Abstract

We propose and design a new kind of all-fiber polarization beam splitter and rotator (PBS and PBR) based on vector-mode-assisted coupling. By embedding a high-contrast-index ring-core between two cores of the conventional fiber couplers, being a three-core coupling structure, the state of polarization (SOP) of fiber-guided modes can be availably controlled, such as polarization splitting and rotating, by transitional coupling through TM01 or TE01 vector mode. Furthermore, the SOP of coupled mode can be rotated with arbitrary angle under different three-core layouts. In particular, by exploiting HE21-assisted coupling case, we can realize full-dimensional SOP rotation for arbitrary polarization input. We give the numerical simulation for the proposed all-fiber PBS and PBR, and investigate the corresponding polarization extinction ratio and polarization rotating purity in detail. The calculation results manifest a favorable performance on SOP management of fiber-guided modes. This vector-mode-assisted coupling might be expected to find potential applications in the polarization-based optical signal processing, multiplexing, and sensing system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

L. Fang and J. Wang, “Full-vectorial mode coupling in optical fibers,” IEEE J. Quantum Electron. 54(2), 6800207 (2018).

2017 (3)

J. Wang, “Data information transfer using complex optical fields: a review and perspective,” Chin. Opt. Lett. 15(3), 030005 (2017).

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

S. Pidishety, B. Srinivasan, and G. Brambilla, “All-fiber fused coupler for stable generation of radially and azimuthally polarized beams,” IEEE Photonics Technol. Lett. 29(1), 31–34 (2017).

2016 (1)

2015 (2)

2013 (3)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

S. Ramachandran, “Optical vortices in fiber,” Nanophotonics 2(5), 455–474 (2013).

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

2012 (1)

2011 (2)

A. Al Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4×4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19(17), 16672–16679 (2011).

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).

2007 (1)

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

2006 (1)

2004 (1)

2003 (1)

2000 (1)

B. Mukherjee, “WDM optical communication networks: progress and challenges,” IEEE J. Sel. Areas Comm. 18(10), 1810–1824 (2000).

1992 (1)

1985 (1)

I. Yokohama, K. Okamato, and J. Noda, “Fiber-optic polarising beam splitter employing birefringent-fiber coupler,” Electron. Lett. 21(10), 415–416 (1985).

1980 (1)

R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fiber-optic directional coupler,” Electron. Lett. 16(7), 260–261 (1980).

Al Amin, A.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).

Alvarado, J. C.

Amezcua-Correa, R.

Antonio-Lopez, J. E.

Barwicz, T.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Bergh, R. A.

R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fiber-optic directional coupler,” Electron. Lett. 16(7), 260–261 (1980).

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Brambilla, G.

S. Pidishety, B. Srinivasan, and G. Brambilla, “All-fiber fused coupler for stable generation of radially and azimuthally polarized beams,” IEEE Photonics Technol. Lett. 29(1), 31–34 (2017).

Brunet, C.

Cai, X.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Chen, C. L.

Chen, S.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

A. Al Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4×4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19(17), 16672–16679 (2011).

Chen, X.

Dai, D.

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

Du, C.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Fang, L.

L. Fang and J. Wang, “Full-vectorial mode coupling in optical fibers,” IEEE J. Quantum Electron. 54(2), 6800207 (2018).

Gao, G.

Gao, S.

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

Han, Y. G.

He, S.

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

Hernández-Cordero, J.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Huang, L.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Ippen, E.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Jeong, M. Y.

Kartner, F.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Kim, C. S.

Klitis, C.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Koshiba, M.

Kotler, G.

R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fiber-optic directional coupler,” Electron. Lett. 16(7), 260–261 (1980).

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

LaRochelle, S.

Lee, S. B.

Li, A.

Li, J.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Li, S.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Liu, J.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Liu, L.

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

Lopez-Galmiche, G.

Love, J. D.

Messaddeq, Y.

Milione, G.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).

Mo, Q.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Mukherjee, B.

B. Mukherjee, “WDM optical communication networks: progress and challenges,” IEEE J. Sel. Areas Comm. 18(10), 1810–1824 (2000).

Ning, T.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Noda, J.

I. Yokohama, K. Okamato, and J. Noda, “Fiber-optic polarising beam splitter employing birefringent-fiber coupler,” Electron. Lett. 21(10), 415–416 (1985).

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).

Okamato, K.

I. Yokohama, K. Okamato, and J. Noda, “Fiber-optic polarising beam splitter employing birefringent-fiber coupler,” Electron. Lett. 21(10), 415–416 (1985).

Okonkwo, C. M.

Pei, L.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Pidishety, S.

S. Pidishety, B. Srinivasan, and G. Brambilla, “All-fiber fused coupler for stable generation of radially and azimuthally polarized beams,” IEEE Photonics Technol. Lett. 29(1), 31–34 (2017).

Popovic, M.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Rakich, P.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

S. Ramachandran, “Optical vortices in fiber,” Nanophotonics 2(5), 455–474 (2013).

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Riesen, N.

Rusch, L. A.

Saitoh, K.

Sanchez-Mondragon, J.

Sato, Y.

Shaw, H. J.

R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fiber-optic directional coupler,” Electron. Lett. 16(7), 260–261 (1980).

Shieh, W.

Sillard, P.

Smith, H.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Socci, L.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Sorel, M.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Srinivasan, B.

S. Pidishety, B. Srinivasan, and G. Brambilla, “All-fiber fused coupler for stable generation of radially and azimuthally polarized beams,” IEEE Photonics Technol. Lett. 29(1), 31–34 (2017).

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).

Tseng, S. M.

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Ung, B.

Velazquez-Benitez, A. M.

Wang, A.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Wang, J.

L. Fang and J. Wang, “Full-vectorial mode coupling in optical fibers,” IEEE J. Quantum Electron. 54(2), 6800207 (2018).

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

J. Wang, “Data information transfer using complex optical fields: a review and perspective,” Chin. Opt. Lett. 15(3), 030005 (2017).

J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4(5), B14–B28 (2016).

Wang, L.

Watts, M.

T. Barwicz, M. Watts, M. Popovic, P. Rakich, L. Socci, F. Kartner, E. Ippen, and H. Smith, “Polarization-transparent microphotonics devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).

Weng, S.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Wu, L.

J. Wang, L. Pei, S. Weng, L. Wu, L. Huang, T. Ning, and J. Li, “A tunable polarization beam splitter based on magnetic fluids-filled dual-core photonic crystal fiber,” IEEE Photonics J. 9(1), 1–10 (2017).

Xu, D. X.

D. Dai, L. Liu, S. Gao, D. X. Xu, and S. He, “Polarization controlment for silicon photonics integrated circuits,” Laser Photonics Rev. 7(3), 303–328 (2013).

Yang, C.

Yokohama, I.

I. Yokohama, K. Okamato, and J. Noda, “Fiber-optic polarising beam splitter employing birefringent-fiber coupler,” Electron. Lett. 21(10), 415–416 (1985).

Yu, S.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).

Zhang, L.

Zhu, L.

J. Liu, S. Li, L. Zhu, A. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Yu, X. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Appl. Sci. 7(3), 17148 (2018).

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).

Appl. Opt. (1)

Chin. Opt. Lett. (1)

Electron. Lett. (2)

R. A. Bergh, G. Kotler, and H. J. Shaw, “Single-mode fiber-optic directional coupler,” Electron. Lett. 16(7), 260–261 (1980).

I. Yokohama, K. Okamato, and J. Noda, “Fiber-optic polarising beam splitter employing birefringent-fiber coupler,” Electron. Lett. 21(10), 415–416 (1985).

IEEE J. Quantum Electron. (1)

L. Fang and J. Wang, “Full-vectorial mode coupling in optical fibers,” IEEE J. Quantum Electron. 54(2), 6800207 (2018).

IEEE J. Sel. Areas Comm. (1)

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Figures (10)

Fig. 1
Fig. 1 Three types of vector-mode-assisted coupling cases for arbitrary polarization rotation and the feasible fabrication process of three-core coupling structures. (a) TM01-assisted case; (b) TE01-assisted case; (c) Odd HE21-assisted case. (d) All-fiber PBS with three cores being in a line layout. (e) All-fiber PBR with three cores being in a vertical layout. SMF: single-mode fiber; RCF: ring-core fiber.
Fig. 2
Fig. 2 The effective index map of the fundamental mode HE11 in SMF and several high-order vector modes in RCF versus core radii at the wavelength of 1550 nm.
Fig. 3
Fig. 3 Sketch of TM01-assisted all-fiber PBS and coupling paths for two different polarization input. Insets show power flow among different modes shown as red arrows, while the yellow thin arrows represent undesired mode coupling.
Fig. 4
Fig. 4 Polarization-dependent power evolution map of TM01-assisted all-fiber PBS for x-polarized HE11 mode at the central wavelength of 1550 nm. (a) x-polarized HE11 input from core1 and x-polarized HE11 output from core3. (b) y-polarized HE11 input from core1, but remains propagating along the core1.
Fig. 5
Fig. 5 Polarization-dependent power evolution of TM01-assisted all-fiber PBS versus coupling lengths at the central wavelength of 1550 nm under different core distances of (a) d=6 μm, (b) d=8 μm, and (c) d=10 μm.
Fig. 6
Fig. 6 (a) Coupling efficiency of the x-polarized HE11 from core1 to core3 for the designed TM01-assisted all-fiber PBS. Polarization extinction ratios of this PBS under three core distances for polarization output from (b) core1 and (c) core3.
Fig. 7
Fig. 7 Sketch of TM01-assisted PBR for x-polarization input and y-polarization output. Insets show power flow among different modes shown as red arrows, while the yellow thin arrows represent undesired mode coupling.
Fig. 8
Fig. 8 Polarization-dependent power evolution map of TM01-assisted all-fiber PBR for x-polarized HE11 mode at the central wavelength of 1550 nm. (a) x-polarized HE11 input from core1 and is coupled into TM01-assisted mode in core2. (b) TM01-assisted mode is coupled into y-polarized HE11 and output from core3.
Fig. 9
Fig. 9 (a) The polarization-dependent power evolution along the core1 and core3 of the TM01-assisted all-fiber PBR with core distance of d=8μm,. (b) The purity of polarization rotation from the x-polarized HE11 in core1 to y-polarized HE11 in core3 under three core distances. (c) The polarization purity (left y-axis), and required full coupling lengths (right y-axis) versus varied core distances at the wavelength of 1550 nm..
Fig. 10
Fig. 10 Mode input and output with SOP rotation by the HE21-assisted full-dimensional PBR. (a) y-polarized HE11 input from core1 and x-polarized HE11 output from core3. (b) x-polarization input from core1 and y-polarization output from core3. (c) x-polarization input from core3 and y-polarization output from core1. (d) y-polarization input from core3 and x-polarization output core1. The arrows represent modal polarization states.

Equations (12)

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d A 1 dz +j β 1 A 1 =j κ 12 A 2 ,
d A 2 dz +j β 2 A 2 =j κ 21 A 1 +j κ 23 A 3 ,
d A 3 dz +j β 3 A 3 =j κ 32 A 2 ,
A 1 ( z )= κ 23 2 χ 2 + κ 12 2 χ 2 cosγz e jδz j κ 12 2 δ γ χ 2 sinγz e jδz ,
A 2 ( z )=j κ 12 γ sinγz e jδz ,
A 3 ( z )= κ 12 κ 23 χ 2 + κ 12 κ 23 χ 2 cosγz e jδz j κ 12 κ 23 δ γ χ 2 sinγz e jδz ,
P 1 ( z )= | A 1 ( z ) | 2 = cos 4 ( 2 2 κz ),
P 2 ( z )= | A 2 ( z ) | 2 = 1 2 sin 2 ( 2 κz ),
P 3 ( z )= | A 3 ( z ) | 2 = sin 4 ( 2 2 κz ).
ER=10log 10 ( | A 1x | 2 | A 1y | 2 ),
ER=10log 10 ( | A 3y | 2 | A 3x | 2 ),
η= | A 3y | 2 | A 3x | 2 + | A 3y | 2 .

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