Abstract

The induced optical phase change produced when a static pressure is applied to the test arm of an interferometric single-mode fiber optic hydrophone is examined in terms of hydrostatic and radial mechanical models. The expressions for the models are given in terms of a 3-D solution to the equations of elastostatics for multilayered cylinders. The induced phase change is calculated using both models for various values of the diameter and elastic properties of fiber jacket materials. It is shown that the phase change predicted from the 3-D approach for each model can be adequately described in terms of much simpler 2-D plane strain models. Calculations show that the hydrophone sensitivity of a jacketed fiber is amplified compared with a bare fiber. The largest increase in sensitivity is predicted with the radial model. Calculated sensitivities for the hydrostatic model are shown to correspond closely in value with static pressure sensitivity measurements for the experimental arrangement used here.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
    [CrossRef]
  2. J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
    [CrossRef]
  3. B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
    [CrossRef]
  4. P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
    [CrossRef]
  5. G. B. Hocker, Appl. Opt. 18, 1445 (1979).
    [CrossRef] [PubMed]
  6. J. A. Bucaro, T. R. Hickman, Appl. Opt. 18, 938 (1979).
    [CrossRef] [PubMed]
  7. B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).
  8. For the ITT fiber considered in Secs. V and VI, the dimensions and materials are: outer layer, 250-μm radius of polyester (Hytrel); middle layer, 125-μm radius of silicone rubber; and the center, 40-μm radius of glass.
  9. L. N. G. Filon, Philos. Trans. R. Soc. London Ser. A: 198, 147 (1902).
    [CrossRef]
  10. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Math, Ser. 55 (Dover, New York, 1972), p. 443.
  11. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970), Chap. 4.
  12. J. F. Nye, Physical Properties of Crystals (Oxford U.P., New York, 1976), p. 249.
  13. Radial dimensions of the layers used in the calculations are: 40, 125, and 250 μm for the glass center; silicone rubber and Hytrel layers, respectively. Values of the elastic coefficients for the jacket materials are given in Table III. The values of the elastic coefficients and Pockel’s coefficients for fused silica were taken from Ref. 14. The values of the fractional phase change presented in this Table and in Table I are for an applied pressure of 1 dyn/cm2.
  14. J. Schroeder, Dissertation; Catholic U. of America, Washington, D.C. (1974) (unpublished).
  15. D. W. Phillips, R. A. Pethrick, J. Macromol. Sci. Rev. Macromol. Chem. 16 (1), 1 (1977–1978).
  16. B. Hartmann, J. Jarzynski, J. Acoust. Soc. Am. 56, 1469 (1974).
    [CrossRef]
  17. J. R. Cunningham, D. G. Ivey, J. Appl. Phys. 27, 967 (1956).
    [CrossRef]
  18. W. S. Cramer, I. Silver, Naval Ordnance Laboratory Report 1778 (Feb.1951).
  19. D. I. G. Jones, J. Sound Vib. 33, 451 (1974).
    [CrossRef]
  20. J. Jarzynski, R. Hughes (unpublished results).
  21. R. Hughes, R. Priest (to be published).

1979 (2)

1978 (1)

P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
[CrossRef]

1977 (3)

J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
[CrossRef]

J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
[CrossRef]

B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
[CrossRef]

1974 (2)

B. Hartmann, J. Jarzynski, J. Acoust. Soc. Am. 56, 1469 (1974).
[CrossRef]

D. I. G. Jones, J. Sound Vib. 33, 451 (1974).
[CrossRef]

1956 (1)

J. R. Cunningham, D. G. Ivey, J. Appl. Phys. 27, 967 (1956).
[CrossRef]

1902 (1)

L. N. G. Filon, Philos. Trans. R. Soc. London Ser. A: 198, 147 (1902).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Math, Ser. 55 (Dover, New York, 1972), p. 443.

Bucaro, J. A.

J. A. Bucaro, T. R. Hickman, Appl. Opt. 18, 938 (1979).
[CrossRef] [PubMed]

J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
[CrossRef]

Budiansky, B.

B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).

Buta, P. G.

J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
[CrossRef]

Carome, E. F.

J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
[CrossRef]

Cole, J. H.

J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
[CrossRef]

Cramer, W. S.

W. S. Cramer, I. Silver, Naval Ordnance Laboratory Report 1778 (Feb.1951).

Culshaw, B.

B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
[CrossRef]

Cunningham, J. R.

J. R. Cunningham, D. G. Ivey, J. Appl. Phys. 27, 967 (1956).
[CrossRef]

Dardy, H. D.

J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
[CrossRef]

Davies, D. E. N.

B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
[CrossRef]

Drucker, D. C.

B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).

Filon, L. N. G.

L. N. G. Filon, Philos. Trans. R. Soc. London Ser. A: 198, 147 (1902).
[CrossRef]

Flatley, J. P.

P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
[CrossRef]

Goodier, J. N.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970), Chap. 4.

Hartmann, B.

B. Hartmann, J. Jarzynski, J. Acoust. Soc. Am. 56, 1469 (1974).
[CrossRef]

Hickman, T. R.

Hocker, G. B.

Hughes, R.

J. Jarzynski, R. Hughes (unpublished results).

R. Hughes, R. Priest (to be published).

Ivey, D. G.

J. R. Cunningham, D. G. Ivey, J. Appl. Phys. 27, 967 (1956).
[CrossRef]

Jarzynski, J.

B. Hartmann, J. Jarzynski, J. Acoust. Soc. Am. 56, 1469 (1974).
[CrossRef]

J. Jarzynski, R. Hughes (unpublished results).

Johnson, R. L.

J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
[CrossRef]

Jones, D. I. G.

D. I. G. Jones, J. Sound Vib. 33, 451 (1974).
[CrossRef]

Kingsley, S. A.

B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
[CrossRef]

Kino, G. S.

B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).

Moffett, M. B.

P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., New York, 1976), p. 249.

Pethrick, R. A.

D. W. Phillips, R. A. Pethrick, J. Macromol. Sci. Rev. Macromol. Chem. 16 (1), 1 (1977–1978).

Phillips, D. W.

D. W. Phillips, R. A. Pethrick, J. Macromol. Sci. Rev. Macromol. Chem. 16 (1), 1 (1977–1978).

Priest, R.

R. Hughes, R. Priest (to be published).

Rice, J. R.

B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).

Schroeder, J.

J. Schroeder, Dissertation; Catholic U. of America, Washington, D.C. (1974) (unpublished).

Shajenko, P.

P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
[CrossRef]

Silver, I.

W. S. Cramer, I. Silver, Naval Ordnance Laboratory Report 1778 (Feb.1951).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Math, Ser. 55 (Dover, New York, 1972), p. 443.

Timoshenko, S. P.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970), Chap. 4.

Appl. Opt. (2)

Electron. Lett. (1)

B. Culshaw, D. E. N. Davies, S. A. Kingsley, Electron. Lett. 13, 760 (1977).
[CrossRef]

J. Acoust. Soc. Am. (4)

P. Shajenko, J. P. Flatley, M. B. Moffett, J. Acoust. Soc. Am. 64, 1286 (1978).
[CrossRef]

J. A. Bucaro, H. D. Dardy, E. F. Carome, J. Acoust. Soc. Am. 62, 1302 (1977).
[CrossRef]

J. H. Cole, R. L. Johnson, P. G. Buta, J. Acoust. Soc. Am. 62, 1136 (1977).
[CrossRef]

B. Hartmann, J. Jarzynski, J. Acoust. Soc. Am. 56, 1469 (1974).
[CrossRef]

J. Appl. Phys. (1)

J. R. Cunningham, D. G. Ivey, J. Appl. Phys. 27, 967 (1956).
[CrossRef]

J. Macromol. Sci. Rev. Macromol. Chem. (1)

D. W. Phillips, R. A. Pethrick, J. Macromol. Sci. Rev. Macromol. Chem. 16 (1), 1 (1977–1978).

J. Sound Vib. (1)

D. I. G. Jones, J. Sound Vib. 33, 451 (1974).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

L. N. G. Filon, Philos. Trans. R. Soc. London Ser. A: 198, 147 (1902).
[CrossRef]

Other (10)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Math, Ser. 55 (Dover, New York, 1972), p. 443.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970), Chap. 4.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., New York, 1976), p. 249.

Radial dimensions of the layers used in the calculations are: 40, 125, and 250 μm for the glass center; silicone rubber and Hytrel layers, respectively. Values of the elastic coefficients for the jacket materials are given in Table III. The values of the elastic coefficients and Pockel’s coefficients for fused silica were taken from Ref. 14. The values of the fractional phase change presented in this Table and in Table I are for an applied pressure of 1 dyn/cm2.

J. Schroeder, Dissertation; Catholic U. of America, Washington, D.C. (1974) (unpublished).

J. Jarzynski, R. Hughes (unpublished results).

R. Hughes, R. Priest (to be published).

W. S. Cramer, I. Silver, Naval Ordnance Laboratory Report 1778 (Feb.1951).

B. Budiansky, D. C. Drucker, G. S. Kino, J. R. Rice, Stanford U., “The Pressure Sensitivity of a Clad Optical Fiber” (unpublished).

For the ITT fiber considered in Secs. V and VI, the dimensions and materials are: outer layer, 250-μm radius of polyester (Hytrel); middle layer, 125-μm radius of silicone rubber; and the center, 40-μm radius of glass.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry of the multilayered cylinder of length 2L.

Fig. 2
Fig. 2

Comparison of calculated ratios of the fractional phase change, Δϕϕ0, for a fiber jacketed with a single layer of Hycar rubber as a function of the ratio of jacket radius b to glass center radius a for the radial and hydrostatic models.

Fig. 3
Fig. 3

Calculated contributions to the normalized phase change, Δϕ/ϕ, with the radial model for a Hycar rubber jacketed fiber as a function of the ratio of jacket radius to glass center radius.

Fig. 4
Fig. 4

Calculated contributions to the normalized phase change, Δϕ/ϕ, with the hydrostatic model for a Hycar rubber jacketed fiber as a function of the ratio of jacket radius to glass center radius.

Fig. 5
Fig. 5

Comparison of calculated ratios of the fractional phase change, Δϕϕ0, for a fiber jacketed with a single layer of Teflon as a function of the ratio of jacket radius b to glass center radius a for the radial and hydrostatic models.

Fig. 6
Fig. 6

Calculated contributions to the normalized phase change, Δϕ/ϕ, with the radial model for a Teflon jacketed fiber as a function of the ratio of jacket radius to glass center radius.

Fig. 7
Fig. 7

Calculated contributions to the normalized phase change, Δϕ/ϕ, with the hydrostatic model for a Teflon jacketed fiber as a function of the ratio of jacket radius to glass center radius.

Fig. 8
Fig. 8

Experimental configuration of the fiber-optic interferometer.

Tables (3)

Tables Icon

Table I Comparison of the Absolute Values of the Measured and Calculated Pressure Induced Fractional Phase Changes for Nonjacketed, Single-Jacketed, and Double-jacketed Single-Mode Optical Fibers.

Tables Icon

Table II Comparison of the Calculated Pressure Induced Optical Phase Change as Predicted by Plain Strain and 3-D Elastostatic Solutions for a Layered Cylinder

Tables Icon

Table III Values of Young’s and Bulk Moduli and Poisson’s Ratio for Selected Plastics and Rubbers; the Calculated Amplification in Static Pressure Sensitivity of an Optical Fiber Hydrophone when it is Jacketed with a Single Layer of Plastic or Rubber

Equations (34)

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

u = u 0 ( 2 ) r + u 1 ( 2 ) r + n { - A 1 n ( 2 ) k n I 1 ( ρ n ) - B 1 n ( 2 ) k n K 1 ( ρ n ) - c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] r I 0 ( ρ n ) + c ( 2 ) [ B 2 n ( 2 ) - B 1 n ( 2 ) ] r K 0 ( ρ n ) } cos ( γ n ) ,
w = w 0 ( 2 ) z + n { A 2 n ( 2 ) k n I 0 ( ρ n ) - B 2 n ( 2 ) k n K 0 ( ρ n ) + c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] r I 1 ( ρ n ) + c ( 2 ) [ B 2 n ( 2 ) - B 1 n ( 2 ) ] r K 1 ( ρ n ) } sin ( γ n ) ,
σ r r = 2 [ λ ( 2 ) + μ ( 2 ) ] u 0 ( 2 ) + λ ( 2 ) w 0 ( 2 ) - 2 μ ( 2 ) u 1 ( 2 ) r 2 + n ( - [ d ( 2 ) A 1 n ( 2 ) + e ( 2 ) A 2 n ( 2 ) ] I 0 ( ρ n ) + [ d ( 2 ) B 1 n ( 2 ) + e ( 2 ) B 2 n ( 2 ) ] K 0 ( ρ n ) + 2 μ ( 2 ) { A 1 n ( 2 ) ρ n - c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] ρ n } I 1 ( ρ n ) + 2 μ ( 2 ) { B 1 n ( 2 ) ρ n - c ( 2 ) [ B 2 n ( 2 ) - B 1 n ( 2 ) ] ρ n } K 1 ( ρ n ) ) cos ( γ n ) ,
σ z z = 2 λ ( 2 ) u 0 ( 2 ) + [ λ ( 2 ) + 2 μ ( 2 ) ] w 0 ( 2 ) + n ( { [ λ ( 2 ) + 2 μ ( 2 ) ] A 2 n ( 2 ) - λ ( 2 ) A 1 n ( 2 ) - 2 λ ( 2 ) c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] } I 0 ( ρ n ) + { - [ λ ( 2 ) + 2 μ ( 2 ) ] B 2 n ( 2 ) + λ ( 2 ) B 1 n ( 2 ) - 2 λ ( 2 ) c ( 2 ) [ B 1 n ( 2 ) - B 2 n ( 2 ) ] } K 0 ( ρ n ) + 2 μ ( 2 ) c ( 2 ) ρ n [ A 2 n ( 2 ) - A 1 n ( 2 ) ] I 1 ( ρ n ) - 2 μ ( 2 ) c ( 2 ) ρ n [ B 1 n ( 2 ) - B 2 n ( 2 ) ] K 1 ( ρ n ) ) cos ( γ n ) ,
σ z r = μ ( 2 ) n { [ A 1 n ( 2 ) + A 2 n ( 2 ) ] I 1 ( ρ n ) + [ B 1 n ( 2 ) + B 2 n ( 2 ) ] K 1 ( ρ n ) + 2 c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] ρ n I 0 ( ρ n ) - 2 c ( 2 ) [ B 2 n ( 2 ) - B 1 n ( 2 ) ] ρ n K 0 ( ρ n ) } sin ( γ n ) ,
c ( 2 ) = [ λ ( 2 ) + μ ( 2 ) ] / 2 [ λ ( 2 ) + 2 μ ( 2 ) ] , d ( 2 ) = μ ( 2 ) [ 2 λ ( 2 ) + 3 μ ( 2 ) ] / [ λ ( 2 ) + 2 μ ( 2 ) ] , e ( 2 ) = [ μ ( 2 ) ] 2 / [ λ ( 2 ) + 2 μ ( 2 ) ] .
k n = ( 2 n + 1 ) π / 2 L ,             n = 0 , 1 , 2 .
ρ n = k n r ,             γ n = k n z .
σ r r = - p at r = c , σ z z = 0 ( RM ) , or σ z z = - p ( HM ) , continuity of σ r r at r = b and r = a , continuity of u at r = b and r = a ,
w 0 ( 2 ) z = n α n ( 2 ) sin ( k n z ) ,
α n ( 2 ) = 4 w 0 ( 2 ) ( - 1 ) n L k n 2 ,
w = n { α n ( 2 ) + A 2 n ( 2 ) k n I 0 ( ρ n ) - B 2 n ( 2 ) k n K 0 ( ρ n ) + c ( 2 ) [ A 2 n ( 2 ) - A 1 n ( 2 ) ] r I 1 ( ρ n ) + c ( 2 ) [ B 2 n ( 2 ) - B 1 n ( 2 ) ] r K 1 ( ρ n ) } sin ( γ n )
σ r r = - p at r = c , continuity of σ r r at r = b and r = a , σ z r = 0 at r = c , continuity of σ z r at r = b and r = a , continuity of u and w at r = b and r = c .
u = u 0 ( 2 ) r + u 1 ( 2 ) / r ,
w = w 0 z ,
σ r r = 2 [ λ ( 2 ) + μ ( 2 ) ] u 0 ( 2 ) + λ ( 2 ) w 0 - 2 μ ( 2 ) u 1 ( 2 ) / r 2 ,
σ z z = 2 λ ( 2 ) u 0 ( 2 ) + [ λ ( 2 ) + 2 μ ( 2 ) ] w 0 .
π ( c 2 - b 2 ) { 2 λ ( 2 ) u 0 ( 2 ) + [ λ ( 2 ) + 2 μ ( 2 ) ] w 0 } + π ( b 2 - a 2 ) { 2 λ ( 1 ) u 0 ( 1 ) + [ λ ( 1 ) + 2 μ ( 1 ) ] w 0 } + π a 2 { 2 λ ( 0 ) u 0 ( 0 ) + [ λ ( 0 ) + 2 μ ( 0 ) ] w 0 } = O ( SRM ) = - p ( SHM ) .
σ r r = - p at r = c , continuity of σ r r at r = b and r = a , continuity of u at r = b and r = a .
Δ ϕ ϕ = e z - ½ n 2 [ 2 e r ( p 11 - p 44 ) + e z ( p 11 - 2 p 44 ) ] ,
Δ ϕ ϕ = 1 2 L { - L + L e z d z - n 2 2 [ 2 ( p 11 - p 44 ) - L + L e r d z + ( p 11 - 2 p 44 ) - L + L e z d z ] } .
I 0 ( ρ n ) 1 ,             I 1 ( ρ n ) ½ ρ n ,             ρ n 0 ,
e r = u r = u 0 ( 0 ) + n { [ c 0 ( 0 ) - ½ ] A 1 n ( 0 ) - c 0 ( 0 ) A 2 n ( 0 ) } cos ( γ n ) ,
e z = w z = w 0 ( 0 ) + n A 2 n ( 0 ) cos ( γ n ) .
- L + L cos ( k n z ) d z = 2 k n ( - 1 ) n .
Δ ϕ ϕ = [ 1 - n 2 2 ( p 11 - 2 p 44 ) ] [ w 0 ( 0 ) + n A 2 n ( 0 ) ( - 1 ) n / k n L ] - n 2 ( p 11 - p 44 ) ( u 0 ( 0 ) + n ( - 1 ) n k n L { c 0 ( 0 ) [ A 1 n ( 0 ) - A z n ( 0 ) ] - ½ A 1 n ( 0 ) } .
| Δ ϕ ϕ ( n + 1 ) - Δ ϕ ϕ ( n ) Δ ϕ ϕ ( n ) | < 10 - 3 .
σ = 0.5 - E / 6 K .
w ( 1 ) = - p E ( 1 ) [ 1 - 2 σ ( 1 ) ] ( SHM ) ,             or             w ( 1 ) = 2 p σ ( 1 ) E ( 1 ) ,
π ( b 2 - a 2 ) σ z z ( 1 ) + π a 2 σ z z ( 0 ) = 0.
w - w ( 1 ) σ z z ( 1 ) E ( 1 ) ,
w - w ( 0 ) σ z z ( 0 ) E ( 0 ) .
w - w ( 0 ) - p { 1 - 2 σ ( 0 ) E ( 1 ) - [ 1 - 2 σ ( 0 ) ] E ( 0 ) } [ 1 + a 2 ( b 2 - a 2 ) E ( 0 ) E ( 1 ) ] ,
w - w ( 0 ) p [ 2 σ ( 1 ) E ( 1 ) - 2 σ ( 0 ) E ( 0 ) ] [ 1 + a 2 ( b 2 - a 2 ) E ( 0 ) E ( 1 ) ] .

Metrics