 (λ/Λ=1.6) for the first, second, third and fourth columns, respectively. The small circles indicate the air holes.

The results for the MOF in Fig. 1(a) should be compared with results for a conventional fiber, shown in Fig. 1(b). The numerical values for U and V on the axes are very similar to those in Fig 1(a) indicating that the choice 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]

] is appropriate. The solid curve in this figure refers to the fundamental mode in a step-index fiber with radius a=1 µm, 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 µm (long-dashed curve) and 3 µm (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 µm (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]

]. In contrast, at the shortest wavelength where 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

We have examined the long-wavelength behavior of the fundamental mode in a MOF with infinite cross section using a newly developed numerical method. While previously, based on the properties of MOFs with finite cross section, it was predicted that this mode cuts off at long wavelengths, we have seen no evidence of this up to wavelengths λ/Λ≈1.6. Since this is well beyond the predicted cutoff wavelength we conclude that the fundamental mode for a MOF with infinite cross section does not cut off. We have compared the MOF results with those for a conventional w-fiber, and find qualitatively the same behavior.

Fig. 3. Axial Poynting vector for a w-fiber (top row), and a step-index fiber with infinite cladding (bottom row), for V=1.55,1.23,0.79,0.58 for the first, second, third and fourth column, respectively. The cladding’s inner and outer edges are indicated by white circles.

The analogy with conventional fibres can be straightforwardly analyzed quantitatively; even at λ/Λ=1.6, nb-n fsm≈0.023, and we can therefore understand the modes’ dominant field component using the scalar approximation [1

1. A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

]. 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 K0(k r), where K0 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

(K0(kr12)K0(kρ))2=0.5,
(2)

we find the approximate radius 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.

Having established the absence of a cut-off for the fundamental MOF mode, the estimation of the mode field diameter of Koshiba and Saitoh [8

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

] can now be used at 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.

The analogy between MOFs and conventional fibers was earlier established heuristically, based on the cutoff of the second mode and on some of the properties of the fundamental mode such as the mode field diameter [8

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

, 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]

]. Our findings here for the absence of a cut-off for the fundamental mode in fibers with an infinite cladding, and the cut-off behavior of this mode in fiber with a finite cladding, provide precise and independent evidence for this analogy.

Acknowledgments

The authors thank Profs Philip Russell and Jonathan Knight for interesting and useful discussions regarding this work. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence Program.

References and links

1.

A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

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]

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 64, 046603:1-20 (2001). [CrossRef]

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]

8.

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

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]

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]

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. 52, 791–809 (1999).

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

  1. A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
  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]
  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. 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]
  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 64, 046603:1�??20 (2001). [CrossRef]
  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]
  8. M. Koshiba and K. Saitoh, �??Applicability of classical optical fiber theories to holey fibers,�?? Opt. Lett. 29, 1739-1741 (2004). [CrossRef] [PubMed]
  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]
  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]
  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. 52, 791-809 (1999).
  12. 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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