Multi-mode interferometer-based twist sensor with low temperature sensitivity employing square coreless fibers

Binbin Song; Yinping Miao; Wei Lin; Hao Zhang; Jixuan Wu; Bo Liu

doi:10.1364/OE.21.026806

Multi-mode interferometer-based twist sensor with low temperature sensitivity employing square coreless fibers

Binbin Song, Yinping Miao, Wei Lin, Hao Zhang, Jixuan Wu, and Bo Liu

Optics Express
Vol. 21,
Issue 22,
pp. 26806-26811
(2013)
•https://doi.org/10.1364/OE.21.026806

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Binbin Song, Yinping Miao, Wei Lin, Hao Zhang, Jixuan Wu, and Bo Liu, "Multi-mode interferometer-based twist sensor with low temperature sensitivity employing square coreless fibers," Opt. Express 21, 26806-26811 (2013)

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Related Topics
Table of Contents Category
- Sensors
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber characterization (060.2270)
- Fiber optics sensors (060.2370)

About this Article
History
- Original Manuscript: July 17, 2013
- Revised Manuscript: October 2, 2013
- Manuscript Accepted: October 6, 2013
- Published: October 30, 2013

Accessible

Open Access

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.

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References

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Year
|
Author
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Publication

V. Lemarquand, “Synthesis study of magnetic torque sensors,” IEEE Trans. Magn. 35(6), 4503–4510 (1999).
[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]
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]
C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]
Y. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[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]
O. V. Ivanov, “Fabrication of long-period fiber gratings by twisting a standard single-mode fiber,” Opt. Lett. 30(24), 3290–3292 (2005).
[Crossref] [PubMed]
O. V. Ivanov, “Propagation and coupling of hybrid modes in twisted fibers,” J. Opt. Soc. Am. A 22(4), 716–723 (2005).
[Crossref] [PubMed]
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]
V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]
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]
J. Wo, M. Jiang, M. Malnou, Q. Sun, J. Zhang, P. P. Shum, and D. Liu, “Twist sensor based on axial strain insensitive distributed Bragg reflector fiber laser,” Opt. Express 20(3), 2844–2850 (2012).
[Crossref] [PubMed]
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]
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]
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]
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]
O. Frazão, J. Viegas, P. Caldas, J. L. Santos, F. M. Araújo, L. A. Ferreira, and F. Farahi, “All-fiber Mach-Zehnder curvature sensor based on multimode interference combined with a long-period grating,” Opt. Lett. 32(21), 3074–3076 (2007).
[Crossref] [PubMed]
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]
Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]
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]
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]
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]
Q. Wang and G. Farrell, “All-fiber multimode-interference-based refractometer sensor: proposal and design,” Opt. Lett. 31(3), 317–319 (2006).
[Crossref] [PubMed]

2012 (1)

J. Wo, M. Jiang, M. Malnou, Q. Sun, J. Zhang, P. P. Shum, and D. Liu, “Twist sensor based on axial strain insensitive distributed Bragg reflector fiber laser,” Opt. Express 20(3), 2844–2850 (2012).
[Crossref] [PubMed]

2011 (3)

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]

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]

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)

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. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

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]

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]

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]

2005 (4)

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]

O. V. Ivanov, “Propagation and coupling of hybrid modes in twisted fibers,” J. Opt. Soc. Am. A 22(4), 716–723 (2005).
[Crossref] [PubMed]

Y. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[Crossref]

O. V. Ivanov, “Fabrication of long-period fiber gratings by twisting a standard single-mode fiber,” Opt. Lett. 30(24), 3290–3292 (2005).
[Crossref] [PubMed]

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)

C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

1999 (1)

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

1996 (1)

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]

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.

Andrade-Lucio, J. A.

Araújo, F. M.

Baptista, J. M.

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.

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]

Brambilla, G.

Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

Caldas, P.

Chan, C. C.

Chang, Y. L.

Chen, J. P.

Y. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[Crossref]

Chen, L. H.

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.

C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

Cho, J. Y.

Chung, Y. J.

Dong, X. Y.

Estudillo-Ayala, J. M.

Farahi, F.

Farrell, G.

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

Ferreira, L. A.

Frazao, O.

Frazão, O.

Gao, S.

Gong, T. X.

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]

Guan, B. O.

Guo, Y.

Ibarra-Escamilla, B.

Ibarra-Manzano, O. G.

Ivanov, O. V.

O. V. Ivanov, “Fabrication of long-period fiber gratings by twisting a standard single-mode fiber,” Opt. Lett. 30(24), 3290–3292 (2005).
[Crossref] [PubMed]

O. V. Ivanov, “Propagation and coupling of hybrid modes in twisted fibers,” J. Opt. Soc. Am. A 22(4), 716–723 (2005).
[Crossref] [PubMed]

Jesus, C.

Jiang, M.

Jin, L.

Jin, Y. X.

Jung, Y.

Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

Kai, G. Y.

Kim, B. K.

Kim, H. M.

Kim, T. H.

Kuzin, E. A.

Lee, K. S.

Lemarquand, V.

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

Li, J.

Lim, J. H.

Lin, C. Y.

C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

Liu, D.

Malnou, M.

Oh, K.

Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

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.

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. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[Crossref]

Richardson, D. J.

Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

Rojas-Laguna, R.

Roy, P.

Ruiz-Pinales, J.

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.

Urbanczyk, W.

Vengsarkar, A. M.

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]

Viegas, J.

Wang, L. A.

C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

Wang, Q.

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

Wang, Y. P.

Y. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[Crossref]

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]

Yuan, S. Z.

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.

Zhang, Y. F.

Zhao, Q. D.

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]

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.

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (7)

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]

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]

IEEE Trans. Magn. (1)

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

J. Lightwave Technol. (3)

C. Y. Lin, L. A. Wang, and G. W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

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]

J. Opt. Soc. Am. A (1)

O. V. Ivanov, “Propagation and coupling of hybrid modes in twisted fibers,” J. Opt. Soc. Am. A 22(4), 716–723 (2005).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

Y. P. Wang, J. P. Chen, and Y. J. Rao, “Torsion characteristics of long period fiber gratings induced by high-frequency laser pulses,” J. Opt. Soc. Am. B 22(6), 1167–1172 (2005).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

Opt. Lett. (6)

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]

O. V. Ivanov, “Fabrication of long-period fiber gratings by twisting a standard single-mode fiber,” Opt. Lett. 30(24), 3290–3292 (2005).
[Crossref] [PubMed]

Y. Jung, G. Brambilla, K. Oh, and D. J. Richardson, “Highly birefringent silica microfiber,” Opt. Lett. 35(3), 378–380 (2010).
[Crossref] [PubMed]

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

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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Fig. 1

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

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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

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

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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

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

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Ψ (x, y, z) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} [a_{m n} ψ_{m n}^{x} (x, y) \exp (j β_{m n}^{x} z) + b_{m n} ψ_{m n}^{y} (x, y) \exp (j β_{m n}^{y} z)]

α_{m n} = \frac{\int_{0}^{\infty} \int_{0}^{\infty} Ψ (x, y, 0) ψ_{m n} (x, y) d x d y}{\int_{0}^{\infty} \int_{0}^{\infty} ψ_{m n}^{2} (x, y) d x d y}; (α = a, ψ = ψ^{x} o r α = b, ψ = ψ^{y})

β_{m n} \approx k_{0} n_{0} - \frac{{(m + 1)}^{2} λ π}{4 n_{0} W_{x m}^{e 2}} - \frac{{(n + 1)}^{2} λ π}{4 n_{0} W_{y n}^{e 2}}

W_{e f f} (σ) \approx W_{0} + (\frac{λ}{π}) {(\frac{1}{n_{0}})}^{2 σ} \frac{1}{\sqrt{n_{0}^{2} - 1}}

η_{o u t} = \frac{{| \int_{0}^{\infty} \int_{0}^{\infty} Ψ (x, y, L_{0}) Ψ_{o u t} (x, y) d x d y |}^{2}}{\int_{0}^{\infty} {| Ψ (x, y, L_{0}) |}^{2} d x d y \int_{0}^{\infty} {| Ψ (x, y) |}^{2} d x d y}

λ = \frac{4 (2 p + 1) n_{0}}{\frac{{(u + 1)}^{2}}{W_{x u}^{e 2}} + \frac{{(v + 1)}^{2}}{W_{y v}^{e 2}} - \frac{{(m + 1)}^{2}}{W_{x m}^{e 2}} - \frac{{(n + 1)}^{2}}{W_{y n}^{e 2}}} \frac{1}{L_{0}}, (p is i nteger)

Abstract

References

2012 (1)

2011 (3)

2010 (3)

2009 (1)

2008 (1)

2007 (1)

2006 (2)

2005 (4)

2003 (2)

2002 (1)

2001 (1)

1999 (1)

1996 (1)

1995 (1)

Alvarado-Mendez, E.

Andrade-Lucio, J. A.

Araújo, F. M.

Baptista, J. M.

Bennion, I.

Bhatia, V.

Brambilla, G.

Caldas, P.

Chan, C. C.

Chang, Y. L.

Chen, J. P.

Chen, L. H.

Chen, X.

Chern, G. W.

Cho, J. Y.

Chung, Y. J.

Dong, X. Y.

Estudillo-Ayala, J. M.

Farahi, F.

Farrell, G.

Ferreira, L. A.

Frazao, O.

Frazão, O.

Gao, S.

Gong, T. X.

Gong, Y.

Guan, B. O.

Guo, Y.

Ibarra-Escamilla, B.

Ibarra-Manzano, O. G.

Ivanov, O. V.

Jesus, C.

Jiang, M.

Jin, L.

Jin, Y. X.

Jung, Y.

Kai, G. Y.

Kim, B. K.

Kim, H. M.

Kim, T. H.

Kuzin, E. A.

Lee, K. S.

Lemarquand, V.

Li, J.

Lim, J. H.

Lin, C. Y.

Liu, D.

Malnou, M.

Oh, K.

Pennings, E. C. M.

Quan, Z.

Ran, Y.

Ran, Z. L.

Rao, Y. J.

Richardson, D. J.

Rojas-Laguna, R.

Roy, P.

Ruiz-Pinales, J.

Santos, J. L.

Shum, P. P.

Silva, S. O.

Soldano, L. B.

Statkiewicz-Barabach, G.

Sun, L. P.