## 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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### Equations (36)

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(1)
$$\frac{\mathrm{\Delta}\varphi}{\mathrm{\Delta}P}=\frac{L\beta}{E}(1-2\nu )\hspace{0.17em}\left[\frac{{n}^{2}}{2}(2{p}_{12}+{p}_{11})-1\right],$$
(2)
$${\sigma}_{r}=\frac{A}{{r}^{2}}+B(1+2\hspace{0.17em}\text{log}r)+2C,$$
(3)
$${\sigma}_{\theta}=-\frac{A}{{r}^{2}}+B(3+2\hspace{0.17em}\text{log}r)+2C,$$
(4)
$${\sigma}_{r}=\{\begin{array}{cc}\frac{A}{{r}^{2}}+2C& \text{region}\hspace{0.17em}2,\\ 2D& \text{region}\hspace{0.17em}1,\end{array}$$
(5)
$${\sigma}_{\theta}=\{\begin{array}{cc}-\frac{A}{{r}^{2}}+2C& \text{region}\hspace{0.17em}2,\\ 2D& \text{region}\hspace{0.17em}1,\end{array}$$
(6)
$${\sigma}_{z}=\{\begin{array}{ll}F\hfill & \text{region}\hspace{0.17em}2,\hfill \\ G\hfill & \text{region}\hspace{0.17em}1.\hfill \end{array}$$
(7)
$${\u220a}_{i}=(1/E)[{\sigma}_{i}-\nu ({\sigma}_{j}+{\sigma}_{k})],$$
(8)
$${\u220a}_{r}=\{\begin{array}{ll}\frac{1}{{E}_{1}}[2D(1-{\nu}_{1})-{\nu}_{1}G]\hfill & \text{region}\hspace{0.17em}1,\hfill \\ \frac{1}{{E}_{2}}\left[\frac{A}{{r}^{2}}(1+{\nu}_{2})+2C(1-{\nu}_{2})-{\nu}_{2}F\right]\hfill & \text{region}\hspace{0.17em}2,\hfill \end{array}$$
(9)
$${\u220a}_{z}=\{\begin{array}{ll}\frac{1}{{E}_{1}}(G-4{\nu}_{1}D)\hfill & \text{region}\hspace{0.17em}1,\hfill \\ \frac{1}{{E}_{2}}(F-4{\nu}_{2}C)\hfill & \text{region}\hspace{0.17em}2.\hfill \end{array}$$
(10)
$$u=\{\begin{array}{ll}\frac{1}{{E}_{1}}[2D(1-{\nu}_{1})-{\nu}_{1}G]r\hfill & \text{Region}\hspace{0.17em}1,\hfill \\ \frac{1}{{E}_{2}}\left[-\frac{(1+{\nu}_{2})A}{r}+2C(1-{\nu}_{2})r-{\nu}_{2}Fr\right]\hfill & \text{Region}\hspace{0.17em}2.\hfill \end{array}$$
(11)
$${\sigma}_{r2}({R}_{2})=-P,$$
(12)
$${\sigma}_{r1}({R}_{1})={\sigma}_{r2}({R}_{1}),$$
(13)
$${u}_{r1}({R}_{1})={u}_{r2}({R}_{1}),$$
(14)
$$-\pi {R}_{2}^{2}P=\pi {R}_{1}^{2}{\sigma}_{z1}+\pi ({R}_{2}^{2}-{R}_{1}^{2}){\sigma}_{z2},$$
(15)
$${\u220a}_{z1}={\u220a}_{z2},$$
(16)
$$\left(\frac{A}{{R}_{2}^{2}}\right)+2C=-P,$$
(17)
$${\left(\frac{{R}_{2}}{{R}_{1}}\right)}^{2}\left(\frac{A}{{R}_{2}^{2}}\right)+2C-2D=0,$$
(18)
$$\left(\frac{{E}_{1}}{{E}_{2}}\right)\left[-(1+{\nu}_{2})\hspace{0.17em}{\left(\frac{{R}_{2}}{{R}_{1}}\right)}^{2}\left(\frac{A}{{R}_{2}^{2}}\right)+2C(1-{\nu}_{2})-{\nu}_{2}F\right]+[{\nu}_{1}G-2D(1-{\nu}_{1})]=0,$$
(19)
$$G+\left[{\left(\frac{{R}_{2}}{{R}_{1}}\right)}^{2}-1\right]\hspace{0.17em}F=-{\left(\frac{{R}_{2}}{{R}_{1}}\right)}^{2}P,$$
(20)
$$G-4{\nu}_{1}D+\left(\frac{{E}_{1}}{{E}_{2}}\right)\hspace{0.17em}(4{\nu}_{2}C-F)=0.$$
(21)
$$\left[\begin{array}{ccccc}1& 2& 0& 0& 0\\ {R}^{2}& 2& -2& 0& 0\\ -E(1+{\nu}_{2}){R}^{2}& 2(1-{\nu}_{2})E& -2(1-{\nu}_{1})& -E{\nu}_{2}& {\nu}_{1}\\ 0& 0& 0& {R}^{2}-1& 1\\ 0& 4E{\nu}_{2}& -4{\nu}_{1}& -E& 1\end{array}\right]\hspace{0.17em}\left[\begin{array}{c}X\\ Y\\ Z\\ V\\ W\end{array}\right]=\left[\begin{array}{c}-1\\ 0\\ 0\\ -{R}_{2}\\ 0\end{array}\right].$$
(22)
$$\mathrm{\Delta}\varphi =\beta \mathrm{\Delta}L+L\mathrm{\Delta}\beta \cong \beta {\u220a}_{z}L+L{k}_{0}\mathrm{\Delta}n=\beta {\u220a}_{z}L-\xbdL{k}_{0}{n}^{3}\mathrm{\Delta}(1/{n}^{2}),$$
(23)
$$\mathrm{\Delta}{\left(\frac{1}{{n}^{2}}\right)}_{i}=\sum _{j=1}^{3}{p}_{ij}{\u220a}_{j},$$
(24)
$${p}_{ij}=\left[\begin{array}{ccc}{p}_{11}& {p}_{12}& {p}_{12}\\ {p}_{12}& {p}_{11}& {p}_{12}\\ {p}_{12}& {p}_{12}& {p}_{11}\end{array}\right].$$
(25)
$$\mathrm{\Delta}{\left(\frac{1}{{n}^{2}}\right)}_{x,y}=({p}_{11}+{p}_{12}){\u220a}_{r}+{p}_{12}{\u220a}_{z}.$$
(26)
$$\mathrm{\Delta}{\left(\frac{1}{{n}^{2}}\right)}_{x,y}=-\frac{\mathrm{\Delta}P{n}^{3}}{2{E}_{1}}\{({p}_{11}+{p}_{12})[2Z(1-{\nu}_{1})-{\nu}_{1}W]+{p}_{12}(W-4{\nu}_{1}Z)\},$$
(27)
$$\frac{\mathrm{\Delta}\varphi}{\mathrm{\Delta}P}\frac{{E}_{1}}{\beta L}=W\left\{1+\frac{{n}^{2}}{2}[({p}_{11}+{p}_{12}){\nu}_{1}-{p}_{12}]\right\}+Z\{-4{\nu}_{1}-{n}^{2}[({p}_{11}+{p}_{12})(1-{\nu}_{1})-2{\nu}_{1}{p}_{12}]\}.$$
(28)
$${\frac{\mathrm{\Delta}\varphi}{\mathrm{\Delta}P}\frac{{E}_{1}}{\beta L}|}_{\begin{array}{l}\text{fused}\\ \text{silica}\end{array}}=0.784\hspace{0.17em}W-1.173Z.$$
(29)
$${\frac{\mathrm{\Delta}\varphi}{\mathrm{\Delta}P}\frac{{E}_{1}}{\beta L}|}_{\begin{array}{l}\text{uncoaied}\\ \text{fused}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{silica}\end{array}}=-\mathrm{0.198.}$$
(31)
$$\left(\frac{A}{{{R}_{1}}^{2}}\right)+2C-2D=0,$$
(32)
$$\left(\frac{{E}_{1}}{{E}_{2}}\right)\hspace{0.17em}\left[-(1+{\nu}_{2})\hspace{0.17em}\left(\frac{A}{{R}_{1}^{2}}\right)+2C(1-{\nu}_{2})-{\nu}_{2}F\right]+[{\nu}_{1}G-2D(1-{\nu}_{1})]=0,$$
(34)
$$G-4{\nu}_{1}D+\left(\frac{{E}_{1}}{{E}_{2}}\right)\hspace{0.17em}(4{\nu}_{2}C-F)=0.$$
(36)
$$\left[\begin{array}{ccc}1& -2& 0\\ -E(1+{\nu}_{2})& -2(1-{\nu}_{1})& {\nu}_{1}\\ 0& -4{\nu}_{1}& 1\end{array}\right]\hspace{0.17em}\left[\begin{array}{c}X\\ Z\\ W\end{array}\right]=\left[\begin{array}{c}1\\ E(1-2{\nu}_{2})\\ E(2{\nu}_{2}-1)\end{array}\right].$$