Abstract

The performance of a multifiber optical lever was geometrically analyzed by extending the Cook and Hamm model [Appl. Opt. 34, 5854–5860 (1995)] for a basic seven-fiber optical lever. The generalized relationships between sensitivity and the displacement detection limit to the fiber core radius, illumination irradiance, and coupling angle were obtained by analyses of three various types of light source, i.e., a parallel beam light source, an infinite plane light source, and a point light source. The analysis of the point light source was confirmed by a measurement that used the light source of a light-emitting diode. The sensitivity of the fiber-optic lever is inversely proportional to the fiber core radius, whereas the receiving light power is proportional to the number of illuminating and receiving fibers. Thus, the bundling of the finer fiber with the larger number of illuminating and receiving fibers is more effective for improving sensitivity and the displacement detection limit.

© 1996 Optical Society of America

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References

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  1. W. E. Frank, “Detection and measurement device having a small flexible fiber transmission line,” U. S. patent3,273,447 (20September1966).
  2. C. D. Kissinger, “Fiber optic proximity probe,” U. S. patent3,327,584 (27June1967).
  3. A. Shimamoto, K. Tanaka, “Optical fiber bundle displacement sensor using an ac modulated light source with subnanometer resolution and low thermal drift,” Appl. Opt. 34, 5854–5860 (1995).
    [CrossRef] [PubMed]
  4. R. O. Cook, C. W. Hamm, “Fiber optic lever displacement transducer,” Appl. Opt. 18, 3230–3241 (1979).
    [CrossRef] [PubMed]
  5. D. L. Shealy, H. M. Berg, “Simulation of optical coupling from surface emitting LED’s,” Appl. Opt. 22, 1722–1730 (1983).
    [CrossRef] [PubMed]

1995 (1)

1983 (1)

1979 (1)

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Figures (8)

Fig. 1
Fig. 1

Schematic of a basic optical fiber displacement sensor.

Fig. 2
Fig. 2

Schematic of the model for the analysis.

Fig. 3
Fig. 3

Model for the correction for the finite bundle size.

Fig. 4
Fig. 4

Normalized total subtended power, P RT P IT , versus the generalized distance k − 1 relationship for two bundle boundary situations, i.e., the infinite bundle–fiber size ratio (X/x 0 = ∞) and the typical finite one (X/x 0 = 24.4).

Fig. 5
Fig. 5

Relationship between the normalized total subtended power, P RT P IT , and normalized distance y tan α/x 0 under a condition of α = 35°.

Fig. 6
Fig. 6

Normalized experimental subtended power against standoff distance y as a function of alignments of the fiber bundle against the target plane. The calculated result is indicated by a solid curve.

Fig. 7
Fig. 7

Sensitivity S ( y ) ( d P ¯ R T / d y ) against standoff distance y according to Fig. 6.

Fig. 8
Fig. 8

Analytical model for a bundle of gathered illuminating fibers surrounded by receiving fibers.

Equations (50)

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P R ( ϕ , y ) = β p ( ϕ , y ) A ( ϕ , y ) ,
β = n R / ( n I + n R 1 ) .
A ( ϕ , y ) = 6 ( m , n A m n m A m 0 ) .
k = 2 y tan ϕ / x 0 + 1 .
A ¯ m n ( k ) = A m n ( k ) / π x 0 2 = 0 ( k C m n 1 ) ,
= ( 1 / π ) [ Ψ m n k 2 + sin 1 ( k sin Ψ m n ) C m n k sin Ψ m n ]
× ( C m n 1 k C m n 2 + 1 ) , = ( 1 / π ) [ Ψ m n k 2 + π sin 1 ( k sin Ψ m n ) C m n k sin Ψ m n ]
× ( C m n 2 + 1 k C m n + 1 ) , = ( 1 / π ) { π Ψ m n ( k 2 ) 2 sin 1 × [ ( k 2 ) sin Ψ m n ]
+ C m n ( k 2 ) sin Ψ m n }
× ( C m n + 1 k 2 + C m n 2 + 1 ) = ( 1 / π ) { Ψ m n ( k 2 ) 2 + sin 1 × [ ( k 2 ) sin Ψ m n ] + C m n ( k 2 ) × sin Ψ m n ] }
× ( 2 + C m n 2 + 1 k C m n + 3 ) , = 0 ( C m n + 3 k ) ,
Ψ m n = cos 1 [ ( k 2 + C m n 2 1 ) / 2 k C m n ] × ( k C m n + 1 ) ,
= cos 1 { [ ( k 2 ) 2 + C m n 2 1 ] / 2 k C m n } × ( C m n + 1 k ) .
C m n = c ( m 2 + n 2 + m n ) 1 / 2 ,
c = l / x 0 = ( 2 x 0 + 2 x c + x s ) / x 0 .
p ¯ ( ϕ , y ) = p ( k ) / p I = 1 / ( 2 k 2 4 k + 4 ) ( k 2 ) ,
= 1 / ( 4 k 4 ) ( 2 k ) ,
ρ ( ϕ , y ) = ρ ( K )
= ( K 2 4 K + 5 ) / ( 2 K 2 4 K + 4 )
× ( K 2 ) , = ( 3 K ) / 4 ( 2 K 3 ) ,
= 0 ( 3 K ) ,
K = 2 y tan ϕ / X + 1 .
ρ ( ϕ , y ) = ρ ( k ) = 1 ( k 1 ) ( x 0 / X ) + ( 1 / 2 ) ( k 1 ) 2 ( x 0 / X ) 2 1 + ( k 1 ) 2 ( x 0 / X ) 2
× ( k 1 + X / x 0 ) , = 2 ( 1 / 4 ) ( k 1 ) ( x 0 / X )
× ( 1 + X / x 0 k 1 + 2 X / x 0 ) , = 0 ( 1 + 2 X / x 0 k ) .
P R T ( ϕ , y ) = P R T ( k ) = n I P R ( ϕ , y ) ρ ( ϕ , y ) = p ¯ ( k ) A ¯ ( k ) ρ ( k ) β P I T ( k ) ,
P max / β P I T = γ ( c ) 2 π / 3 c 2 .
d p I = 2 π p 0 sin ϕ cos ϕ d ϕ ,
P R T = β P I T sin 2 α 0 α sin 2 ϕ ρ ( ϕ , y ) p ¯ ( ϕ , y ) A ¯ ( ϕ , y ) d ϕ ,
p I ( ϕ i j ) = P 0 π cos 2 ϕ i j ( y g 2 + x i j 2 ) = P 0 cos 4 ϕ i j π y g 2 ,
P R T = 0 X i P R 2 γ ( c ) π x d x π x 0 2 ,
= β P 0 γ ( c ) sin 2 tan 1 ( X i / y g ) 0 tan 1 ( X i / y g ) × sin 2 ϕ p ¯ ( ϕ , y ) A ¯ ( ϕ , y ) ρ d ϕ .
P I T = γ ( c ) 0 tan 1 ( X i / y g ) 2 P 0 sin ϕ cos ϕ d ϕ ,
= γ ( c ) P 0 sin 2 tan 1 ( X i / y g ) .
P R T = β P I T sin 2 tan 1 ( X i / y g ) 0 tan 1 ( X i / y g ) × sin 2 ϕ p ¯ ( ϕ , y ) A ¯ ( ϕ , y ) ρ d ϕ .
N = ( Q 1 / P max + Q 2 / P max ) 1 / 2 B 1 / 2 S ( y ) 1 ,
P max ργβ P I T .
S ( y ) = d P ¯ R T / d y = ( d P ¯ R T / d k ) ( d k / d y ) .
S ( y ) 2 tan ϕ / x 0 .
S ( y ) 1 . 2 tan α / x 0
S ( y ) 1 . 2 X I / ( y g x 0 )
N 0 . 15 c cot ϕ [ ( n I + n R 1 ) Q 1 B σ I σ R n I n R ρ p I ] 1 / 2 ,
N 0 . 14 c csc α cot α [ ( n I + n R 1 ) Q 1 B σ I σ R n I n R ρ p 0 ] 1 / 2
N 0 . 44 y g c { [ 1 + ( y g / X I ) 2 ] ( n I + n R 1 ) Q 1 B σ I σ R n I n R ρ P 0 } 1 / 2
P R / P I T = ργ ( c I ) ( k 2 1 ) / ( 2 k 2 4 k + 4 ) ( k 2 ) ,
= ργ ( c I ) ( k + 1 ) / 4 ( k 2 ) ,
= ργ ( c I ) ( k 1 ) ,
ρ ( K ) = 1 ( k X / X I ) ,
= 1 ( X / X I ) 2 / ( 4 k 4 ) ( X / X I k X / X I + 2 ) ,
= 0 ( k X / X I + 2 ) ,

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