## Design strategies and rigorous conditions for single-polarization single-mode waveguides

Optics Express, Vol. 16, Issue 19, pp. 15170-15184 (2008)

http://dx.doi.org/10.1364/OE.16.015170

Acrobat PDF (451 KB)

### 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

*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

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]

*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

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

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]

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

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

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. 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]

_{01}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]

_{01}modes [35]). (The scalar approximation applies in the high-frequency limit to holey fibers as well [36–38

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]

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]

## 2. Sufficient conditions for two-polarization waveguides

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]

### 2.1. Index-guided modes

*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

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]

*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

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

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]

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

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]

*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:

**H**(

*x*,

*y*,

*z*)

*e*

^{iβz}^{-}

*(via Bloch’s theorem) [36], where*

^{iωt}*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

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

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]

*β*there is a fundamental (minimum-

*ω*) space-filling mode at a frequency

*ω*(

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

**H**

*[36, 41*

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

*Foundations of Photonic Crystal Fibres* (Imperial College Press, London, 2005). [CrossRef]

*ω*(

_{c}*β*), the only solutions in the cladding are evanescent modes that decay exponentially in the transverse directions [36, 41

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

*Foundations of Photonic Crystal Fibres* (Imperial College Press, London, 2005). [CrossRef]

*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].

*ω*<

*ω*, the existence of a guided mode can be shown by demonstrating that

_{c}*ω*has an upper bound <

*ω*. Using the variational (min–max) theorem for Hermitian eigenproblems [36], we derived [3

_{c}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]

*β*: a guided mode

*must*exist whenever

*xy*and one period in

*z*, and

*ω*→0,

*β*→0) limit of

**D**

*.*

_{c}*ω*

_{min}(

*β*) is given by the following Rayleigh quotient for any divergence-free trial function

**H**[i.e., (∇+

*iβ*

**ẑ**)·

**H**= 0]:

### 2.2. Two-polarization waveguides

*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}*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*, the variational theorem can be generalized as follows [4

_{n}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]

*𝓗*is any

_{n}*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}*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**

*(and*

_{c}**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.

**H**

^{(1,2)}

*, it is*

_{c}*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}*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).

**D**

*| everywhere–this is true for a homogeneous cladding (where |*

_{c}**D**

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

_{c}*β*limit where a scalar approximation becomes valid [36–38

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]

*∫*

**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.)

**D**

*| 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*

_{c}*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

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]

*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.

*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).

*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

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]

*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

*both*polarizations have cutoffs for index-guiding, where the cutoffs are different because of some asymmetry and hence there is a single-polarization region.

**D**

*| is different for the two polarizations (due to cladding inhomogeneity) so that Eq. (2) can be satisfied for one polarization but not the other.*

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

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]

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

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

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. 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]

**16**, 9261–9275 (08). [CrossRef]

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]

### 4.1. Strategy (i)

*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

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

*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

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]

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. 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. 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]

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]

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

*ε*> 0) and enlarged holes (Δ

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

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

*Foundations of Photonic Crystal Fibres* (Imperial College Press, London, 2005). [CrossRef]

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]

*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

56. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric Coaxial Waveguide,” Science **289**, 415–419 (2000). [CrossRef] [PubMed]

### 4.2. Strategy (ii)

*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)}

*but not for the orthogonal fundamental mode. Experimentally, this has been achieved/proposed using stress birefringence [25*

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

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

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]

### 4.3. Strategy (iii)

**25**, 2700–2705 (2007). [CrossRef]

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. 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]

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]

*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.

*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

**16**, 9261–9275 (08). [CrossRef]

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]

### 4.4. Strategy (iv)

**D**

^{(1,2)}

*| 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*

_{c}*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}**D**

^{(1,2)}

*, and it is only necessary to find one pair that has an asymmetrical |*

_{c}**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

**16**, 9261–9275 (08). [CrossRef]

*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

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]

*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

*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.

*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

## References and links

1. | R. Ramaswami and K. N. Sivarajan, |

2. | T. F. Morse and A. Méndez, |

3. | K. K. Lee, Y. Avniel, and S. G. Johnson, “Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides,” Opt. Express |

4. | A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. |

5. | H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis |

6. | T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. |

7. | T. Okoshi, K. Oyamada, M. Nishimura, and H. Yakato, “Side tunnel fibre: An approach to polarization-maintaining optical waveguiding schemes,” Electron. Lett. |

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. |

9. | K. Tajima, M. Ohashi, and Y. Sasaki, “A new single-polarization optical fiber,” IEEE J. Lightwave Technol. |

10. | M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. |

11. | A. Ferrando and J. J. Miret, “Single-polarization single-mode intraband guidance in supersquare photonic crystal fibers,” Appl. Phys. Lett. |

12. | K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Tech. Lett. |

13. | H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. |

14. | J. R. Folkenberg, M. D. Nielsen, and C. Jakobsen, “Broadband single-polarization photonic crystal fiber,” Opt. Lett. |

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. |

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 |

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 |

18. | J. Ju, W. Jin, and M. S. Demokan, “Design of single-polarization single-mode photonic crystal fiber at 1.30 and 1.55 |

19. | F. Zhang, M. Zhang, X. Liu, and P. Ye, “Design of wideband single-polarization single-mode photonic crystal fiber,” IEEE J. Lightwave Technol. |

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 |

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 |

22. | M. Eguchi and Y. Tsuji, “Single-mode single-polarization holey fiber using anisotropic fundamental space-filling mode,” Opt. Lett. |

23. | D. Chen and L. Shen, “Highly birefringent elliptical-hole photonic crystal fibers with double defect,” IEEE J. Lightwave Technol. |

24. | R. A. Waldron, |

25. | W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. |

26. | A. W. Snyder and F. Ruöhl, “Single-mode, single-polarization fibers made of birefringent material,” J. Opt. Soc. Am. |

27. | Y. Chen, “Tapered polarizing anisotropic fibers,” Opt. Lett. |

28. | F. F. Ruöhl and D. Wong, “True single-polarization design for bow-tie optical fibers,” Opt. Lett. |

29. | K. S. Chiang, “Stress-induced birefringence fibers designed for single-polarization single-mode operation,” IEEE J. Lightwave Technol. |

30. | M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” IEEE J. Lightwave Technol. |

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 |

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 |

33. | E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. |

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 |

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

36. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

37. | A.-S. Bonnet-Bendhia and R. Djellouli, “High-frequency asymptotics of guided modes in optical fibres,” IMA J. Applied Math. |

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

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. |

40. | P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in |

41. | P. Russell, “Photonic crystal fibers,” Science |

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

43. | F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, |

44. | J. A. Kong, |

45. | C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE |

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 |

47. | P. Kuchment, “The Mathematics of Photonic Crystals,” in |

48. | K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. |

49. | T. Inui, Y. Tanabe, and Y. Onodera, |

50. | M. Tinkham, |

51. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E |

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 |

53. | R. G. Hunsperger, |

54. | C.-L. Chen, |

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

56. | M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An All-Dielectric Coaxial Waveguide,” Science |

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. |

58. | L. Wang and D. Yang, “Highly birefringent elliptical-hole rectangular lattice photonic crystal fibers with modified air holes near the core,” Opt. Express |

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 |

**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

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- H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, "Absolutely single polarization photonic crystal fiber," IEEE Photon. Tech. Lett. 16, 182-184 (2004). [CrossRef]
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- 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]
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- F. Zhang, M. Zhang, X. Liu, and P. Ye, "Design of wideband single-polarization single-mode photonic crystal fiber," J. Lightwave Technol. 25, 1184-1189 (2007). [CrossRef]
- M. Szpulak, T. Martynkien, J. Olszewski, W. Urbanóczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, "Single-polarization single-mode photonic band gap fiber," Acta Phys. Pol. A 111, 239-245 (2007).
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- 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]
- D. Chen and L. Shen, "Highly birefringent elliptical-hole photonic crystal fibers with double defect," J. Lightwave Technol. 25, 2700-2705 (2007). [CrossRef]
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- Y. Chen, "Tapered polarizing anisotropic fibers," Opt. Lett. 13, 598-600 (1988). [CrossRef]
- F. F. Ruöhl and D. Wong, "True single-polarization design for bow-tie optical fibers," Opt. Lett. 14, 648-650 (1989). [CrossRef]
- K. S. Chiang, "Stress-induced birefringence fibers designed for single-polarization single-mode operation," J. Lightwave Technol. 7, 436-441 (1989). [CrossRef]
- M. J. Steel and R. M. Osgood, "Polarization and dispersive properties of elliptical-hole photonic crystal fibers," J. Lightwave Technol. 19, 495-503 (2001). [CrossRef]
- 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]
- 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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- 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]
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