Abstract

Fiber optic acoustic sensors with composite structures are analyzed and shown to offer greatly increased acoustic sensitivity. The composite structure consists of an optical fiber coated with or embedded in an elastic material of lower elastic modulus. Sensitivity increases of 10–100 times are indicated. Important advantages of this technique in practical acoustic sensors are also described.

© 1979 Optical Society of America

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References

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  1. J. A. Bucaro et al., Appl. Opt. 16, 1761 (1977).
    [Crossref] [PubMed]
  2. J. A. Bucaro, E. F. Carome, Appl. Opt. 17, 330 (1978).
    [Crossref] [PubMed]
  3. G. B. Hocker, Appl. Opt. 18, 1445 (1979).
    [Crossref] [PubMed]
  4. Our most recent experimental results for the pressure sensitivity of fused silica single-mode fiber, more accurate than those in Ref. 3, are within about 10% of the value predicted by our analysis.
  5. G. B. Hocker, “Fiber Optic Acoustic Sensors with Increased Sensitivity by Use of Composite Structure,” to be published in “Optics Letters,” October1979.
    [Crossref]
  6. S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).
  7. The strains in the z-direction must be equal in regions 1 and 2 and independent of z, r, and θ. Since the components of ∊z caused by the transverse stresses σr and σθ have a radial dependence, which cancels in region 1 and is independent of r in region 2 [see Eqs. (4), (5)], the stresses in the z-direction must be constant in regions 1 and 2, but are not equal to each other.
  8. For other glasses, the above values can be somewhat different. Although exact values of the strain-optic coefficients are difficult to find, we can take typical values as perhaps: n = 1.50; p11 = p12 = 0.25; and ν = 0.20. This would lead to (Δϕ)/(ΔP)·E1/(βL) for the uncoated fiber is particularly sensitive to the fiber material parameters because it is the difference between two terms of nearly the same value. Physically, this is because the pressure-induced phase shift is due to two effects, the length change and the index change, and these effects are opposite in sign and similar in magnitude. Thus, the phase shift is more sensitive to the assumed material parameters than either of the two individual effects which produce it.

1979 (1)

1978 (1)

1977 (1)

Bucaro, J. A.

Carome, E. F.

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Hocker, G. B.

G. B. Hocker, Appl. Opt. 18, 1445 (1979).
[Crossref] [PubMed]

G. B. Hocker, “Fiber Optic Acoustic Sensors with Increased Sensitivity by Use of Composite Structure,” to be published in “Optics Letters,” October1979.
[Crossref]

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Appl. Opt. (3)

Other (5)

Our most recent experimental results for the pressure sensitivity of fused silica single-mode fiber, more accurate than those in Ref. 3, are within about 10% of the value predicted by our analysis.

G. B. Hocker, “Fiber Optic Acoustic Sensors with Increased Sensitivity by Use of Composite Structure,” to be published in “Optics Letters,” October1979.
[Crossref]

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

The strains in the z-direction must be equal in regions 1 and 2 and independent of z, r, and θ. Since the components of ∊z caused by the transverse stresses σr and σθ have a radial dependence, which cancels in region 1 and is independent of r in region 2 [see Eqs. (4), (5)], the stresses in the z-direction must be constant in regions 1 and 2, but are not equal to each other.

For other glasses, the above values can be somewhat different. Although exact values of the strain-optic coefficients are difficult to find, we can take typical values as perhaps: n = 1.50; p11 = p12 = 0.25; and ν = 0.20. This would lead to (Δϕ)/(ΔP)·E1/(βL) for the uncoated fiber is particularly sensitive to the fiber material parameters because it is the difference between two terms of nearly the same value. Physically, this is because the pressure-induced phase shift is due to two effects, the length change and the index change, and these effects are opposite in sign and similar in magnitude. Thus, the phase shift is more sensitive to the assumed material parameters than either of the two individual effects which produce it.

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Figures (4)

Fig. 1
Fig. 1

Cross section of optical fiber (region 1) coated with elastic medium (region 2). The fiber has Young’s modulus E1, Poisson’s ratio ν1, index of refraction n, and strain-optic coefficients p11 and p12. The coating material has Young’s modulus E2 and Poisson’s ratio ν2.

Fig. 2
Fig. 2

Acoustic sensitivity (Δϕ)/(ΔPE1/(βL) for a coated fused silica fiber vs ratio of radii R, with ratio of elastic moduli E as a parameter.

Fig. 3
Fig. 3

Dependence of acoustic sensitivity (Δϕ)/(ΔPE1/(βL) for a coated fused silica fiber on Poisson’s ratio in the coating material ν2 with E = 20 and R = 10.

Fig. 4
Fig. 4

Acoustic sensitivity for fused silica fiber embedded in elastic medium vs elastic modulus ratio E, with Poisson’s ratio in embedding medium ν2 as a parameter.

Equations (36)

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Δ ϕ Δ P = L β E ( 1 - 2 ν ) [ n 2 2 ( 2 p 12 + p 11 ) - 1 ] ,
σ r = A r 2 + B ( 1 + 2 log r ) + 2 C ,
σ θ = - A r 2 + B ( 3 + 2 log r ) + 2 C ,
σ r = { A r 2 + 2 C region 2 , 2 D region 1 ,
σ θ = { - A r 2 + 2 C region 2 , 2 D region 1 ,
σ z = { F region 2 , G region 1.
i = ( 1 / E ) [ σ i - ν ( σ j + σ k ) ] ,
r = { 1 E 1 [ 2 D ( 1 - ν 1 ) - ν 1 G ] region 1 , 1 E 2 [ A r 2 ( 1 + ν 2 ) + 2 C ( 1 - ν 2 ) - ν 2 F ] region 2 ,
z = { 1 E 1 ( G - 4 ν 1 D ) region 1 , 1 E 2 ( F - 4 ν 2 C ) region 2.
u = { 1 E 1 [ 2 D ( 1 - ν 1 ) - ν 1 G ] r Region 1 , 1 E 2 [ - ( 1 + ν 2 ) A r + 2 C ( 1 - ν 2 ) r - ν 2 F r ] Region 2.
σ r 2 ( R 2 ) = - P ,
σ r 1 ( R 1 ) = σ r 2 ( R 1 ) ,
u r 1 ( R 1 ) = u r 2 ( R 1 ) ,
- π R 2 2 P = π R 1 2 σ z 1 + π ( R 2 2 - R 1 2 ) σ z 2 ,
z 1 = z 2 ,
( A R 2 2 ) + 2 C = - P ,
( R 2 R 1 ) 2 ( A R 2 2 ) + 2 C - 2 D = 0 ,
( E 1 E 2 ) [ - ( 1 + ν 2 ) ( R 2 R 1 ) 2 ( A R 2 2 ) + 2 C ( 1 - ν 2 ) - ν 2 F ] + [ ν 1 G - 2 D ( 1 - ν 1 ) ] = 0 ,
G + [ ( R 2 R 1 ) 2 - 1 ] F = - ( R 2 R 1 ) 2 P ,
G - 4 ν 1 D + ( E 1 E 2 ) ( 4 ν 2 C - F ) = 0.
[ 1 2 0 0 0 R 2 2 - 2 0 0 - E ( 1 + ν 2 ) R 2 2 ( 1 - ν 2 ) E - 2 ( 1 - ν 1 ) - E ν 2 ν 1 0 0 0 R 2 - 1 1 0 4 E ν 2 - 4 ν 1 - E 1 ] [ X Y Z V W ] = [ - 1 0 0 - R 2 0 ] .
Δ ϕ = β Δ L + L Δ β β z L + L k 0 Δ n = β z L - ½ L k 0 n 3 Δ ( 1 / n 2 ) ,
Δ ( 1 n 2 ) i = j = 1 3 p i j j ,
p i j = [ p 11 p 12 p 12 p 12 p 11 p 12 p 12 p 12 p 11 ] .
Δ ( 1 n 2 ) x , y = ( p 11 + p 12 ) r + p 12 z .
Δ ( 1 n 2 ) x , y = - Δ P n 3 2 E 1 { ( p 11 + p 12 ) [ 2 Z ( 1 - ν 1 ) - ν 1 W ] + p 12 ( W - 4 ν 1 Z ) } ,
Δ ϕ Δ P E 1 β L = W { 1 + n 2 2 [ ( p 11 + p 12 ) ν 1 - p 12 ] } + Z { - 4 ν 1 - n 2 [ ( p 11 + p 12 ) ( 1 - ν 1 ) - 2 ν 1 p 12 ] } .
Δ ϕ Δ P E 1 β L | fused silica = 0.784 W - 1.173 Z .
Δ ϕ Δ P E 1 β L | uncoaied fused silica = - 0.198.
2 C = - P ,
( A R 1 2 ) + 2 C - 2 D = 0 ,
( E 1 E 2 ) [ - ( 1 + ν 2 ) ( A R 1 2 ) + 2 C ( 1 - ν 2 ) - ν 2 F ] + [ ν 1 G - 2 D ( 1 - ν 1 ) ] = 0 ,
F = - P ,
G - 4 ν 1 D + ( E 1 E 2 ) ( 4 ν 2 C - F ) = 0.
F = 2 C = - P ,
[ 1 - 2 0 - E ( 1 + ν 2 ) - 2 ( 1 - ν 1 ) ν 1 0 - 4 ν 1 1 ] [ X Z W ] = [ 1 E ( 1 - 2 ν 2 ) E ( 2 ν 2 - 1 ) ] .

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