Abstract

A magnetic field sensing system based on V-shaped groove filled with magnetic fluids is developed in this work. The propagation direction of the emergent light after the V-shaped groove (or the position of the emergent light on the detecting plane) is related to the strength of the externally applied magnetic field. The analytical expressions for the sensing system are derived in detail. The sensitivity and other sensing properties of the sensing system are investigated numerically and experimentally. The sensing mechanism is analyzed and ascribed to the magnetically tunable refractive index of magnetic fluids.

© 2012 Optical Society of America

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2011 (17)

Y. Zhao, Y. Zhang, R. Lv, and Q. Wang, “Novel optical devices based on the tunable refractive index of magnetic fluid and their characteristics,” J. Magn. Magn. Mater. 323, 2987–2996 (2011).
[CrossRef]

C. Z. Fan, E. J. Liang, and J. P. Huang, “Optical properties in the soft photonic crystals based on ferrofluids,” J. Phys. D 44, 325003 (2011).

S. Pu, X. Bai, and L. Wang, “Temperature dependence of photonic crystals based on thermoresponsive magnetic fluids,” J. Magn. Magn. Mater. 323, 2866–2871 (2011).
[CrossRef]

W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

X. Bai, S. Pu, and L. Wang, “Optical relaxation properties of magnetic fluids under externally magnetic fields,” Opt. Commun. 284, 4929–4935 (2011).
[CrossRef]

X. Bai, S. Pu, L. Wang, X. Wang, G. Yu, and H. Ji, “Tunable magneto-optic modulation based on magnetically responsive nanostructured magnetic fluid,” Chin. Phys. B 20, 107501 (2011).

Y. Miao, Y. Liu, B. Liu, K. Zhang, H. Zhang, and Q. Zhao, “Intensity-modulated temperature sensor based on the photonic crystal fibers filled with magnetic fluid,” Proc. SPIE 7753, 775347 (2011).

J. Dai, M. Yang, X. Li, H. Liu, and X. Tong, “Magnetic field sensor based on magnetic fluid clad etched fiber Bragg grating,” Opt. Fiber Technol. 17, 210–213 (2011).
[CrossRef]

R. Patel, “Mechanism of chain formation in nanofluid based MR fluids,” J. Magn. Magn. Mater. 323, 1360–1363 (2011).
[CrossRef]

M. Li and Y. Li, “Fiber-optic temperature sensor based on interaction of temperature-dependent refractive index and absorption of germanium film,” Appl. Opt. 50, 231–236 (2011).
[CrossRef]

D. Pagliero, Y. Li, S. Fisher, and C. A. Meriles, “Approach to high-frequency, cavity-enhanced Faraday rotation in fluids,” Appl. Opt. 50, 648–654 (2011).
[CrossRef]

H. Zhang, Y. Dong, J. Leeson, L. Chen, and X. Bao, “High sensitivity optical fiber current sensor based on polarization diversity and a Faraday rotation mirror cavity,” Appl. Opt. 50, 924–929 (2011).
[CrossRef]

P. Zu, C. C. Chan, L. W. Siang, Y. Jin, Y. Zhang, L. H. Fen, L. Chen, and X. Dong, “Magneto-optic fiber Sagnac modulator based on magnetic fluids,” Opt. Lett. 36, 1425–1427 (2011).

A. Candiani, W. Margulis, C. Sterner, M. Konstantaki, and S. Pissadakis, “Phase-shifted Bragg microstructured optical fiber gratings utilizing infiltrated ferrofluids,” Opt. Lett. 36, 2548–2550 (2011).
[CrossRef]

L.-X. Chen, X.-G. Huang, J.-H. Zhu, G.-C. Li, and S. Lan, “Fiber magnetic-field sensor based on nanoparticle magnetic fluid and Fresnel reflection,” Opt. Lett. 36, 2761–2763 (2011).
[CrossRef]

R. Patel and R. V. Mehta, “Ferrodispersion: a promising candidate for an optical capacitor,” Appl. Opt. 50, G17–G22 (2011).
[CrossRef]

J. Li, X. Qiu, Y. Lin, X. Liu, J. Fu, H. Miao, Q. Zhang, and T. Zhang, “Oscillatory-like relaxation behavior of light transmitted through ferrofluids,” Appl. Opt. 50, 5780–5787 (2011).
[CrossRef]

2010 (6)

S. Pu, M. Dai, and G. Sun, “Longitudinal field-induced polarized light transmittance of magnetic fluids,” Opt. Commun. 283, 4012–4016 (2010).
[CrossRef]

Q.-F. Dai, H.-D. Deng, W.-R. Zhao, J. Liu, L.-J. Wu, S. Lan, and A. V. Gopal, “All-optical switching mediated by magnetic nanoparticles,” Opt. Lett. 35, 97–99 (2010).
[CrossRef]

H. Bhatt, R. Patel, and R. V. Mehta, “Magnetically induced Mie resonance in a magnetic sphere suspended in a ferrofluid,” J. Opt. Soc. Am. A 27, 873–877 (2010).
[CrossRef]

T. Hu, Y. Zhao, X. Li, J. Chen, and Z. Lv, “Novel optical fiber current sensor based on magnetic fluid,” Chin. Opt. Lett. 8, 392–394 (2010).

T.-Z. Zhang, J. Li, H. Miao, Q.-M. Zhang, J. Fu, and B.-C. Wen, “Enhancement of the field modulation of light transmission through films of binary ferrofluids,” Phys. Rev. E 82, 021403 (2010).

L. Zhang and J. Huang, “Photonic band structure of three-dimensional colloidal crystals with field-induced lattice structure transformation,” Chin. Phys. B 19, 024213 (2010).

2009 (4)

S. Pu and M. Liu, “Tunable photonic crystals based on MnFe2O4 magnetic fluids by magnetic fields,” J. Alloys Compd. 481, 851–854 (2009).
[CrossRef]

S. Pu, L. Yao, F. Guan, and M. Liu, “Threshold-tunable optical limiters based on nonlinear refraction in ferrosols,” Opt. Commun. 282, 908–913 (2009).
[CrossRef]

T. Verbiest and J. Wouters, “Magnetic field sensing based on Faraday rotation in inorganic/polymer hybrid materials,” Proc. SPIE 7467, 74670B (2009).

C.-Y. Hong, J.-J. Chieh, S.-Y. Yang, H.-C. Yang, and H.-E. Horng, “Simultaneous identification of the low-field-induced tiny variation of complex refractive index for anisotropic and opaque magnetic-fluid thin film by a stable heterodyne Mach-Zehnder interferometer,” Appl. Opt. 48, 5604–5611 (2009).
[CrossRef]

2008 (6)

S. Pu and X. Chen, “Dispersion stability requirements of the nanostructured magnetic liquid,” J. Univ. Shanghai Sci. Technol. 30, 335–338 (2008), in Chinese.

R. Patel, V. K. Aswal, and R. V. Upadhyay, “Magneto-optically induced retardation and relaxation study in a mixed system of magnetic fluid and cationic micelles,” J. Magn. Magn. Mater. 320, 3366–3369 (2008).
[CrossRef]

T. Liu, X. Chen, Z. Di, J. Zhang, X. Li, and J. Chen, “Measurement of the magnetic field-dependent refractive index of magnetic fluids in bulk,” Chin. Opt. Lett. 6, 195–197(2008).
[CrossRef]

R. V. Mehta, R. Patel, B. Chudasama, and R. V. Upadhyay, “Experimental investigation of magnetically induced unusual emission of light from a ferrodispersion,” Opt. Lett. 33, 1987–1989 (2008).
[CrossRef]

J. Li, Y. Huang, X. Liu, Y. Lin, Q. Li, and R. Gao, “Coordinated chain motion resulting in intensity variation of light transmitted through ferrofluid film,” Phys. Lett. A 372, 6952–6955 (2008).
[CrossRef]

R. V. Mehta, R. J. Patel, B. N. Chudasama, H. B. Desai, and R. V. Upadhyay, “Effect of dielectric and magnetic contrast on the photonic band gap in ferrodispersion,” Magnetohydrodynamics 44, 69–74 (2008).

2007 (1)

J. Li, Y. Huang, X. Liu, Y. Lin, L. Bai, and Q. Li, “Effect of aggregates on the magnetization property of ferrofluids: A model of gaslike compression,” Sci. Tech. Adv. Mater. 8, 448–454 (2007).

2006 (1)

Z. Di, X. Chen, S. Pu, X. Hu, and Y. Xia, “Magnetic-field-induced birefringence and particle agglomeration in magnetic fluids,” Appl. Phys. Lett. 89, 211106 (2006).

2005 (3)

C. D. Perciante and J. A. Ferrari, “Faraday current sensor with temperature monitoring,” Appl. Opt. 44, 6910–6912 (2005).
[CrossRef]

S. Pu, X. Chen, Y. Chen, W. Liao, L. Chen, and Y. Xia, “Measurement of the refractive index of a magnetic fluid by the retroreflection on the fiber-optic end face,” Appl. Phys. Lett. 86, 171904 (2005).
[CrossRef]

S. Pu, X. Chen, L. Chen, W. Liao, Y. Chen, and Y. Xia, “Tunable magnetic fluid grating by applying a magnetic field,” Appl. Phys. Lett. 87, 021901 (2005).

2004 (3)

P. Trivedi, R. Patel, K. Parekh, R. V. Upadhyay, and R. V. Mehta, “Magneto-optical effects in temperature-sensitive ferrofluids,” Appl. Opt. 43, 3619–3622 (2004).
[CrossRef]

C.-Y. Hong, H. E. Horng, and S. Y. Yang, “Tunable refractive index of magnetic fluids and its applications,” Phys. Status Solidi C 1, 1604–1609 (2004).

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 (2)

C.-Y. Hong, S. Y. Yang, H. E. Horng, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3852 (2003).
[CrossRef]

Y. F. Chen, S. Y. Yang, W. S. Tse, H. E. Horng, C.-Y. Hong, and H. C. Yang, “Thermal effect on the field-dependent refractive index of the magnetic fluid film,” Appl. Phys. Lett. 82, 3481–3483 (2003).
[CrossRef]

2002 (1)

R. P. Bhatt, “Magnetic-fluid-based smart centrifugal switch,” J. Magn. Magn. Mater. 252, 347–349 (2002).
[CrossRef]

2000 (1)

O. Baltag, D. Costandache, and A. Salceanu, “Tilt measurement sensor,” Sens. Actuators 81(1–3), 336–339, 2000.
[CrossRef]

1988 (1)

Anton, E.

Suprun, E. Anton, Simonenko, and V. Dmitri, “Location tracking device,” U.S. patent 7,292,223 (6 November, 2007).

Aswal, V. K.

R. Patel, V. K. Aswal, and R. V. Upadhyay, “Magneto-optically induced retardation and relaxation study in a mixed system of magnetic fluid and cationic micelles,” J. Magn. Magn. Mater. 320, 3366–3369 (2008).
[CrossRef]

Bai, L.

J. Li, Y. Huang, X. Liu, Y. Lin, L. Bai, and Q. Li, “Effect of aggregates on the magnetization property of ferrofluids: A model of gaslike compression,” Sci. Tech. Adv. Mater. 8, 448–454 (2007).

Bai, X.

X. Bai, S. Pu, and L. Wang, “Optical relaxation properties of magnetic fluids under externally magnetic fields,” Opt. Commun. 284, 4929–4935 (2011).
[CrossRef]

X. Bai, S. Pu, L. Wang, X. Wang, G. Yu, and H. Ji, “Tunable magneto-optic modulation based on magnetically responsive nanostructured magnetic fluid,” Chin. Phys. B 20, 107501 (2011).

S. Pu, X. Bai, and L. Wang, “Temperature dependence of photonic crystals based on thermoresponsive magnetic fluids,” J. Magn. Magn. Mater. 323, 2866–2871 (2011).
[CrossRef]

Baltag, O.

O. Baltag, D. Costandache, and A. Salceanu, “Tilt measurement sensor,” Sens. Actuators 81(1–3), 336–339, 2000.
[CrossRef]

Bao, X.

Bhatt, H.

Bhatt, R. P.

R. P. Bhatt, “Magnetic-fluid-based smart centrifugal switch,” J. Magn. Magn. Mater. 252, 347–349 (2002).
[CrossRef]

Candiani, A.

A. Candiani, W. Margulis, C. Sterner, M. Konstantaki, and S. Pissadakis, “Phase-shifted Bragg microstructured optical fiber gratings utilizing infiltrated ferrofluids,” Opt. Lett. 36, 2548–2550 (2011).
[CrossRef]

P. Childs, A. Candiani, and S. Pissadakis, “Optical fiber cladding ring magnetic field sensor,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2011), pp. 929–931.

Cao, Z.

W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

Cease, T. W.

T. W. Cease and P. Johnston, “A magneto-optic current transducer,” in Proceedings of IEEE Conference on Transactions on Power Delivery (IEEE, 1990), pp. 548–555.

Chan, C. C.

Chen, J.

Chen, L.

H. Zhang, Y. Dong, J. Leeson, L. Chen, and X. Bao, “High sensitivity optical fiber current sensor based on polarization diversity and a Faraday rotation mirror cavity,” Appl. Opt. 50, 924–929 (2011).
[CrossRef]

P. Zu, C. C. Chan, L. W. Siang, Y. Jin, Y. Zhang, L. H. Fen, L. Chen, and X. Dong, “Magneto-optic fiber Sagnac modulator based on magnetic fluids,” Opt. Lett. 36, 1425–1427 (2011).

S. Pu, X. Chen, Y. Chen, W. Liao, L. Chen, and Y. Xia, “Measurement of the refractive index of a magnetic fluid by the retroreflection on the fiber-optic end face,” Appl. Phys. Lett. 86, 171904 (2005).
[CrossRef]

S. Pu, X. Chen, L. Chen, W. Liao, Y. Chen, and Y. Xia, “Tunable magnetic fluid grating by applying a magnetic field,” Appl. Phys. Lett. 87, 021901 (2005).

Chen, L.-X.

Chen, X.

W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

T. Liu, X. Chen, Z. Di, J. Zhang, X. Li, and J. Chen, “Measurement of the magnetic field-dependent refractive index of magnetic fluids in bulk,” Chin. Opt. Lett. 6, 195–197(2008).
[CrossRef]

S. Pu and X. Chen, “Dispersion stability requirements of the nanostructured magnetic liquid,” J. Univ. Shanghai Sci. Technol. 30, 335–338 (2008), in Chinese.

Z. Di, X. Chen, S. Pu, X. Hu, and Y. Xia, “Magnetic-field-induced birefringence and particle agglomeration in magnetic fluids,” Appl. Phys. Lett. 89, 211106 (2006).

S. Pu, X. Chen, L. Chen, W. Liao, Y. Chen, and Y. Xia, “Tunable magnetic fluid grating by applying a magnetic field,” Appl. Phys. Lett. 87, 021901 (2005).

S. Pu, X. Chen, Y. Chen, W. Liao, L. Chen, and Y. Xia, “Measurement of the refractive index of a magnetic fluid by the retroreflection on the fiber-optic end face,” Appl. Phys. Lett. 86, 171904 (2005).
[CrossRef]

Chen, Y.

S. Pu, X. Chen, Y. Chen, W. Liao, L. Chen, and Y. Xia, “Measurement of the refractive index of a magnetic fluid by the retroreflection on the fiber-optic end face,” Appl. Phys. Lett. 86, 171904 (2005).
[CrossRef]

S. Pu, X. Chen, L. Chen, W. Liao, Y. Chen, and Y. Xia, “Tunable magnetic fluid grating by applying a magnetic field,” Appl. Phys. Lett. 87, 021901 (2005).

Chen, Y. F.

Y. F. Chen, S. Y. Yang, W. S. Tse, H. E. Horng, C.-Y. Hong, and H. C. Yang, “Thermal effect on the field-dependent refractive index of the magnetic fluid film,” Appl. Phys. Lett. 82, 3481–3483 (2003).
[CrossRef]

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]

Chieh, J.-J.

Childs, P.

P. Childs, A. Candiani, and S. Pissadakis, “Optical fiber cladding ring magnetic field sensor,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2011), pp. 929–931.

Chudasama, B.

Chudasama, B. N.

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S. Pu, M. Dai, and G. Sun, “Longitudinal field-induced polarized light transmittance of magnetic fluids,” Opt. Commun. 283, 4012–4016 (2010).
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Deng, H.-D.

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R. V. Mehta, R. J. Patel, B. N. Chudasama, H. B. Desai, and R. V. Upadhyay, “Effect of dielectric and magnetic contrast on the photonic band gap in ferrodispersion,” Magnetohydrodynamics 44, 69–74 (2008).

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C.-Y. Hong, S. Y. Yang, H. E. Horng, and H. C. Yang, “Control parameters for the tunable refractive index of magnetic fluid films,” J. Appl. Phys. 94, 3849–3852 (2003).
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W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

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J. Li, X. Qiu, Y. Lin, X. Liu, J. Fu, H. Miao, Q. Zhang, and T. Zhang, “Oscillatory-like relaxation behavior of light transmitted through ferrofluids,” Appl. Opt. 50, 5780–5787 (2011).
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Y. Miao, Y. Liu, B. Liu, K. Zhang, H. Zhang, and Q. Zhao, “Intensity-modulated temperature sensor based on the photonic crystal fibers filled with magnetic fluid,” Proc. SPIE 7753, 775347 (2011).

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Y. Miao, Y. Liu, B. Liu, K. Zhang, H. Zhang, and Q. Zhao, “Intensity-modulated temperature sensor based on the photonic crystal fibers filled with magnetic fluid,” Proc. SPIE 7753, 775347 (2011).

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R. V. Mehta, R. J. Patel, B. N. Chudasama, H. B. Desai, and R. V. Upadhyay, “Effect of dielectric and magnetic contrast on the photonic band gap in ferrodispersion,” Magnetohydrodynamics 44, 69–74 (2008).

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W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

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X. Bai, S. Pu, and L. Wang, “Optical relaxation properties of magnetic fluids under externally magnetic fields,” Opt. Commun. 284, 4929–4935 (2011).
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Z. Di, X. Chen, S. Pu, X. Hu, and Y. Xia, “Magnetic-field-induced birefringence and particle agglomeration in magnetic fluids,” Appl. Phys. Lett. 89, 211106 (2006).

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[CrossRef]

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Y. F. Chen, S. Y. Yang, W. S. Tse, H. E. Horng, C.-Y. Hong, and H. C. Yang, “Thermal effect on the field-dependent refractive index of the magnetic fluid film,” Appl. Phys. Lett. 82, 3481–3483 (2003).
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X. Bai, S. Pu, L. Wang, X. Wang, G. Yu, and H. Ji, “Tunable magneto-optic modulation based on magnetically responsive nanostructured magnetic fluid,” Chin. Phys. B 20, 107501 (2011).

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W. Yuan, C. Yin, P. Xiao, X. Wang, J. Sun, S. Huang, X. Chen, and Z. Cao, “Microsecond-scale switching time of magnetic fluids due to the optical trapping effect in waveguide structure,” Microfluid Nanofluid, 1–5 (2011).

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Figures (10)

Fig. 1.
Fig. 1.

Cross-section view of the magnetic field sensing system based on V-shaped groove filled with magnetic fluids.

Fig. 2.
Fig. 2.

Height of the emergent light ( Y ) as a function of refractive index of MF ( n MF ) under several different parameters: (a) different 2 θ 0 , (b) different L , (c) different n 0 , (d) different θ , (e) different h 0 , (f) different n 1 , and (g) different d 0 .

Fig. 3.
Fig. 3.

(a) Comparison of the Y - n MF curves calculated with the simplified [Eq. (8)] and the exactly analytical expressions [Eq. (7)] and (b) Difference ( Δ ) between the two sets of equations as a function of n MF

Fig. 4.
Fig. 4.

Schematic drawing of the V-shaped groove.

Fig. 5.
Fig. 5.

Schematics of experimental setup for studying the sensing system based on the V-shaped groove filled with MF.

Fig. 6.
Fig. 6.

Sensing property of the system at several included angles of the V-shaped groove: (a)  2 θ 0 = 1.42 ° , (b)  2 θ 0 = 1.77 ° , (c)  2 θ 0 = 2.49 ° , (d)  2 θ 0 = 2.84 ° , (e)  2 θ 0 = 3.2 ° , and (f)  2 θ 0 = 3.55 ° and the strength of magnetic field H as a function of current I .

Fig. 7.
Fig. 7.

Variation range of the height of the transmitted light Δ Y as a function of the included angle of the V-shaped groove 2 θ 0 .

Fig. 8.
Fig. 8.

Diagram of geometric optical path ignoring the thickness of the glass slide.

Fig. 9.
Fig. 9.

Offset within the MF between the transmitted lights when ignoring and considering the influence of thickness of the glass slide.

Fig. 10.
Fig. 10.

Offset on the detecting plane when considering and ignoring the influence of thickness of the glass slide.

Equations (31)

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n 0 sin θ = n 1 sin θ 1 = n MF sin θ ,
n MF sin ( 2 θ 0 θ ) = n 1 sin θ 1 = n 0 sin θ 0 ,
n 0 sin θ = n MF sin θ ,
n MF sin ( 2 θ 0 θ ) = n 0 sin θ 0 .
y = [ L ( P + t g θ 0 sin θ ) h 0 + h 0 t g θ 0 ( P t g θ 0 sin θ ) ( P + t g θ 0 sin θ ) t g ( π 2 θ 0 ) ( P t g θ 0 sin θ ) ] × sin 2 θ 0 P Q t g θ 0 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) + sin 2 θ 0 t g θ 0 P Q t g θ 0 + h 0 ( P + t g θ 0 sin θ ) + h 0 t g θ 0 ( P t g θ 0 sin θ ) ( P + t g θ 0 sin θ ) t g ( π 2 θ 0 ) ( P t g θ 0 sin θ ) t g ( π 2 θ 0 ) .
Δ y = [ sin 2 θ 0 P Q 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) d 0 n 0 sin 2 θ 0 P n 0 Q n 1 2 n 0 2 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) d 0 d 0 P ( n 1 2 n 0 2 sin 2 θ sin θ n 0 sin θ cos θ ) cos θ n 1 2 n 0 2 sin 2 θ ( cos 2 θ 0 P + sin θ sin 2 θ 0 ) ] × 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) cos θ 0 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) + sin θ 0 ( sin 2 θ 0 P Q ) .
Y = y + Δ y .
Y = 2 θ 0 L n MF + h 0 3 θ 0 L .
h 1 = h 0 + ( L 2 + h 0 t g θ 0 ) t g ( θ 0 θ ) ,
h 1 = L 2 t g ( π 2 θ 0 ) ,
L 1 = L L 2 ,
y = L 1 t g ( θ 0 θ 0 ) + L 2 t g ( π 2 θ 0 ) .
t g ( θ 0 θ ) = t g θ 0 t g θ 1 + t g θ 0 t g θ ,
t g θ = sin θ cos θ = n 0 sin θ n MF 2 n 0 2 sin 2 θ ,
t g ( θ 0 θ 0 ) = t g θ 0 t g θ 0 1 + t g θ 0 t g θ 0 ,
sin θ 0 = sin 2 θ 0 n MF 2 n 0 2 sin 2 θ cos 2 θ 0 sin θ ,
cos θ 0 = 1 [ sin 2 2 θ 0 ( n MF 2 n 0 2 sin 2 θ ) + cos 2 2 θ 0 sin 2 θ sin 4 θ 0 sin θ n MF 2 n 0 2 sin 2 θ ] .
y = [ L ( P + t g θ 0 sin θ ) h 0 + h 0 t g θ 0 ( P t g θ 0 sin θ ) ( P + t g θ 0 sin θ ) t g ( π 2 θ 0 ) ( P t g θ 0 sin θ ) ] × sin 2 θ 0 P Q t g θ 0 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0    sin θ P ) 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) + sin 2 θ 0 t g θ 0 P Q t g θ 0 + h 0 ( P + t g θ 0 sin θ ) + h 0 t g θ 0 ( P t g θ 0 sin θ ) ( P + t g θ 0 sin θ ) t g ( π 2 θ 0 ) ( P t g θ 0 sin θ ) t g ( π 2 θ 0 ) .
D = d cos θ cos θ ,
d = d 0 cos θ 1 sin ( θ θ 1 ) .
D = d 0 n MF 2 n 0 2 sin 2 θ n MF cos θ n 1 2 n 0 2 sin 2 θ ( n 1 2 n 0 2 sin 2 θ sin θ n 0 sin θ cos θ ) .
Δ y = d cos ( θ 0 θ 0 ) .
B E = d cos θ 0 ,
O 1 O 2 = D cos ( 2 θ 0 θ ) = C D .
tan E O 1 D = ( E B + B C + C D ) / O 1 D = tan θ 0 .
tan B O 2 C = B C / O 2 C = tan θ 1 .
Δ y = [ sin 2 θ 0 P Q 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) d 0 n 0 sin 2 θ 0 P n 0 Q n 1 2 n 0 2 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) d 0 d 0 P ( n 1 2 n 0 2 sin 2 θ sin θ n 0 sin θ cos θ ) cos θ n 1 2 n 0 2 sin 2 θ ( cos 2 θ 0 P + sin θ sin 2 θ 0 ) ] × 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) cos θ 0 1 ( sin 2 2 θ 0 P 2 + Q 2 sin 4 θ 0 sin θ P ) + sin θ 0 ( sin 2 θ 0 P Q ) .
Y = [ L h 0 P θ 0 + P θ 0 3 P ( 1 θ 0 2 ) + 2 θ 0 θ ] × 2 θ 0 P θ θ 0 1 ( 2 θ 0 P θ ) 2 1 ( 2 θ 0 P θ ) 2 + 2 θ 0 2 P θ θ 0 + h 0 P ( 1 + θ 0 2 ) P ( 1 θ 0 2 ) + 2 θ 0 θ + [ 2 θ 0 P θ 1 ( 2 θ 0 P θ ) 2 2 θ 0 P θ 1.5 2 ( 2 θ 0 P θ ) 2 P ( 1.5 2 θ 2 θ θ ) 1.5 2 θ 2 ( P + θ 2 θ 0 ) ] × 1 ( 2 θ 0 P θ ) 2 1 ( 2 θ 0 P θ ) 2 + θ 0 ( 2 θ 0 P θ ) .
Y = [ L h 0 θ 0 ] × 2 θ 0 n MF θ θ 0 1 ( 2 θ 0 n MF θ ) 2 1 ( 2 θ 0 n MF θ ) 2 + 2 θ 0 2 n MF θ θ 0 + h 0 + [ 2 θ 0 n MF θ 1 ( 2 θ 0 n MF θ ) 2 2 θ 0 n MF θ 1.5 2 ( 2 θ 0 n MF θ ) 2 θ 3 ] × 1 ( 2 θ 0 n MF θ ) 2 1 ( 2 θ 0 n MF θ ) 2 + θ 0 ( 2 θ 0 n MF θ ) .
Y = ( 2 θ 0 L 2 h 0 θ 0 2 + 2 θ 0 3 ) n MF + h 0 3 θ 0 L + 3 h 0 θ 0 2 4 θ 0 3 .
Y = 2 θ 0 L n MF + h 0 3 θ 0 L .

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