Source: http://aoot.osa.org/ao/abstract.cfm?uri=ao-51-8-1010
Timestamp: 2019-04-19 10:29:09+00:00

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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.
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Fig. 1. Cross-section view of the magnetic field sensing system based on V-shaped groove filled with magnetic fluids.
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. 4. Schematic drawing of the V-shaped groove.
Fig. 5. Schematics of experimental setup for studying the sensing system based on the V-shaped groove filled with MF.
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. 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. Diagram of geometric optical path ignoring the thickness of the glass slide.
Fig. 9. Offset within the MF between the transmitted lights when ignoring and considering the influence of thickness of the glass slide.
Fig. 10. Offset on the detecting plane when considering and ignoring the influence of thickness of the glass slide.
(4) n MF sin ( 2 θ 0 − θ ′ ) = n 0 sin θ 0 ′ .
(5) 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 ) .
(6) 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 ) .
(7) Y = y + Δ y .
(8) Y = 2 θ 0 L n MF + h 0 − 3 θ 0 L .
(A4) y = L 1 t g ( θ 0 ′ − θ 0 ) + L 2 t g ( π 2 − θ 0 ) .
(A9) 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 ] .
(A10) 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 ) .
(B2) d = d 0 cos θ 1 sin ( θ − θ 1 ) .
(B3) 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 ) .
(B4) Δ y = d ′ cos ( θ 0 ′ − θ 0 ) .
(B6) O 1 O 2 = D cos ( 2 θ 0 − θ ′ ) = C D .
(B7) tan ∠ E O 1 D = ( E B + B C + C D ) / O 1 D = tan θ 0 ′ .
(B8) tan ∠ B O 2 C = B C / O 2 C = tan θ 1 ′ .
(B9) 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 ) .
(C1) 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 ) .
(C2) 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 ) .
(C3) 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 .
(C4) Y = 2 θ 0 L n MF + h 0 − 3 θ 0 L .

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