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

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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).
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    [CrossRef]

2007 (5)

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]

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).

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

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]

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]

2006 (3)

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]

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).

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]

2005 (8)

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]

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]

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]

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]

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]

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]

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

2004 (1)

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]

2003 (2)

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

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

2001 (5)

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]

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]

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]

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]

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]

2000 (1)

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]

1997 (1)

1996 (1)

H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis 27 (1996).

1995 (1)

1994 (1)

A.-S. Bonnet-Bendhia and R. Djellouli, “High-frequency asymptotics of guided modes in optical fibres,” IMA J. Applied Math. 52, 271–287 (1994).
[CrossRef]

1991 (1)

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]

1990 (1)

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]

1989 (4)

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

K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
[CrossRef]

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

F. F. Ruöhl and D. Wong, “True single-polarization design for bow-tie optical fibers,” Opt. Lett. 14, 648–650 (1989).
[CrossRef]

1988 (1)

Y. Chen, “Tapered polarizing anisotropic fibers,” Opt. Lett. 13, 698–600 (1988).
[CrossRef]

1983 (2)

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]

A. W. Snyder and F. Ruöhl, “Single-mode, single-polarization fibers made of birefringent material,” J. Opt. Soc. Am. 73, 1165–1174 (1983).
[CrossRef]

1982 (2)

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

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]

1976 (1)

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

1974 (1)

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]

1964 (1)

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).

Avniel, Y.

Bamberger, A.

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]

Barkou, S.

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]

Berghmans, F.

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).

Berkey, G. E.

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]

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]

Birks, T. A.

Bjarklev, A.

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]

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[CrossRef]

Bjarklev, A. S.

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[CrossRef]

Bonnet, A. S.

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]

Bonnet-Bendhia, A.-S.

A.-S. Bonnet-Bendhia and R. Djellouli, “High-frequency asymptotics of guided modes in optical fibres,” IMA J. Applied Math. 52, 271–287 (1994).
[CrossRef]

Botten, L.

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]

Botten, L. C.

Broeng, J.

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]

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[CrossRef]

Chen, C.-L.

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

Chen, D.

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

Chen, J. C.

Chen, X.

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]

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]

Chen, Y.

Y. Chen, “Tapered polarizing anisotropic fibers,” Opt. Lett. 13, 698–600 (1988).
[CrossRef]

Chiang, K. S.

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

de Llano, M.

K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
[CrossRef]

de Sterke, C. M.

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]

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]

Demokan, M. S.

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]

Devenyi, A.

Djellouli, R.

A.-S. Bonnet-Bendhia and R. Djellouli, “High-frequency asymptotics of guided modes in optical fibres,” IMA J. Applied Math. 52, 271–287 (1994).
[CrossRef]

Eguchi, M.

Eickhoff, W.

Elachi, C.

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

Engeness, T. D.

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).
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Figures (8)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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.
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.

Fig. 5.
Fig. 5.

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

Fig. 6.
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.

Fig. 7.
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.

Fig. 8.
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.

Equations (5)

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Δ ( x , y , z ) ε 1 ε c 1 .
D c * . Δ ( x , y , z ) D c < 0 ,
ω min 2 ( β ) c 2 [ ( + i β z ̂ ) × H ] * · 1 ε [ ( + z ̂ ) × H ] H * · H Q ( H ) ,
ω n 2 sup Q ( H ) ,
H n

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