Abstract

An all-fiber twist sensor based on multimode interferometer (MMI) has been proposed and fabricated by splicing both ends of a section of square no-core fiber (NCF) with a single mode fiber. We have investigated the transmission spectral characteristics of the square fiber under different applied twisting angles. Within a torsion angle range of -360°~360°, the wavelength and transmission sensitivities are 1.28615nm/(rad × m−1) and 0.11863%/ (rad × m−1), respectively. Moreover due to the trivial thermal expansion coefficient of pure silica fiber, the proposed twist sensor has a low temperature sensitivity, which is desirable to solve the temperature cross sensitivity.

© 2013 Optical Society of America

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References

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2012 (1)

2011 (3)

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

J. Li, L. P. Sun, S. Gao, Z. Quan, Y. L. Chang, Y. Ran, L. Jin, and B. O. Guan, “Ultrasensitive refractive-index sensors based on rectangular silica microfibers,” Opt. Lett. 36(18), 3593–3595 (2011).
[Crossref] [PubMed]

Y. Gong, T. Zhao, Y. J. Rao, and Y. Wu, “All-fiber curvature sensor based on multimode interference,” IEEE Photon. Technol. Lett. 23(11), 679–681 (2011).
[Crossref]

2010 (3)

2009 (1)

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (2)

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

Q. Wang and G. Farrell, “All-fiber multimode-interference-based refractometer sensor: proposal and design,” Opt. Lett. 31(3), 317–319 (2006).
[Crossref] [PubMed]

2005 (4)

2003 (2)

Y. J. Rao, Y. P. Wang, Z. L. Ran, and T. Zhu, “Novel fiber-optic sensors based on long-period fiber gratings written by high-frequency CO2 laser pulses,” J. Lightwave Technol. 21(5), 1320–1327 (2003).
[Crossref]

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

2002 (1)

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

2001 (1)

1999 (1)

V. Lemarquand, “Synthesis study of magnetic torque sensors,” IEEE Trans. Magn. 35(6), 4503–4510 (1999).
[Crossref]

1996 (1)

1995 (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Alvarado-Mendez, E.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Andrade-Lucio, J. A.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Araújo, F. M.

Baptista, J. M.

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

O. Frazão, S. O. Silva, J. M. Baptista, J. L. Santos, G. Statkiewicz-Barabach, W. Urbanczyk, and J. Wojcik, “Simultaneous measurement of multiparameters using a Sagnac interferometer with polarization maintaining side-hole fiber,” Appl. Opt. 47(27), 4841–4848 (2008).
[Crossref] [PubMed]

Bennion, I.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

Bhatia, V.

Brambilla, G.

Caldas, P.

Chan, C. C.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Chang, Y. L.

Chen, J. P.

Chen, L. H.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Chen, X.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

Chern, G. W.

Cho, J. Y.

J. Y. Cho, J. H. Lim, and K. S. Lee, “Optical fiber twist sensor with two orthogonally oriented mechanically induced long-period grating sections,” IEEE Photon. Technol. Lett. 17(2), 453–455 (2005).
[Crossref]

Chung, Y. J.

H. M. Kim, T. H. Kim, B. K. Kim, and Y. J. Chung, “Temperature-insensitive torsion sensor with enhanced sensitivity by use of a highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 22(20), 1539–1541 (2010).
[Crossref]

Dong, X. Y.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

Estudillo-Ayala, J. M.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Farahi, F.

Farrell, G.

Ferreira, L. A.

Frazao, O.

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

Frazão, O.

Gao, S.

Gong, T. X.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Gong, Y.

Y. Gong, T. Zhao, Y. J. Rao, and Y. Wu, “All-fiber curvature sensor based on multimode interference,” IEEE Photon. Technol. Lett. 23(11), 679–681 (2011).
[Crossref]

Y. Gong, T. Zhao, Y. J. Rao, Y. Wu, and Y. Guo, “A ray-transfer-matrix model for hybrid fiber Fabry-Perot sensor based on graded-index multimode fiber,” Opt. Express 18(15), 15844–15852 (2010).
[Crossref] [PubMed]

Guan, B. O.

Guo, Y.

Ibarra-Escamilla, B.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Ibarra-Manzano, O. G.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Ivanov, O. V.

Jesus, C.

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

Jiang, M.

Jin, L.

Jin, Y. X.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Jung, Y.

Kai, G. Y.

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

Kim, B. K.

H. M. Kim, T. H. Kim, B. K. Kim, and Y. J. Chung, “Temperature-insensitive torsion sensor with enhanced sensitivity by use of a highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 22(20), 1539–1541 (2010).
[Crossref]

Kim, H. M.

H. M. Kim, T. H. Kim, B. K. Kim, and Y. J. Chung, “Temperature-insensitive torsion sensor with enhanced sensitivity by use of a highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 22(20), 1539–1541 (2010).
[Crossref]

Kim, T. H.

H. M. Kim, T. H. Kim, B. K. Kim, and Y. J. Chung, “Temperature-insensitive torsion sensor with enhanced sensitivity by use of a highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 22(20), 1539–1541 (2010).
[Crossref]

Kuzin, E. A.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Lee, K. S.

J. Y. Cho, J. H. Lim, and K. S. Lee, “Optical fiber twist sensor with two orthogonally oriented mechanically induced long-period grating sections,” IEEE Photon. Technol. Lett. 17(2), 453–455 (2005).
[Crossref]

Lemarquand, V.

V. Lemarquand, “Synthesis study of magnetic torque sensors,” IEEE Trans. Magn. 35(6), 4503–4510 (1999).
[Crossref]

Li, J.

Lim, J. H.

J. Y. Cho, J. H. Lim, and K. S. Lee, “Optical fiber twist sensor with two orthogonally oriented mechanically induced long-period grating sections,” IEEE Photon. Technol. Lett. 17(2), 453–455 (2005).
[Crossref]

Lin, C. Y.

Liu, D.

Malnou, M.

Oh, K.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Quan, Z.

Ran, Y.

Ran, Z. L.

Rao, Y. J.

Richardson, D. J.

Rojas-Laguna, R.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Roy, P.

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

Ruiz-Pinales, J.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Santos, J. L.

Shum, P. P.

Silva, S. O.

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Statkiewicz-Barabach, G.

Sun, L. P.

Sun, Q.

Torres-Cis-neros, M.

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Urbanczyk, W.

Vengsarkar, A. M.

Viegas, J.

Wang, L. A.

Wang, Q.

Wang, Y. P.

Wo, J.

Wojcik, J.

Wu, Y.

Y. Gong, T. Zhao, Y. J. Rao, and Y. Wu, “All-fiber curvature sensor based on multimode interference,” IEEE Photon. Technol. Lett. 23(11), 679–681 (2011).
[Crossref]

Y. Gong, T. Zhao, Y. J. Rao, Y. Wu, and Y. Guo, “A ray-transfer-matrix model for hybrid fiber Fabry-Perot sensor based on graded-index multimode fiber,” Opt. Express 18(15), 15844–15852 (2010).
[Crossref] [PubMed]

Yuan, S. Z.

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

Zhang, J.

Zhang, L.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

Zhang, W. G.

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

Zhang, Y. F.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Zhao, Q. D.

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

Zhao, T.

Y. Gong, T. Zhao, Y. J. Rao, and Y. Wu, “All-fiber curvature sensor based on multimode interference,” IEEE Photon. Technol. Lett. 23(11), 679–681 (2011).
[Crossref]

Y. Gong, T. Zhao, Y. J. Rao, Y. Wu, and Y. Guo, “A ray-transfer-matrix model for hybrid fiber Fabry-Perot sensor based on graded-index multimode fiber,” Opt. Express 18(15), 15844–15852 (2010).
[Crossref] [PubMed]

Zhou, K.

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

Zhu, T.

Zu, P.

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (7)

P. Zu, C. C. Chan, Y. X. Jin, T. X. Gong, Y. F. Zhang, L. H. Chen, and X. Y. Dong, “A temperature-insensitive twist sensor by using low-birefringence photonic-crystal-fiber-based Sagnac interferometer,” IEEE Photon. Technol. Lett. 23(13), 920–922 (2011).
[Crossref]

O. Frazao, C. Jesus, J. M. Baptista, J. L. Santos, and P. Roy, “Fiber-optic interferometric torsion sensor based on a two-LP-mode operation in birefringent fiber,” IEEE Photon. Technol. Lett. 21(17), 1277–1279 (2009).
[Crossref]

X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber Bragg grating with 81 tilted structure,” IEEE Photon. Technol. Lett. 18, 2596–2598 (2006).
[Crossref]

W. G. Zhang, G. Y. Kai, X. Y. Dong, S. Z. Yuan, and Q. D. Zhao, “Temperature-independent FBG-type torsion sensor based on combinatorial torsion beam,” IEEE Photon. Technol. Lett. 14(8), 1154–1156 (2002).
[Crossref]

J. Y. Cho, J. H. Lim, and K. S. Lee, “Optical fiber twist sensor with two orthogonally oriented mechanically induced long-period grating sections,” IEEE Photon. Technol. Lett. 17(2), 453–455 (2005).
[Crossref]

H. M. Kim, T. H. Kim, B. K. Kim, and Y. J. Chung, “Temperature-insensitive torsion sensor with enhanced sensitivity by use of a highly birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 22(20), 1539–1541 (2010).
[Crossref]

Y. Gong, T. Zhao, Y. J. Rao, and Y. Wu, “All-fiber curvature sensor based on multimode interference,” IEEE Photon. Technol. Lett. 23(11), 679–681 (2011).
[Crossref]

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Opt. Lasers Eng. (1)

J. M. Estudillo-Ayala, J. Ruiz-Pinales, R. Rojas-Laguna, J. A. Andrade-Lucio, O. G. Ibarra-Manzano, E. Alvarado-Mendez, M. Torres-Cis-neros, B. Ibarra-Escamilla, and E. A. Kuzin, “Analysis of a Sagnac interferometer with low-birefringence twisted fiber,” Opt. Lasers Eng. 39(5-6), 635–643 (2003).
[Crossref]

Opt. Lett. (6)

Other (1)

D. A. Gonzalez, C. Jauregui, A. Quintela, F. J. Madruga, P. Marquez, and J. M. Lopez-Higuera, “Torsion-induced effects on UV long-period fiber gratings,” In Second European Workshop on Optical Fibre Sensors. International Society for Optics and Photonics. (192–195) (2004).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Schematic experiment setup of the twist sensor (a) experimental setup; (b) Enlarged view of dotted circle.

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Fig. 2
Fig. 2

Transmission spectral characteristics of the MMI under different torsion angles ranging from −360°-360°, inset is the enlarged view of the dip in red dotted frame.

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Fig. 3
Fig. 3

Transmission response of the dip to the torsion angle: (a) wavelength response; (b) transmission response.

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Fig. 4
Fig. 4

Transmission spectral characteristics of the MMI for different temperatures, inset is the enlarged view of the dip in red dotted frame.

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Fig. 5
Fig. 5

Transmission-dependent transmission response of the dip, above is the wavelength response; below is transmission response.

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Equations (6)

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Ψ(x,y,z)= ∑ m=0 M ∑ n=0 N [ a mn ψ mn x (x,y)exp( j β mn x z )+ b mn ψ mn y (x,y)exp(j β mn y z) ]
α mn = ∫ 0 ∞ ∫ 0 ∞ Ψ( x,y,0 ) ψ mn ( x,y )dxdy ∫ 0 ∞ ∫ 0 ∞ ψ mn 2 ( x,y )dxdy ;(α=a,ψ= ψ x orα=b,ψ= ψ y )
β mn ≈ k 0 n 0 − ( m+1 ) 2 λπ 4 n 0 W xm e2 − ( n+1 ) 2 λπ 4 n 0 W yn e2
W eff (σ)≈ W 0 +( λ π ) ( 1 n 0 ) 2σ 1 n 0 2 −1
η out = | ∫ 0 ∞ ∫ 0 ∞ Ψ( x,y, L 0 ) Ψ out (x,y)dxdy | 2 ∫ 0 ∞ | Ψ( x,y, L 0 ) | 2 dxdy ∫ 0 ∞ | Ψ( x,y ) | 2 dxdy
λ= 4(2p+1) n 0 ( u+1 ) 2 W xu e2 + ( v+1 ) 2 W yv e2 − ( m+1 ) 2 W xm e2 − ( n+1 ) 2 W yn e2 1 L 0 , (p is integer) 

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