Abstract

An ultrasensitive magnetic field sensor based on a compact in-fiber Mach–Zehnder interferometer (MZI) created in twin-core fiber (TCF) is proposed, and its performance is experimentally demonstrated. A section of TCF was spliced between two sections of standard single-mode fibers, and then a microchannel was drilled through one core of the TCF by means of femtosecond laser micromachining. The TCF with one microchannel was then immersed in a water-based Fe3O4 magnetic fluid (MF), forming a direct component of the light propagation path, and then sealed in a capillary tube, achieving a magnetic sensing element, which merges the advantages of an MZI with an MF. Experiments were conducted to investigate the magnetic response of the proposed sensor. The developed magnetic field sensor exhibits a linear response within a measurement range from 5 to 9.5 mT and an ultrahigh sensitivity of 20.8 nm/mT, which, to our best knowledge, is 2 orders of magnitude greater than other previously reported magnetic sensors. The proposed sensor is expected to offer significant potential for detecting weak magnetic fields.

© 2016 Chinese Laser Press

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References

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  1. J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (5)

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

M. Deng, D. Liu, and D. Li, “Magnetic field sensor based on asymmetric optical fiber taper and magnetic fluid,” Sens. Actuators A 211, 55–59 (2014).
[Crossref]

S. Dong, S. Pu, and H. Wang, “Magnetic field sensing based on magnetic-fluid-clad fiber-optic structure with taper-like and lateral-offset fusion splicing,” Opt. Express 22, 19108–19116 (2014).
[Crossref]

2013 (5)

2012 (1)

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

2010 (2)

2005 (1)

2004 (1)

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

2003 (1)

C. Y. Hong, S. Y. Yang, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3853 (2003).
[Crossref]

Abdelaziz, S.

Chan, C. C.

J. Zheng, X. Dong, P. Zu, L.-Y. Shao, C. C. Chan, Y. Cui, and P. P. Shum, “Magnetic field sensor using tilted fiber grating interacting with magnetic fluid,” Opt. Express 21, 17863–17868 (2013).
[Crossref]

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Chen, J. P.

Chen, L. H.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Chen, Y.

Chieh, J. J.

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

Cui, Y.

Dai, Y.

Deng, M.

M. Deng, D. Liu, and D. Li, “Magnetic field sensor based on asymmetric optical fiber taper and magnetic fluid,” Sens. Actuators A 211, 55–59 (2014).
[Crossref]

M. Deng, X. Sun, M. Han, and D. Li, “Compact magnetic-field sensor based on optical microfiber Michelson interferometer and Fe3O4 nanofluid,” Appl. Opt. 52, 734–741 (2013).
[Crossref]

Dong, S.

Dong, X.

J. Zheng, X. Dong, P. Zu, L.-Y. Shao, C. C. Chan, Y. Cui, and P. P. Shum, “Magnetic field sensor using tilted fiber grating interacting with magnetic fluid,” Opt. Express 21, 17863–17868 (2013).
[Crossref]

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Gao, R.

Han, M.

Han, Q.

Hong, C. Y.

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

C. Y. Hong, S. Y. Yang, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3853 (2003).
[Crossref]

Horng, H. E.

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

Hu, L.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Jiang, Y.

Jin, Y.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Lan, X.

Lew, W. S.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Li, D.

M. Deng, D. Liu, and D. Li, “Magnetic field sensor based on asymmetric optical fiber taper and magnetic fluid,” Sens. Actuators A 211, 55–59 (2014).
[Crossref]

M. Deng, X. Sun, M. Han, and D. Li, “Compact magnetic-field sensor based on optical microfiber Michelson interferometer and Fe3O4 nanofluid,” Appl. Opt. 52, 734–741 (2013).
[Crossref]

Li, Z.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Liao, C.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Liew, H. F.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Lin, W.

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

Liu, B.

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Liu, D.

M. Deng, D. Liu, and D. Li, “Magnetic field sensor based on asymmetric optical fiber taper and magnetic fluid,” Sens. Actuators A 211, 55–59 (2014).
[Crossref]

Liu, S.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Y. Wang, M. Yang, D. N. Wang, S. Liu, and P. Lu, “Fiber in-line Mach–Zehnder interferometer fabricated by femtosecond laser micromachining for refractive index measurement with high sensitivity,” J. Opt. Soc. Am. B 27, 370–374 (2010).
[Crossref]

Liu, T.

Liu, Y.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Lu, P.

Miao, Y.

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

Pu, S.

Rao, Y. J.

Shao, L.-Y.

Shum, P. P.

Song, B.

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Sun, X.

Wang, D. N.

Wang, H.

Wang, Q.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Wang, Y.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Y. Wang, “Review of long period fiber gratings written by CO2 laser,” J. Appl. Phys. 108, 081101 (2010).

Y. Wang, M. Yang, D. N. Wang, S. Liu, and P. Lu, “Fiber in-line Mach–Zehnder interferometer fabricated by femtosecond laser micromachining for refractive index measurement with high sensitivity,” J. Opt. Soc. Am. B 27, 370–374 (2010).
[Crossref]

Wang, Y. P.

Wong, W. C.

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Wu, J.

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Xiao, H.

Xu, G.

Yang, H. C.

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

C. Y. Hong, S. Y. Yang, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3853 (2003).
[Crossref]

Yang, K.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Yang, M.

Yang, S. Y.

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

C. Y. Hong, S. Y. Yang, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3853 (2003).
[Crossref]

Yao, J.

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Yin, G.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Yuan, Y.

Zhang, H.

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

Zhang, K.

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

Zheng, J.

Zhong, X.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Zhou, J.

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

Zu, P.

J. Zheng, X. Dong, P. Zu, L.-Y. Shao, C. C. Chan, Y. Cui, and P. P. Shum, “Magnetic field sensor using tilted fiber grating interacting with magnetic fluid,” Opt. Express 21, 17863–17868 (2013).
[Crossref]

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. Wu, Y. Miao, B. Song, W. Lin, H. Zhang, K. Zhang, B. Liu, and J. Yao, “Low temperature sensitive intensity-interrogated magnetic field sensor based on modal interference in thin-core fiber and magnetic fluid,” Appl. Phys. Lett. 104, 252402 (2014).
[Crossref]

S. Y. Yang, J. J. Chieh, H. E. Horng, C. Y. Hong, and H. C. Yang, “Origin and applications of magnetically tunable refractive index of magnetic fluid films,” Appl. Phys. Lett. 84, 5204–5206 (2004).
[Crossref]

IEEE Photon. J. (1)

P. Zu, C. C. Chan, W. S. Lew, L. Hu, Y. Jin, H. F. Liew, L. H. Chen, W. C. Wong, and X. Dong, “Temperature-insensitive magnetic field sensor based on nanoparticle magnetic fluid and photonic crystal fiber,” IEEE Photon. J. 4, 491–498 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Wu, Y. Miao, W. Lin, K. Zhang, B. Song, H. Zhang, B. Liu, and J. Yao, “Dual-direction magnetic field sensor based on core-offset microfiber and ferrofluid,” IEEE Photon. Technol. Lett. 26, 1581–1584 (2014).
[Crossref]

J. Appl. Phys. (2)

Y. Wang, “Review of long period fiber gratings written by CO2 laser,” J. Appl. Phys. 108, 081101 (2010).

C. Y. Hong, S. Y. Yang, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3853 (2003).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Express (3)

Opt. Lett. (2)

Sens. Actuators A (1)

M. Deng, D. Liu, and D. Li, “Magnetic field sensor based on asymmetric optical fiber taper and magnetic fluid,” Sens. Actuators A 211, 55–59 (2014).
[Crossref]

Sens. Actuators B (1)

Z. Li, Y. Wang, C. Liao, S. Liu, J. Zhou, X. Zhong, Y. Liu, K. Yang, Q. Wang, and G. Yin, “Temperature-insensitive refractive index sensor based on in-fiber Michelson interferometer,” Sens. Actuators B 199, 31–35 (2014).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of the proposed magnetic field sensor.
Fig. 2.
Fig. 2. (a) Optical microscope image of the cross-sectional morphology of SMF and TCF, including the dimensions of the elliptical TCF cores and the splicing point between SMF 1 and TCF. (b) Schematic diagram of the femtosecond laser micromachining system. The insert image shows an optical microscope image of the drilled microchannel through Core 2 .
Fig. 3.
Fig. 3. Transmission spectra of the pristine TCF and TCF with a microchannel filled with either air or an MF.
Fig. 4.
Fig. 4. Schematic diagram of magnetic field response measurement.
Fig. 5.
Fig. 5. Variation of the fringe dip wavelength with respect to an applied magnetic field, divided into sluggish area, high-sensitive area, low-sensitive area, and saturated area.
Fig. 6.
Fig. 6. (a) Transmission spectral evolution with an increasing applied magnetic field in the linear response region from 5 to 9.5 mT. (b) Variation of the fringe dip wavelength and dip intensity with respect to an applied magnetic field.

Tables (1)

Tables Icon

Table 1. Comparisons of the Proposed TCF-Based MZI with Other Magnetic Field Sensors

Equations (5)

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

I = I core 1 + I core 2 + 2 I core 1 I core 2 cos ( 2 π L Δ n λ + ϕ 0 ) ,
λ m = 2 π L Δ n ( 2 m + 1 ) π ,
FSR = λ 2 Δ n L .
δ λ 2 L ( Δ n + δ n ) 2 m + 1 2 L Δ n 2 m + 1 = 2 L δ n 2 m + 1 .
ϵ MF = ϵ col ϵ liq + [ ϵ col ( 1 f ) + ϵ liq ( f 1 ) ] 2 + 4 ( 1 + f ) 2 ϵ col ϵ liq 2 ( 1 + f ) ,

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