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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 15170–15184
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Design strategies and rigorous conditions for single-polarization single-mode waveguides

Karen K. Y. Lee, Yehuda Avniel, and Steven G. Johnson  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 15170-15184 (2008)
http://dx.doi.org/10.1364/OE.16.015170


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Abstract

We establish rigorous necessary analytical conditions for the existence of single-polarization single-mode (SPSM) bandwidths in index-guided microstructured waveguides (such as photonic-crystal fibers). These conditions allow us to categorize designs for SPSM waveguides into four strategies, at least one of which seems previously unexplored. Conversely, we obtain rigorous sufficient conditions for the existence of two cutoff-free index-guided modes in a wide variety of microstructured dielectric waveguides with arbitrary periodic claddings, based on the existence of a degenerate fundamental mode of the cladding (a degenerate light line). We show how such a degenerate light line, in turn, follows from the symmetry of the cladding.

© 2008 Optical Society of America

1. Introduction

In a perfectly cylindrical fiber, the fundamental mode is doubly degenerate and there are two orthogonal “polarizations” with the same dispersion relation, and hence travelling at the same velocity down the fiber. Due to mechanical and thermal stresses induced during the fabrication process and by environmental conditions, there is often a slight asymmetry in the fiber geometry, which breaks the degeneracy of the two polarization modes [1

1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

]. The two modes then travel at slightly different velocities, causing pulse broadening via polarization-mode dispersion (PMD) [1

1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

]. Polarization-maintaining (PM) fibers [2

2. T. F. Morse and A. Méndez, Specialty Optical Fibers Handbook (Academic Press, 2007).

] are optical fibers that can faithfully preserve and transmit the polarization state of the light that is launched into it under practical operating conditions, to alleviate the problem of PMD and provide a known polarization output at the end face of the fiber (useful for coupling to polarization-sensitive devices such as most integrated optics). Effectively single-polarization behavior can also be observed in hollow metallic tubes [33

33. E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

] and in photonic-bandgap fibers, both of which can operate in a non-degenerate lowest-loss mode (e.g. the TE01 mode) [34

34. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Optics Express 9, 748–779 (2001). [CrossRef] [PubMed]

]–even when these waveguides are multimode, the higher loss of the other modes effectively filters them out. The strictest guarantees of polarization maintenance, however, are achieved in SPSM fibers, in which only one polarization mode is guided, rather than having two (or more) guided polarization modes that are merely birefringent and difficult to convert between.

To be more precise, the two “polarizations” merely refer to the two lowest-frequency guided modes. They may or may not correspond to two orthogonal linear polarizations, and may or may not be degenerate. The terminology came about from the scalar limit: in conventional homogeneous-cladding fibers, the core has a very slightly increased index from the cladding, and in this “weakly-guiding” limit, the scalar approximation [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

] applies and the two lowest-frequency solutions are purely polarized in two orthogonal directions [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

, 36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

] (known as the “linearly polarized” LP01 modes [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

]). (The scalar approximation applies in the high-frequency limit to holey fibers as well [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

38

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

].) More generally, the two lowest-frequency modes are two linearly independent guided-mode solutions that satisfy the usual field-orthogonality relationships [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

, 36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

], but are neither purely polarized in one direction nor are 90° rotations of one another (e.g. in a holey fiber with sixfold-symmetry). With sufficient symmetry, the two “polarization” solutions are degenerate [39

39. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001). [CrossRef]

]. We shall return to the details of the effect of symmetry on the structure of the guided modes in the subsequent discussion.

In Sec. 2, we first derive rigorous sufficient conditions to obtain two cutoff-free guided modes in a wide range of microstructured fibers. In many cases, these conditions allow one to rigorously predict the existence of two guided “polarizations” without calculation, merely from the fact that the waveguide core was created by strictly increasing the refractive index. This theorem depended on the existence of a doubly degenerate space-filling fundamental mode of the waveguide cladding (in the absence of the waveguide core), and in Sec. 3 we discuss under what conditions this degeneracy follows from the symmetry of the cladding. The contrapositive of our sufficient conditions for two guided modes are necessary conditions for SPSM waveguides, and in Sec. 4 we apply these necessary conditions to divide old and new designs for SPSM waveguides into four categories. Finally, we conclude in Sec. 5 with some remarks about open questions and future directions.

2. Sufficient conditions for two-polarization waveguides

In this section, we will derive and discuss sufficient conditions for a waveguide to have at least two linearly independent index-guided modes. (The contrapositive of this will be necessary conditions for a single-polarization waveguide, which are discussed in Sec. 4.) These sufficient conditions are a generalization of our earlier proof of the existence of at least one index-guided mode under certain conditions [3

3. K. K. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express 16, 9261–9275 (08). [CrossRef]

]. We begin by reviewing the definition of index-guided modes and the class of waveguides that we consider in this paper, a well as reprising our earlier results.

Fig. 1. Schematics of various types of dielectric waveguides in which our theorem is applicable. Light propagates in the z direction (along which the structure is either uniform or periodic) and is confined in the xy direction by a higher-index core compared to the surrounding (homogeneous or periodic) cladding.

2.1. Index-guided modes

Loosely speaking, index-guiding is the phenomenon of light being confined along a waveguide consisting of core with a higher “average” index of refraction surrounded by a cladding with a lower “average” index of refraction. (In this paper, we do not consider other guiding mechanisms such as photonic band gaps [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

,40

40. P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in Waves in Periodic and Random Media, vol. 339 of Contemporary Mathematics, pp. 105–115 (AMS, Providence, RI, 2003).

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

] or metallic waveguides [44

44. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1975).

].) A schematic of several such dielectric waveguides is shown in Fig. 1. In particular, we suppose that the waveguide is described by a relative permittivity (square of the index) ε(x,y, z)=εc(x,y,z)+Δε(x,y,z) such that: ε, εc, and Δε are periodic in z (the propagation direction) with period a (a→0 for the common case of a waveguide with a constant cross-section); that the cladding permittivity εc is periodic in xy (e.g. in a photonic-crystal fiber), with a homogeneous cladding (e.g. in a conventional fiber) as a special case; and the core is formed by a change Δε in some region of the xy plane, sufficiently localized that |1/ε-1/εc|<∞(integrated over the xy plane and the unit cell in z). This includes a very wide variety of dielectric waveguides, from conventional fibers [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

] [Fig. 1(a)] to photonic-crystal “holey” fibers [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 41

41. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

] [Fig. 1(b)] to waveguides with a periodic “grating” along the propagation direction [Fig. 1(c)] such as fiber-Bragg gratings [1

1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

] and other periodic waveguides [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 45

45. C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE 64, 1666–1698 (1976). [CrossRef]

, 46

46. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995). [CrossRef]

]. We exclude metallic structures (i.e, we require ε > 0) and make the approximation of lossless materials (real ε). We allow anisotropic materials, in which case ε must be a 3 × 3 Hermitian matrix to be lossless. For convenience, we define:

Δ(x,y,z)ε1εc1.
(1)

In a waveguide as defined above, the solutions of Maxwell’s equations (both guided and non-guided) can be written in the form of eigenmodes H(x,y, z)e iβz - iωt (via Bloch’s theorem) [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

], where ω is the frequency, β is the propagation constant, and the magnetic-field envelopeH(x,y, z) is periodic in z with period a (or is independent of z in the common case of a constant cross section, a→0). A plot of ω versus β for all eigenmodes is the “dispersion relation” of the waveguide, one example of which is shown in Fig. 2. In the absence of the core (i.e. if Δε = 0), the (non-localized) modes propagating in the infinite cladding form the “light cone” of the structure [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 41

41. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

]; and at each real β there is a fundamental (minimum-ω) space-filling mode at a frequency ωc(β) with a corresponding field envelope H c [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 41

41. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

]. Such a light cone is shown as a shaded triangular region in Fig. 2. Below the “light line” ωc(β), the only solutions in the cladding are evanescent modes that decay exponentially in the transverse directions [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 41

41. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

, 47

47. P. Kuchment, “The Mathematics of Photonic Crystals,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematics, pp. 207–272 (SIAM, Philadelphia, 2001).

]. Therefore, once the core is introduced (Δε≠0), any new solutions with ω < ωc must be guided modes, since they are exponentially decaying in the cladding far from the core: these are the index-guided modes (if any). Such guided modes are shown as lines below the light cone in Fig. 2: in this case, both a lowest-lying (“fundamental”) guided mode with no low-frequency cutoff (although it approaches the light line asymptotically as ω→0) and higher-order guided modes with low-frequency cutoffs are visible. In this particular case, there are actually two non-degenerate cutoff-free guided modes corresponding roughly to two polarizations; the fields are not purely polarized, so the two polarizations can be more precisely distinguished in terms of their even/odd symmetry with respect to the mirror planes of the waveguide [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

].

Fig. 2. Example dispersion relation for a 3D rectangular waveguide in air of width a and height 0.4a (inset), showing the light cone, the light line, and fundamental and second (cutoff-free) guided modes, and higher-order modes with cutoffs.

Since a mode is guided if ω < ωc, the existence of a guided mode can be shown by demonstrating that ω has an upper bound <ωc. Using the variational (min–max) theorem for Hermitian eigenproblems [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

], we derived [3

3. K. K. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express 16, 9261–9275 (08). [CrossRef]

] the following sufficient condition for the existence of an index-guided mode in a dielectric waveguide at a given β: a guided mode must exist whenever

Dc*.Δ(x,y,z)Dc<0,
(2)

where the integral is over xy and one period in z, and Dc=iω(+iβẑ)×Hc is the displacement field of the cladding’s fundamental mode. From this condition, we immediately obtained a number of useful special cases:

• There must be a cutoff-free guided mode if Δ is negative-definite everywhere. (i.e., if we only increase the index to make the core).

• More generally, a guided mode has no long-wavelength cutoff if Eq. (2) is satisfied for the quasi-static (ω→0, β→0) limit of D c.

In particular, this was proved using the variational/min–max theorem, which states that an upper bound for the lowest eigenfrequencyω min(β) is given by the following Rayleigh quotient for any divergence-free trial function H [i.e., (∇+ H = 0]:

ωmin2(β)c2[(+iβẑ)×H]*·1ε[(+ẑ)×H]H*·HQ(H),
(3)

2.2. Two-polarization waveguides

In many important cases, the cladding fundamental mode ωc(β) is doubly degenerate (two linearly independent “polarizations” with the same frequencyωc)–this is independent of whether the guided mode is doubly degenerate, which depends on the symmetry of the core as well as of the cladding. When ωc is doubly degenerate, one obtains an index-guided mode if Eq. (2) is true for any of the degenerate fundamental modes D c (because any one of these modes could have been used in the proof from Ref. 3). If Eq. (2) holds for all of the degenerate fundamental field patterns D c, then one is guaranteed to have at least two index-guided modes (a two-polarization waveguide). We now prove this statement, a generalization of a result in Ref. 4 for homogeneous claddings. In the subsequent section, we will give symmetry conditions to have a doubly degenerate light line ωc, but here we simply assume that to be the case.

The variational theorem [Eq. (3)] gave us an upper bound for the lowest-frequency mode’s ω in terms of the Rayleigh quotient Q(H) for any divergenceless trial function H. In order to obtain an upper bound for the n-th mode’s frequency ωn, the variational theorem can be generalized as follows [4

4. A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990). [CrossRef]

]:

ωn2supQ(H),
Hn
(4)

where 𝓗n is any n-dimensional subspace of divergence-free vector fields. That is, the supremum of the Rayleigh quotient of any n-dimensional subspace is an upper bound for the n-th eigenfrequency. Equality is achieved when 𝓗n is the span of the n lowest-ω modes, and for this subspace the maximum of the Rayleigh quotient is ω 2 n.

The consequence of Eq. (4), here, is that if we can find any two-dimensional subspace 𝓗 2 of divergence-free trial fields such that Q(H) is below the light line for every field H in the subspace, then the second eigenfrequency ω 2 must also lie below the light line, and hence there must be two guided modes. We can find such a subspace, assuming that the fundamental mode H c (and D c) of the cladding is doubly degenerate, if Eq. (2) is satisfied for both degenerate modes and all their linear combinations. We construct the subspace 𝓗 2 as follows. Given two linearly independent cladding fundamental modes H (1) c and H (2) c, we construct the corresponding trial functions H (1,2) as in Ref. 3. Because this construction is linear, H=c 1 H (1)+c 2 H (2) is then the trial function constructed from H c = c 1 H (1) c+c 2 H (2) c for any constants c 1 and c 2. Because H c is also a cladding fundamental mode, and satisfies Eq. (2) by assumption, then Q(H)<ω 2 c by exactly the same proof as in Ref. 3. Hence Q < ω 2 c for every H in 𝓗 2=span{H (1),H (2)}, and there are at least two index-guided modes.

Given a doubly-degenerate cladding fundamental mode H (1,2) c, it is not in general sufficient for Eq. (2) to be satisfied only for any two of these modes; it must be satisfied for all their linear combinations as assumed above. The reason is that, given the displacement fields D (1,2) c and some linear combination D c = c 1 D (1) c+c 2 D (2) c, when substituted into Eq. (2) there are cross terms 2ℜ[(D (1) c)*·ΔD (2) c] that may be positive. On the other hand, if Eq. (2) holds for two degenerate cladding fundamental modes that one has orthogonalized in the sense that (D (1) c)*·ΔD (2) c = 0, then it holds for all linear combinations and the existence of at least two index-guided modes follows.

The easiest case, of course, is the one in which Δ is nonpositive-definite (e.g. if Δε≥0 everywhere), in which case Eq. (2) always holds. As we will describe below, a holey photoniccrystal fiber with sufficient symmetry always has a doubly-degenerate fundamental cladding mode, and it follows that filling in a hole (or otherwise strictly increasing ε) will always result in two cutoff-free index-guided modes (which are also degenerate if the core has sufficient symmetry, but may be nondegenerate otherwise).

Another simple common case is the one in which all degenerate modes have the same displacement-field magnitude |D c| everywhere–this is true for a homogeneous cladding (where |D c| is a constant), and also for an arbitrary cladding in the large-β limit where a scalar approximation becomes valid [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

38

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

]. Then, if the materials are isotropic, so that D * c·ΔD c = Δ|D c|2, Eq. (2) will hold for all degenerate modes (if it holds for any of them) and one is guaranteed two index-guided modes. (This reproduces the result proved by Ref. 4 for homogeneous claddings.)

In general, |D c| is not the same for different degenerate modes of an inhomogeneous cladding. However, if the degeneracy is due to cladding symmetry as described below, and the core Δε preserves this symmetry, then Eq. (2) is equivalent for all degenerate modes. That is, in symmetric structures (with three-fold, four-fold, or six-fold symmetry as described below), it is sufficient for Eq. (2) to hold for one of the degenerate modes, from which it follows that it holds for all of the degenerate modes. (The reason for this is that D * c·ΔD c is invariant under symmetry operations/rotations of D c if Δ is invariant and hence commutes with the rotation.) In this case, again, one is guaranteed at least two index-guided modes in the (symmetric) core, and in fact these two modes must themselves be doubly degenerate (because they cannot be orthogonal to the trial functions, and hence cannot belong to a different irreducible representation). So, for example, a typical holey fiber formed by a triangular lattice of circular air holes (with six-fold symmetry) and a missing-hole waveguide core [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

] is guaranteed analytically to have a degenerate pair of cutoff-free index-guided modes.

3. Symmetry and the degeneracy of the light line

Under what conditions is the cladding fundamental mode doubly degenerate? Usually, such degeneracy is a consequence of symmetry, and in particular is a consequence of the cladding symmetry group having a two-dimensional irreducible representation [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 39

39. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001). [CrossRef]

, 49

49. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Heidelberg, 1996).

]. (Any degeneracy that does not result from symmetry is known as “accidental,” but this is something of a misnomer since accidental degeneracies are very unlikely to arise by chance [50

50. M. Tinkham, Group Theory and Quantum Mechanics (Dover, New York, 2003).

].) For example, two-dimensional irreducible representations arise when the cladding has three-fold (C 3v), four-fold (C 4v), or six-fold (C 6v) symmetry [49

49. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Heidelberg, 1996).

], as depicted schematically in Fig. 3(a–c). Even if the symmetry group has a two-dimensional representation, this would not seem to guarantee that the fundamental mode will fall into this representation and be doubly degenerate, but it is easy to check whether this is the case by a small calculation using the unit-cell of the cladding–in particular, the common “holey fiber” claddings of a square or triangular lattice of symmetrical air holes in dielectric both have doubly degenerate fundamental cladding modes.

Fig. 3. . Example microstructured optical fiber cladding structures with three-fold, fourfold, six-fold and cylindrical rotation symmetries. Claddings with these symmetries are guaranteed to have a doubly-degenerate light line, at least in the long-wavelength limit for cases (a–c).
Fig. 4. First two bands (Bloch modes) of a holey-fiber cladding (triangular lattice, period a, of radius 0.3a air holes in index-1.45 silica) plotted around the boundary of the irreducible Brillouin zone (inset). Each pair of bands is plotted for a fixed value of β:β = 0.001, 0.2, 0.5, and 1.0 in units of 2π/a. The bands are doubly degenerate, by symmetry, at the Γ point, and, because this is the lowest-frequency mode at each β, it is the (doubly degenerate) fundamental mode of the cladding and defines the light line.

Also, in the common case of a homogeneous, isotropic cladding (C ∞v symmetry), the cladding fundamental mode is known analytically to be the two orthogonal linear polarizations (which fall into a two-dimensional irreducible representation).

However, by considering the relationship between an arbitrary periodic (symmetrical) cladding and the homogeneous case more carefully, it turns out that there is a guarantee that a sufficiently symmetrical cladding (one with a two-dimensional irreducible representation) will have a doubly degenerate fundamental mode for all sufficiently long wavelengths. This guarantee is implied by continuity considerations for the eigenmode’s irreducible representation, which force the fundamental-mode symmetry to be determined by the long-wavelength quasi-static limit. As an example to illustrate this argument, consider the fundamental mode of the typical holey fiber cladding, a triangular lattice of air holes in silica, which is plotted in the cladding Brillouin zone [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

] for several values of β in Fig. 4. As β goes to zero, the fundamental mode at the Γ point must go to zero frequency: this is the long-wavelength “quasistatic” solution, and in this limit the structure can be replaced by an effective homogeneous medium [51

51. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005). [CrossRef]

]. Moreover, the rotational symmetry of the structure implies that the effective homogeneous medium must be isotropic, and hence must have a doubly degenerate fundamental mode consisting of two orthogonal polarizations. But two orthogonal polarizations are described by one of the two-dimensional irreducible representations of the six-fold symmetry group [49

49. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Heidelberg, 1996).

], which means that the exact solution at Γ must also fall into this representation. The reason is that, as β is increased and the Γ-point mode moves up in frequency, the corresponding field pattern must change continuously–it cannot discontinuously jump from one symmetry representation to another. Therefore, as long as the fundamental mode is the mode at the Γ point, it must be doubly degenerate. The only way that the fundamental mode could conceivably become non-degeneratewould be if, for some sufficiently short wavelength, the frequency at some other point in the Brillouin zone (e.g. M or K) became lower than the frequency at Γ. It may be that this is possible, although we do not observe it to occur for this structure. Regardless, the conclusion remains that, at least for sufficiently long wavelengths (once the ω at Γ becomes the lowest), the cladding fundamental mode must be doubly degenerate. The same conclusion holds for every other crystalline symmetry group (three-fold, four-fold, or six-fold symmetry) in which there is a two-dimensional irreducible representation.

Fig. 5. Schematic single-polarization waveguides embodying strategies (i)–(iv).

4. Four strategies to design SPSM waveguides

Above, we derived a sufficient condition to have two linearly independent guided modes in the waveguide. The contrapositive of this is a necessary condition to have only a single guided mode: one must violate the conditions of the theorem to obtain such a waveguide. This is a useful result, because truly single-polarization single-mode (SPSM) waveguides are the most robust way to obtain a polarization-maintaining waveguide, and have been proposed by many authors for this application [6

6. T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80). [CrossRef]

23

23. D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. 25, 2700–2705 (2007). [CrossRef]

, 25

25. W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. 7, 629–631 (1982). [CrossRef] [PubMed]

32

32. S. Kim, U. C. Paek, and K. Oh, “New defect design in index guiding holey fiber for uniform birefringence and negative flat dispersion over a wide spectral range,” Opt. Express 13, 6039–6050 (2005). [CrossRef] [PubMed]

]. One can divide the mechanisms for violating the conditions for two-mode guidance into four strategies for single-polarization waveguides:

(i) Violate Eq. (2) entirely, or the underlying conditions of the theorem (e.g. employ a non-periodic cladding, such as an asymmetrical substrate). With one notable exception, this typically means that both polarizations have cutoffs for index-guiding, where the cutoffs are different because of some asymmetry and hence there is a single-polarization region.

(ii) Utilize anisotropic media (a tensor Δ) so that Eq. (2) is satisfied only for one orientation of the degenerate cladding fundamental modes.

(iii) Use an asymmetrical cladding so that the cladding fundamental mode is nondegenerate.

(iv) Use a symmetrical cladding with a doubly degenerate fundamental mode, but one in which |D c| is different for the two polarizations (due to cladding inhomogeneity) so that Eq. (2) can be satisfied for one polarization but not the other.

4.1. Strategy (i)

In this category, we include all SPSM structures that violate the conditions of our theorem entirely, including waveguides that do not rely upon index-guiding. Perhaps the most common such technique to achieve an SPSM fiber is to design a waveguide in which both polarizations have cutoffs, albeit at different frequencies because of some asymmetry. In order to do this, one must violate the conditions of our theorem, which otherwise would guarantee that a guided mode exists. One way to do this is to use a non-periodic cladding, such as an asymmetrical substrate (e.g. an silicon-on-insulator waveguide with air above), which leads to a long-wavelength cutoff for all guided modes of a waveguide [53

53. R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1982).

, 54

54. C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006). [CrossRef]

]. With a periodic (or homogeneous) cladding, our theorem implies that a waveguide in which all modes are cut off must have a Δε that is negative in some regions (since Δε≥0 yields a cutoff-free guided mode). To accomplish this, the oldest technique is a W-profile fiber, in which a higher-index core is surrounded by a depressed-index inner cladding, leading to a cutoff [55

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

] and to an SPSM bandwidth if some asymmetry is introduced [8

8. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983). [CrossRef]

, 10

10. M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991). [CrossRef]

, 15

15. M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005). [CrossRef]

]. Other geometries include side-pit or bow-tie fibers, in which the depressed-index regions are located asymmetrically on two sides of the core (instead of surrounding it as in W-profile fibers) [6

6. T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80). [CrossRef]

, 7

7. T. Okoshi, K. Oyamada, M. Nishimura, and H. Yakato, “Side tunnel fibre: An approach to polarization-maintaining optical waveguiding schemes,” Electron. Lett. 18, 824–826 (1982). [CrossRef]

, 9

9. K. Tajima, M. Ohashi, and Y. Sasaki, “A new single-polarization optical fiber,” IEEE J. Lightwave Technol. 7, 1499–1503 (1989). [CrossRef]

, 16

16. M.-J. Li, D. A. Nolan, G. E. Berkey, X. Chen, J. Koh, D. T. Walton, J. Wang, W. A. Wood, and L. A. Zenteno, “High-performance single-polarization optical fibers,” Proc. SPIE 5623, 612–621 (2005). [CrossRef]

]. More recently, photonic-crystal holey fibers have used combinations of removed or shrunk holes (Δε > 0) and enlarged holes (Δε < 0) to cut off both guided modes [11

11. A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystal fibers,” Appl. Phys. Lett. 78, 3184–3186 (2001). [CrossRef]

14

14. J. R. Folkenberg, M. D. Nielsen, and C. Jakobsen, “Broadband single-polarization photonic crystal fiber,” Opt. Lett. 30, 1446–1448 (2005). [CrossRef] [PubMed]

, 17

17. X. Liu, F. Zhang, M. Zhang, and P. Ye, “A novel single-mode single-polarization photonic crystal fiber using resonant absorption effect,” Proc. SPIE 6351, 63,511K (2006).

20

20. M. Szpulak, T. Martynkien, J. Olszewski, W. Urbanóczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, “Single-polarization single-mode photonic band gap fiber,” Acta Physica Polonica A 111, 239–245 (2007).

, 22

22. M. Eguchi and Y. Tsuji, “Single-mode single-polarization holey fiber using anisotropic fundamental space-filling mode,” Opt. Lett. 32, 2112–2114 (2007). [CrossRef] [PubMed]

, 23

23. D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. 25, 2700–2705 (2007). [CrossRef]

]. Fibers that guide light by a photonic bandgap [36

36. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

, 41

41. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

43

43. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

], e.g. in a hollow core, also have long-wavelength cutoffs of the gap-guided modes for all polarizations–in the long-wavelength limit, any periodic dielectric structure can be described by an effective homogeneous material [51

51. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005). [CrossRef]

] and hence has no bandgap. (Gap-guided modes fall outside the confines of our theorem because they do not lie below the cladding light cone.)

4.2. Strategy (ii)

Another strategy for SPSM fibers is to use an anisotropic ε. The most obvious technique would be to use an Δε that is positive for one polarization and negative (or zero) for the other, which therefore will guide only one polarization (which is cutoff-free). In the context of our theorem, this strategy appears as an anisotropic (tensor) Δ, such that Eq. (2) is satisfied for one degenerate fundamental cladding mode D (1) c but not for the orthogonal fundamental mode. Experimentally, this has been achieved/proposed using stress birefringence [25

25. W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. 7, 629–631 (1982). [CrossRef] [PubMed]

29

29. K. S. Chiang, “Stress-induced birefringence fibers designed for single-polarization single-mode operation,” IEEE J. Lightwave Technol. 7, 436–441 (1989). [CrossRef]

] or by liquid crystal filling the core of a holey fiber [21

21. D. C. Zografopoulos, E. E. Kriezis, and T. D. Tsiboukis, “Photonic crystal-liquid crystal fibers for single-polarization or high-birefringent guidance,” Opt. Express 14, 914–925 (2006). [CrossRef] [PubMed]

]. Note that by varying various parameters, the design in Ref. 21 can have a cutoff for both guided modes (strategy (i))

Fig. 6. Fig. 6. Dispersion relations of structures with an asymmetric cladding with no two-dimensional irreducible presentation. In both cases, the fundamental polarization is rigorously cutoff-free, with a single-polarization region below the second-mode cutoff at β = 0.82 and β = 0.5 respectively. Here, we plot the mode frequency ω as ω c-ω, where ωc is the light-line frequency–this difference is positive for a guided mode.

4.3. Strategy (iii)

In order to obtain a cutoff-free SPSM region without using anisotropic materials, perhaps the simplest strategy is to use a periodic cladding with an asymmetrical unit cell so that the fundamental cladding mode is nondegenerate. For example, one could use a triangular or square lattice of elliptical holes [23

23. D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. 25, 2700–2705 (2007). [CrossRef]

, 30

30. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” IEEE J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

32

32. S. Kim, U. C. Paek, and K. Oh, “New defect design in index guiding holey fiber for uniform birefringence and negative flat dispersion over a wide spectral range,” Opt. Express 13, 6039–6050 (2005). [CrossRef] [PubMed]

], or a rectangular lattice of circular holes [19

19. F. Zhang, M. Zhang, X. Liu, and P. Ye, “Design of wideband single-polarization single-mode photonic crystal fiber,” IEEE J. Lightwave Technol. 25, 1184–1189 (2007). [CrossRef]

], or many other possibilities. Then, filling in a hole (or some similar Δε≥0 core), as in the structure of Ref. 30, guarantees at least one cutoff-free guided mode by our theorem, but there is no such expectation of a second cutoff-free mode.

In fact, we conjecture that the second guided mode will always have a long-wavelength cutoff in the case where the fundamental cladding mode is nondegenerate (that is, that this is a sufficient condition for SPSM guidance). This prediction is borne out by numerical calculations for a variety of structures, such as the triangular lattice of elliptical holes or rectangular lattice of cylindrical holes, in both cases with a missing-hole core, shown in Fig. 6. Ref. 30 pointed out the existence of an SPSM region for a triangular lattice of elliptical holes, but did not note the lack of a cutoff (which is difficult to establish numerically [3

3. K. K. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express 16, 9261–9275 (08). [CrossRef]

], 52

52. S. Wilcox, L. Botten, C. M. 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 (2005). [CrossRef] [PubMed]

]). Our theorem establishes the lack of a cutoff for the fundamental mode, and provides a necessary condition for the second mode to have a cutoff (i.e., an SPSM region), but not a sufficient condition for SPSM. An intuitive argument for why the second mode should have a cutoff is that, in the long-wavelength regime, the guided modes asymptotically approach a corresponding extended mode of the cladding, but the second guided mode in this case approaches a cladding mode that is above the nondegenerate cladding light line (and hence is not guided below some cutoff where it crosses the light line).

Fig. 7. |D (1,2) c|2 field patterns of the two degenerate fundamental space-filling modes of the cladding of the structure in Fig. 8. We can utilize the asymmetry of the degenerate cladding mode and design an asymmetric core such that Eq. (2) is satisfied for one cladding mode but not the other.

Several authors have suggested SPSM waveguides based on a combination of nondegenerate claddings (e.g. elliptical holes) and cores with both positive and negative Δε to cut off both polarizations [19

19. F. Zhang, M. Zhang, X. Liu, and P. Ye, “Design of wideband single-polarization single-mode photonic crystal fiber,” IEEE J. Lightwave Technol. 25, 1184–1189 (2007). [CrossRef]

, 23

23. D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. 25, 2700–2705 (2007). [CrossRef]

], in some sense combining strategies (i) and (iii). Not apparent in that work is the fact that the negative Δε regions (enlarged holes) were superfluous, and an asymmetrical cladding alone is sufficient to attain SPSM. Other authors have used elliptical holes or rectangular lattices to achieve birefringence rather than SPSM [57

57. D. Mogilevtsev, J. Broeng, S. Barkou, and A. Bjarklev, “Design of polarization-preserving photonic crystal fibres with elliptical pores,” J. Opt. A: Pure Appl. Opt. 3, 141–143 (2001). [CrossRef]

, 58

58. L. Wang and D. Yang, “Highly birefringent elliptical-hole rectangular lattice photonic crystal fibers with modified air holes near the core,” Opt. Express 15, 8892–8897 (2007). [CrossRef] [PubMed]

].

4.4. Strategy (iv)

Fig. 8. Dispersion relation of a structure with an asymmetrical core in a symmetrical cladding of circular air holes (radius 0.47a in a hexagonal lattice with n = 1.87). The core is formed by two small cylinders of Δ=±0.18, respectively, shown in the inset as light and dark circles in the veins between two pairs of air holes. Here, we plot the mode frequency ω as ωc-ω, where ωc is the light-line frequency–this difference is positive for a guided mode.

5. Concluding remarks

Some questions remain as to what the necessary and sufficient conditions for SPSM fiber designs are. For example, does a nondegenerate light line guarantee that the second mode necessarily have a cutoff? We conjecture that this is the case, but have only an intuitive argument: since the fundamental cladding modes are nondegenerate, and the guided modes approach the cladding modes as β→0, we expect the second guided mode to asymptotically approach the second cladding mode for small β, and hence it should intersect the light line as a nonzero frequency.

Acknowledgment

This research was supported in part by the U. S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract W911NF-07-D-0004.

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2.

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K. K. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express 16, 9261–9275 (08). [CrossRef]

4.

A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990). [CrossRef]

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6.

T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80). [CrossRef]

7.

T. Okoshi, K. Oyamada, M. Nishimura, and H. Yakato, “Side tunnel fibre: An approach to polarization-maintaining optical waveguiding schemes,” Electron. Lett. 18, 824–826 (1982). [CrossRef]

8.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983). [CrossRef]

9.

K. Tajima, M. Ohashi, and Y. Sasaki, “A new single-polarization optical fiber,” IEEE J. Lightwave Technol. 7, 1499–1503 (1989). [CrossRef]

10.

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991). [CrossRef]

11.

A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystal fibers,” Appl. Phys. Lett. 78, 3184–3186 (2001). [CrossRef]

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K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Tech. Lett. 15, 1384–1386 (2003). [CrossRef]

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J. R. Folkenberg, M. D. Nielsen, and C. Jakobsen, “Broadband single-polarization photonic crystal fiber,” Opt. Lett. 30, 1446–1448 (2005). [CrossRef] [PubMed]

15.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005). [CrossRef]

16.

M.-J. Li, D. A. Nolan, G. E. Berkey, X. Chen, J. Koh, D. T. Walton, J. Wang, W. A. Wood, and L. A. Zenteno, “High-performance single-polarization optical fibers,” Proc. SPIE 5623, 612–621 (2005). [CrossRef]

17.

X. Liu, F. Zhang, M. Zhang, and P. Ye, “A novel single-mode single-polarization photonic crystal fiber using resonant absorption effect,” Proc. SPIE 6351, 63,511K (2006).

18.

J. Ju, W. Jin, and M. S. Demokan, “Design of single-polarization single-mode photonic crystal fiber at 1.30 and 1.55 µm,” IEEE J. Lightwave Technol. 24, 825–830 (2006). [CrossRef]

19.

F. Zhang, M. Zhang, X. Liu, and P. Ye, “Design of wideband single-polarization single-mode photonic crystal fiber,” IEEE J. Lightwave Technol. 25, 1184–1189 (2007). [CrossRef]

20.

M. Szpulak, T. Martynkien, J. Olszewski, W. Urbanóczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, “Single-polarization single-mode photonic band gap fiber,” Acta Physica Polonica A 111, 239–245 (2007).

21.

D. C. Zografopoulos, E. E. Kriezis, and T. D. Tsiboukis, “Photonic crystal-liquid crystal fibers for single-polarization or high-birefringent guidance,” Opt. Express 14, 914–925 (2006). [CrossRef] [PubMed]

22.

M. Eguchi and Y. Tsuji, “Single-mode single-polarization holey fiber using anisotropic fundamental space-filling mode,” Opt. Lett. 32, 2112–2114 (2007). [CrossRef] [PubMed]

23.

D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. 25, 2700–2705 (2007). [CrossRef]

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30.

M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” IEEE J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

31.

Y. C. Liu and Y. Lai, “Optical birefringence and polarization dependent loss of square- and rectangular-lattice holey fibers with elliptical air holes:numerical analysis,” Opt. Express 13, 225–235 (2005). [CrossRef] [PubMed]

32.

S. Kim, U. C. Paek, and K. Oh, “New defect design in index guiding holey fiber for uniform birefringence and negative flat dispersion over a wide spectral range,” Opt. Express 13, 6039–6050 (2005). [CrossRef] [PubMed]

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43.

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52.

S. Wilcox, L. Botten, C. M. 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 (2005). [CrossRef] [PubMed]

53.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1982).

54.

C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006). [CrossRef]

55.

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

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57.

D. Mogilevtsev, J. Broeng, S. Barkou, and A. Bjarklev, “Design of polarization-preserving photonic crystal fibres with elliptical pores,” J. Opt. A: Pure Appl. Opt. 3, 141–143 (2001). [CrossRef]

58.

L. Wang and D. Yang, “Highly birefringent elliptical-hole rectangular lattice photonic crystal fibers with modified air holes near the core,” Opt. Express 15, 8892–8897 (2007). [CrossRef] [PubMed]

59.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005). [CrossRef]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(130.2790) Integrated optics : Guided waves
(060.4005) Fiber optics and optical communications : Microstructured fibers
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 22, 2008
Revised Manuscript: September 1, 2008
Manuscript Accepted: September 1, 2008
Published: September 11, 2008

Citation
Karen K. Y. Lee, Yehuda Avniel, and Steven G. Johnson, "Design strategies and rigorous conditions for single-polarization single-mode waveguides," Opt. Express 16, 15170-15184 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15170


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References

  1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).
  2. T. F. Morse and A. Mendez, Specialty Optical Fibers Handbook (Academic Press, 2007).
  3. K. K. Lee, Y. Avniel, and S. G. Johnson, "Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides," Opt. Express 16, 9261-9275 (08). [CrossRef]
  4. A. Bamberger and A. S. Bonnet, "Mathematical analysis of the guided modes of an optical fiber," SIAM J. Math. Anal. 21, 1487-1510 (1990). [CrossRef]
  5. H. P. Urbach, "Analysis of the domain integral operator for anisotropic dielectric waveguides," J. Math. Anal. 27,204-220 (1996).
  6. T. Okoshi and K. Oyamoda, "Single-polarization single-mode optical fibre with refractive-index pits on both sides of core," Electron. Lett.  16,712-713 (1980). [CrossRef]
  7. T. Okoshi, K. Oyamada, M. Nishimura, and H. Yakato, "Side tunnel fibre: An approach to polarizationmaintaining optical waveguiding schemes," Electron. Lett. 18, 824-826 (1982). [CrossRef]
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