## 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

A *single-polarization single-mode* (SPSM) waveguide is a waveguide that is *truly single-mode* in the sense of supporting only a single guided-mode solution (rather than two or more, commonly corresponding to two polarizations as in standard “single-mode” fibers [1]); such waveguides are important, for example, as polarization-maintaining fibers (PMFs) [2]. (In contrast to a merely birefringent fiber, where two polarizations are guided but travel at different speeds [1], an SPSM fiber completely removes the possibility of coupling one polarization into the other.) In this paper, we derive rigorous necessary conditions to obtain SPSM waveguides, and identify different categories of such designs, especially focusing on those that yield cutoff-free single-polarization regions with isotropic materials (compared to most previous designs that either employ birefringent materials or have long-wavelength cutoffs in both polarizations). The latter categories require an inhomogeneous fiber cladding, such as in a photoniccrystal fiber (a periodic cladding), as opposed to traditional dielectric waveguides surrounded by asymptotically homogeneous cladding materials. More specifically, generalizing a previous paper that derived sufficient conditions for index-guided modes in microstructured dielectric waveguides [3] as well as an earlier theorem for homogeneous-cladding waveguides [4, 5], we derive sufficient conditions for a dielectric waveguide to support *two* index-guided modes when the cladding has a doubly degenerate light-line mode (usually as a consequence of symmetry). This is an entirely analytical result that provides rigorous guarantees for a wide range of microstructured waveguide geometries. In consequence, we are able to categorize single-polarization waveguides into four categories: (i) those that violate the conditions of our theorem entirely, typically resulting in a cutoff for both polarizations; (ii) those that employ anisotropic materials to guide one polarization and not the other; (iii) those using an asymmetrical cladding structure (e.g. an asymmetrical photonic crystal) that does not have a doubly degenerate light-line mode; and (iv) those with a symmetrical periodic cladding that exploit the asymmetry of the light-line’s two polarizations to guide one polarization but not the other. Most previous single-polarization designs fall into category (i) [6–23] (also including non-index-guiding structures such as coaxial metallic guides [24]) or category (ii) [21,25–29], and we will give examples of designs in categories (iii) [30–32] and (iv).

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]. The two modes then travel at slightly different velocities, causing pulse broadening via polarization-mode dispersion (PMD) [1]. Polarization-maintaining (PM) fibers [2] 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] and in photonic-bandgap fibers, both of which can operate in a non-degenerate lowest-loss mode (e.g. the TE_{01} mode) [34]–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] applies and the two lowest-frequency solutions are purely polarized in two orthogonal directions [35, 36] (known as the “linearly polarized” LP_{01} modes [35]). (The scalar approximation applies in the high-frequency limit to holey fibers as well [36–38].) More generally, the two lowest-frequency modes are two linearly independent guided-mode solutions that satisfy the usual field-orthogonality relationships [35, 36], 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]. 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]. 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.

#### 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 [36,40–43] or metallic waveguides [44].) 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*)=*ε _{c}*(

*x*,

*y*,

*z*)+Δ

*ε*(

*x*,

*y*,

*z*) such that:

*ε*,

*ε*, and Δ

_{c}*ε*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

*ε*is periodic in

_{c}*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 Δ

*ε*in some region of the

*xy*plane, sufficiently localized that

*∫*|1/

*ε*-1/

*ε*|<∞(integrated over the

_{c}*xy*plane and the unit cell in

*z*). This includes a very wide variety of dielectric waveguides, from conventional fibers [35] [Fig. 1(a)] to photonic-crystal “holey” fibers [36, 41–43] [Fig. 1(b)] to waveguides with a periodic “grating” along the propagation direction [Fig. 1(c)] such as fiber-Bragg gratings [1] and other periodic waveguides [36, 45, 46]. 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:

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], where

*ω*is the frequency,

*β*is the propagation constant, and the magnetic-field envelope

**H**(

*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 Δ

*ε*= 0), the (non-localized) modes propagating in the infinite cladding form the “light cone” of the structure [36, 41–43]; and at each real

*β*there is a fundamental (minimum-

*ω*) space-filling mode at a frequency

*ω*(

_{c}*β*) with a corresponding field envelope

**H**

*[36, 41–43]. Such a light cone is shown as a shaded triangular region in Fig. 2. Below the “light line”*

_{c}*ω*(

_{c}*β*), the only solutions in the cladding are evanescent modes that decay exponentially in the transverse directions [36, 41–43, 47]. Therefore, once the core is introduced (Δ

*ε*≠0), any new solutions with

*ω*<

*ω*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

_{c}*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].

Since a mode is guided if *ω* < *ω _{c}*, the existence of a guided mode can be shown by demonstrating that

*ω*has an upper bound <

*ω*. Using the variational (min–max) theorem for Hermitian eigenproblems [36], we derived [3] the following sufficient condition for the existence of an index-guided mode in a dielectric waveguide at a given

_{c}*β*: a guided mode

*must*exist whenever

where the integral is over *xy* and one period in *z*, and ${\mathbf{D}}_{c}=\frac{i}{\omega}\left(\nabla +i\beta \hat{\mathbf{z}}\right)\times {\mathbf{H}}_{c}$ 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 eigenfrequency*ω*
_{min}(*β*) is given by the following Rayleigh quotient for any divergence-free trial function **H** [i.e., (∇+*iβ*
**ẑ**)·**H** = 0]:

where the integral is over the *xy* cross-section and the unit cell in *z*. We were then able to construct a trial function **H**, via a linear operation on the (unknown) cladding fundamental mode, such that this upper bound *Q*(**H**) is provably less than *ω*
^{2}
* _{c}* when Eq. (2) is satisfied, and hence the lowest eigenvalue corresponds to a guided mode beneath the light cone. This construction, inspired by a related proof of localization from quantum mechanics [48], requires a somewhat delicate limit analysis [3] that we will not reproduce here.

#### 2.2. Two-polarization waveguides

In many important cases, the *cladding* fundamental mode *ω _{c}*(

*β*) is doubly degenerate (two linearly independent “polarizations” with the same frequency

*ω*)–this is independent of whether the

_{c}*guided*mode is doubly degenerate, which depends on the symmetry of the core as well as of the cladding. When

*ω*is doubly degenerate, one obtains an index-guided mode if Eq. (2) is true for any of the degenerate fundamental modes

_{c}**D**

*(because any one of these modes could have been used in the proof from Ref. 3). If Eq. (2) holds for*

_{c}*all*of the degenerate fundamental field patterns

**D**

*, then one is guaranteed to have at least*

_{c}*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

*ω*, but here we simply assume that to be the case.

_{c}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 *ω _{n}*, the variational theorem can be generalized as follows [4]:

where *𝓗 _{n}* 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

*𝓗*is the span of the

_{n}*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**

*) of the*

_{c}*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)}

*and*

_{c}**H**

^{(2)}

*, we construct the corresponding trial functions*

_{c}**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)}

*for any constants*

_{c}*c*

_{1}and

*c*

_{2}. Because

**H**

*is also a cladding fundamental mode, and satisfies Eq. (2) by assumption, then*

_{c}*Q*(

**H**)<

*ω*

^{2}

*by exactly the same proof as in Ref. 3. Hence*

_{c}*Q*<

*ω*

^{2}

*for every*

_{c}**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)}

*and some linear combination*

_{c}**D**

*=*

_{c}*c*

_{1}

**D**

^{(1)}

*+*

_{c}*c*

_{2}

**D**

^{(2)}

*, when substituted into Eq. (2) there are cross terms 2ℜ[*

_{c}*∫*(

**D**

^{(1)}

*)*·Δ*

_{c}**D**

^{(2)}

*] 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*

_{c}*∫*(

**D**

^{(1)}

*)*·Δ*

_{c}**D**

^{(2)}

*= 0, then it holds for all linear combinations and the existence of at least two index-guided modes follows.*

_{c}The easiest case, of course, is the one in which Δ is nonpositive-definite (e.g. if Δ*ε*≥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**

*| is a constant), and also for an arbitrary cladding in the large-*

_{c}*β*limit where a scalar approximation becomes valid [36–38]. Then, if the materials are isotropic, so that

*∫*

**D**

^{*}

*·Δ*

_{c}**D**

*=*

_{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 Δ

*ε*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**

^{*}

*·Δ*

_{c}**D**

*is invariant under symmetry operations/rotations of*

_{c}**D**

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

_{c}## 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, 39, 49]. (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].) 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], 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.

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] 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]. 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], 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.

## 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 [6–23, 25–32]. 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.

Almost all previous single-polarization waveguide designs seem to fall into strategies (i) [6–23] and (ii) [21, 25–29], whereas strategies (iii) and (iv) are the only ones that have no long-wavelength cutoff for single-polarization guidance using isotropic materials. Strategy (iii) was previously discussed in work by Steel and Osgood [30–32], although they did not remark upon the the cutoff-free nature of the SPSM region (which is difficult to establish numerically in the absence of an analytical theorem like the one here [3, 52]). We discuss each of these strategies, in turn, below, and give examples embodying strategies (iii) and (iv).

#### 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, 54]. With a periodic (or homogeneous) cladding, our theorem implies that a waveguide in which all modes are cut off *must* have a Δ*ε* that is negative in some regions (since Δ*ε*≥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] and to an SPSM bandwidth if some asymmetry is introduced [8, 10, 15]. 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, 7, 9, 16]. More recently, photonic-crystal holey fibers have used combinations of removed or shrunk holes (Δ*ε* > 0) and enlarged holes (Δ*ε* < 0) to cut off both guided modes [11–14, 17–20, 22, 23]. Fibers that guide light by a photonic bandgap [36, 41–43], 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] 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.)

However, there is one notable example of a waveguide that violates the conditions of our theorem and yet has a cutoff-free SPSM region: a coaxial metallic waveguide with its non-degenerate cutoff-free TEM mode [24]. This case lies outside the conditions of the theorem because it requires metallic (*ε* < 0) materials and does not operate by index-guiding. Related single-polarization modes can also be confined by photonic bandgaps in symmetrical coaxial structures [56], but these modes have a long-wavelength cutoff (as for all bandgap guidance, as explained above).

#### 4.2. Strategy (ii)

Another strategy for SPSM fibers is to use an anisotropic *ε*. The most obvious technique would be to use an Δ*ε* 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 [25–29] or by liquid crystal filling the core of a holey fiber [21]. Note that by varying various parameters, the design in Ref. 21 can have a cutoff for both guided modes (strategy (i))

#### 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, 30–32], or a rectangular lattice of circular holes [19], or many other possibilities. Then, filling in a hole (or some similar Δ*ε*≥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], 52]). 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).

Several authors have suggested SPSM waveguides based on a combination of nondegenerate claddings (e.g. elliptical holes) and cores with both positive and negative Δ*ε* to cut off both polarizations [19, 23], in some sense combining strategies (i) and (iii). Not apparent in that work is the fact that the negative Δ*ε* 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, 58].

#### 4.4. Strategy (iv)

A fourth strategy is suggested by the fact that the cladding fundamental-mode field pattern |**D**
^{(1,2)}
* _{c}*| is neither uniform in space nor symmetrical even for a symmetrical cladding with a doubly degenerate light line, as long as the cladding is inhomogeneous. This is shown in Fig. 7, for the doubly degenerate fundamental modes of a triangular lattice of air holes in silica. Because of this, it is possible to arrange a core composed of positive and negative Δ

*ε*so that Eq. (2) is true for one fundamental cladding mode but not the other. If we do this in the longwavelength limit, then we will again obtain a cutoff-free SPSM region. An example of this is shown in the inset of Fig. 8: we form a core by a small cylinder of Δ=-0.18 (Δ

*ε*> 0) in a region where |

**D**

^{(1)}

*| is peaked (and where |*

_{c}**D**

^{(2)}

*| is small) and a small cylinder of Δ=+0.18 (Δ*

_{c}*ε*< 0) in a region where |

**D**

^{(2)}

*| is peaked (and where |*

_{c}**D**

^{(1)}

*| is small). This yields a cutoff-free SPSM region shown in Fig. 8. This strategy is fundamentally different from the previous three in the sense that it is cutoff-free unlike (i), uses isotropic materials unlike (ii), and uses a symmetrical cladding with a degenerate fundamental mode unlike (iii).*

_{c}There are several subtleties to this approach. First, there are many choices of linear combinations of the degenerate cladding modes **D**
^{(1,2)}
* _{c}*, and it is only necessary to find one pair that has an asymmetrical |

**D**

^{(1,2)}

*|. In the case of Fig. 7, the field patterns were chosen corresponding to two “orthogonal” polarizations, or more technically to be even and odd with respect to orthogonal mirror planes [36]; the fact that the same modes could be combined into “circular” polarizations*

_{c}**D**

^{(1)}

*±*

_{c}*i*

**D**

^{(2)}

*with identical |*

_{c}**D**

^{(±)}

*| patterns is irrelevant because the theorem for two guided modes required Eq. (2) to be satisfied for*

_{c}*all*linear combinations. Another difficulty arises because the condition for two guided modes is only a sufficient condition, not a necessary one–as remarked upon in our previous work [3], it becomes overly strong when large Δ

*ε*regions are present, in which case we suspect that a weaker (necessary) condition would involve the polarizability of the defects as in our previous work on perturbation theory [59]. Because of this, when Δ

*ε*becomes large (e.g. if the defects are created by enlarging and/or shrinking air holes), violating Eq. (2) for one of the fundamental modes is not enough to predict whether the second guided mode has a cutoff, and numerical calculations are required.

## 5. Concluding remarks

We have demonstrated the necessary conditions to obtain single-polarization single-mode (SPSM) waveguides, by generalizing previous results to show the sufficient conditions for the existence of *two* index-guided modes for general microstructured dielectric fibers. According to these conditions, we have categorized single-polarization waveguides into four groups: (i) those that violate the conditions of our theorem entirely, typically resulting in a cutoff for both polarizations; (ii) those that employ anisotropic materials to guide one polarization and not the other; (iii) those using an asymmetrical cladding structure (e.g. an asymmetrical photonic crystal) that does not have a doubly degenerate light-line mode; and (iv) those with a symmetrical periodic cladding that exploit the asymmetry of the light-line’s two polarizations to guide one polarization but not the other. We have shown that the latter two categories can guarantee cutoff-free single-polarization regions with isotropic materials without relying on birefringent materials. They require an inhomogeneous fiber cladding, such as in a photonic-crystal fiber (a periodic cladding), as opposed to traditional dielectric waveguides surrounded by asymptotically homogeneous cladding materials.

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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