Abstract

A novel type of polarization converters (PCs) based on highly birefringent (Hi-Bi) microfibers is presented. Analytical formulation based on the Jones Matrix method and a numerical code based on the Full Vectorial Finite Difference Beam Propagation Method are developed to analyze the polarization evolutions in such PCs. Two different design configurations, namely the “one-side” and “two-side” perturbation configurations, are studied by use of the two methods, and the results obtained agree well with each others. The PCs can be flexibly designed to have different operating wavelengths, spectral bandwidths, and devices lengths. A particular PC based on an elliptical microfiber demonstrates a bandwidth of ∼ 600 nm around 1550 nm with a device length of ∼ 150 μm.

© 2013 Optical Society of America

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2011 (1)

2010 (3)

2009 (3)

K. Bayat, S. K. Chaudhuri, S. Safavi-Naeini, “Ultra-compact photonic crystal based polarization rotator,” Opt. Express 17, 7145–7158 (2009).
[CrossRef] [PubMed]

H. F. Xuan, W. Jin, M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17, 21882–21890 (2009).
[CrossRef] [PubMed]

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

2008 (2)

2007 (1)

2005 (1)

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

G. Kakarantzas, A. Ortigosa-Blanch, T. a. Birks, P. S. J. Russell, L. Farr, F. Couny, B. J. Mangan, “Structural rocking filters in highly birefringent photonic crystal fiber,” Opt. Lett. 28, 158–160 (2003).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (3)

J. Broeng, S. E. Barkou, T. Søndergaard, A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96–98 (2000).
[CrossRef]

S. Obayya, B. Rahman, H. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photonics Technol. Lett. 12, 1346–1348 (2000).
[CrossRef]

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

1998 (2)

J. Shibayama, M. Sekiguchi, J. Yamauchi, H. Nakano, “Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method,” Electron. Comm. Jpn. 2 81, 1–9 (1998).

J. Yamauchi, G. Takahashi, H. Nakano, “Full-vectorial beam-propagation method based on the McKee-Mitchell scheme with improved finite-difference formulas,” J. Lightwave Technol. 16, 2458–2464 (1998).
[CrossRef]

1995 (1)

1993 (1)

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” J. Lightwave Technol. 29, 2639–2649 (1993).

1992 (1)

W. P. Huang, M. Z. Mao, “Polarization rotation in periodic loaded rib waveguides,” J. Lightwave Technol. 10, 1825–1831 (1992).
[CrossRef]

1991 (2)

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
[CrossRef] [PubMed]

1986 (1)

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).
[CrossRef]

1984 (2)

R. Bergh, H. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984).
[CrossRef]

R. H. Stolen, A. Ashkin, W. Pleibel, J. M. Dziedzic, “In-line fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
[CrossRef] [PubMed]

Alferness, R.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Anuszkiewicz, A.

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Ashkin, A.

Barkou, S. E.

Bassi, P.

Bauer, J.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Bayat, K.

Bellanca, G.

Bergh, R.

R. Bergh, H. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984).
[CrossRef]

Birks, T. a.

Bjarklev, A.

Brambilla, G.

Broeng, J.

Bruns, J.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Chaudhuri, S. K.

Chen, X.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

Coudray, P.

Couny, F.

De Micheli, M.

Dziedzic, J. M.

El-Mikati, H.

S. Obayya, B. Rahman, H. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photonics Technol. Lett. 12, 1346–1348 (2000).
[CrossRef]

Escoubas, L.

Fang, X.

Farr, L.

Finazzi, V.

Flory, F.

Fogli, F.

Gajda, A.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Gattass, R. R.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Giuntoni, I.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Hadley, G. R.

He, S.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” J. Lightwave Technol. 29, 2639–2649 (1993).

W. P. Huang, M. Z. Mao, “Polarization rotation in periodic loaded rib waveguides,” J. Lightwave Technol. 10, 1825–1831 (1992).
[CrossRef]

Jin, W.

Ju, J.

Kakarantzas, G.

Kaul, R.

Koch, T.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Koren, U.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Kou, J. L.

Lefevre, H.

R. Bergh, H. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984).
[CrossRef]

Liao, C. R.

Liu,

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

Liu, S.

Lou, J.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Lu, Y. Q.

Mangan, B. J.

Mangeat, T.

Mao, M. Z.

W. P. Huang, M. Z. Mao, “Polarization rotation in periodic loaded rib waveguides,” J. Lightwave Technol. 10, 1825–1831 (1992).
[CrossRef]

Marschmeyer, S.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Miller, B. I.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Nakano, H.

J. Yamauchi, G. Takahashi, H. Nakano, “Full-vectorial beam-propagation method based on the McKee-Mitchell scheme with improved finite-difference formulas,” J. Lightwave Technol. 16, 2458–2464 (1998).
[CrossRef]

J. Shibayama, M. Sekiguchi, J. Yamauchi, H. Nakano, “Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method,” Electron. Comm. Jpn. 2 81, 1–9 (1998).

Noda, J.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).
[CrossRef]

Obayya, S.

S. Obayya, B. Rahman, H. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photonics Technol. Lett. 12, 1346–1348 (2000).
[CrossRef]

Okamoto, K.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).
[CrossRef]

Oron, M.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Ortigosa-Blanch, A.

Petermann, K.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Pleibel, W.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Qiu, S. J.

Rahman, B.

S. Obayya, B. Rahman, H. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photonics Technol. Lett. 12, 1346–1348 (2000).
[CrossRef]

Richardson, D.

Richter, H.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Roussel, L.

Russell, P. S. J.

Saccomandi, L.

Safavi-Naeini, S.

Sasaki, Y.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Sekiguchi, M.

J. Shibayama, M. Sekiguchi, J. Yamauchi, H. Nakano, “Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method,” Electron. Comm. Jpn. 2 81, 1–9 (1998).

Shani, Y.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Shaw, H. J.

R. Bergh, H. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984).
[CrossRef]

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Shibayama, J.

J. Shibayama, M. Sekiguchi, J. Yamauchi, H. Nakano, “Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method,” Electron. Comm. Jpn. 2 81, 1–9 (1998).

Søndergaard, T.

Statkiewicz-Barabach, G.

Stolarek, D.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Stolen, R. H.

Sumetsky, M.

Takahashi, G.

Tillack, B.

I. Giuntoni, D. Stolarek, H. Richter, S. Marschmeyer, J. Bauer, A. Gajda, J. Bruns, B. Tillack, K. Petermann, L. Zimmermann, “Deep-UV technology for the fabrication of Bragg gratings on SOI rib waveguides,” IEEE Photonics Technol. Lett. 21, 1894–1896 (2009).
[CrossRef]

Tong, L.

L. Tong, J. Lou, R. R. Gattass, S. He, X. Chen, Liu, E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonic devices,” Nano Lett. 5, 259–262 (2005).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Trillo, S.

Urbanczyk, W.

Wang, D. N.

Wojcik, J.

Xu, C. L.

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” J. Lightwave Technol. 29, 2639–2649 (1993).

Xu, F.

Xuan, H. F.

Yamauchi, J.

J. Yamauchi, G. Takahashi, H. Nakano, “Full-vectorial beam-propagation method based on the McKee-Mitchell scheme with improved finite-difference formulas,” J. Lightwave Technol. 16, 2458–2464 (1998).
[CrossRef]

J. Shibayama, M. Sekiguchi, J. Yamauchi, H. Nakano, “Eigenmode analysis of optical waveguides by an improved finite-difference imaginary-distance beam propagation method,” Electron. Comm. Jpn. 2 81, 1–9 (1998).

Young, M. G.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Zhang, M.

Zimmermann, L.

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Figures (12)

Fig. 1
Fig. 1

Modal properties of an elliptical Hi-Bi microfiber calculated by use of the FV-FDBPM code and the FEM software. (a) Dispersion curves of the four lowest-order modes for b/a = 0.5; (b) The three-region model of the air-clad Hi-Bi microfiber. This model has been used to model a practical Hi-Bi microfiber fabricated previously [9]. (c)–(f) mode intensity distributions and (g)–(j) field vectors of the four modes. (c) and (g): oHE11 mode; (d) and (h): eHE11 mode; (e) and (i): oHE21 mode; (f) and (j): eEH01 mode.

Fig. 2
Fig. 2

Two configurations of Hi-Bi microfiber-based PCs: (a) Fiber surface perturbed from “one-side”, (b) Fiber surface perturbed from “two-sides”, (c) the cross-section of a perturbed region.

Fig. 3
Fig. 3

3D view of field components ( o HE 11 y) of the eigenmode oHE11 in (a) a perfect elliptical microfiber and (b) a perturbed elliptical microfiber. The cross-section of the perturbed microfiber is shown in Fig. 2(c) and b/a=0.5.

Fig. 4
Fig. 4

Contour plotting of the field components ( o HE 11 y) and the electric field vectors of the fundamental oHE11 mode in the perfect (a) and the perturbed microfiber (b).

Fig. 5
Fig. 5

The value of parameter α for different perturbation geometries. (a) Different azimuth angle θ and (b) corresponding value of α; (c) Different depth δ and (d) corresponding value of α.

Fig. 6
Fig. 6

Schematics demonstrating the different waveguide geometries and birefringent principal axes. Elliptical microfiber with one-side ((a) and (c)) and two-side ((b) and (d)) perturbations. (x,y) coordinates are aligned to the principal axes of the original elliptical microfiber, they are defined in Fig. 2(c) ; ( V s , V f ) and ( V s +, V f +) are the principal axes of the perturbed sections, and they can be obtained respectively by rotating (x,y) clockwise and anti-clockwise by an angle γ.

Fig. 7
Fig. 7

Evolution of electric field distributions of two orthogonal polarizations in a microfiber-based PC.

Fig. 8
Fig. 8

Evolution of light power in two polarizations with propagation distance in microfiber-based PCs. (a) a “two-side” configuration, and (b) the “one-side” configuration.

Fig. 9
Fig. 9

Polarization evolutions for different perturbation azimuth angle θ (defined in Fig. 5(a)) (a) Normalized power of the coupled polarization as function of propagating distance for different θ values. (b) Number of periods needed to achieve complete polarization conversion for different θ values.

Fig. 10
Fig. 10

Polarization evolutions for different perturbation depth δ (defined in Fig. 5(c)) in a Hi-bi microfiber PC with two-side deformations. (a) Normalized power of coupled polarization as function of propagation distance for different values of δ. (b) Number of periods needed to achieve complete polarization conversion as function of δ.

Fig. 11
Fig. 11

(a) Birefringence as the functions of wavelength for various fiber semi-major diameters a (b = 0.5a); (b) pitches required for phase matching between the two polarizations as function of wavelength.

Fig. 12
Fig. 12

Power exchange between two polarizations as the function of light wavelength. (a) Λ = 36 μm and only the coupled polarization (Py) is shown for the two-side configuration. (b) Λ = 42 μm and only the coupled power is shown for the one-side configuration. The parameters of the Hi-Bi microfiber are same as in Fig. 8.

Equations (12)

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Λ = λ Δ n eff
α = < o HE 11 , e HE 11 > | o HE 11 | | e HE 11 | = | Ω o HE 11 y e HE 11 y d Ω + Ω o HE 11 x e HE 11 x d Ω | | Ω o HE 11 o HE 11 * d Ω Ω e HE 11 e HE 11 * d Ω | 0.5
γ = π / 2 arccos ( α )
ψ = n γ , n = 1 or 2
[ V x V y ] = W d L R ( ψ ) W d R R ( ψ ) W d L R ( ψ ) W d R R ( ψ ) [ V x V y ]
R ( ψ ) = [ cos ψ sin ψ sin ψ cos ψ ]
W d x = e i ϕ x [ exp ( i Γ x / 2 ) 0 0 exp ( i Γ x / 2 ) ]
W Λ = W d L R ( ψ ) W d R R ( ψ )
W Λ = [ cos 2 ψ sin 2 ψ sin 2 ψ cos 2 ψ ]
V = [ cos 2 ψ sin 2 ψ sin 2 ψ cos 2 ψ ] [ 1 0 ] [ cos 2 ψ sin 2 ψ ]
P = sin 2 ( 2 ψ N )
N c = π / ( 4 ψ )

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