## Abstract

For the fused tapering optical fibers, a physics–mathematics equation of dynamic-shape curves is
deduced. The closed-form solution of the equation can be acquired by use of its
initial, boundary, and volume-conserved conditions. The shape curves in different tapering
processes can be described by the solution conveniently, accurately, and integratively.
A large quantity of experimental data has been provided and shown to have a coincidence with
the theoretical results properly. Especially, the initial length of the fused zone is known, and the
solution can predict the whole tapering shape curve with an arbitrary elongating length.

© 2006 Optical Society of America

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### Equations (16)

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(1)
$$\mathrm{\Delta}{V}_{R}=\pi \left\{\left[{R}^{2}({x}^{\prime}+\mathrm{\Delta}{x}^{\prime})-{R}^{2}\left({x}^{\prime}\right)\right]/2+{R}^{2}\left({x}^{\prime}\right)\right\}\mathrm{\Delta}{x}^{\prime}\mathrm{.}$$
(2)
$$\mathrm{\Delta}{V}_{R}=\pi {R}^{2}\left({x}^{\prime}\right)\mathrm{\Delta}{x}^{\prime}\mathrm{.}$$
(3)
$$\mathrm{\Delta}{V}_{f}=\pi {L}_{f}\left\{{R}^{2}\left(x\right)-[R{\left(x\right)-\mathrm{\Delta}R\left(x\right)]}^{2}\right\}/2;$$
(4)
$$\mathrm{\Delta}{V}_{f}=\pi {L}_{f}R\left(x\right){R}^{\prime}\left(x\right)\mathrm{\Delta}x\mathrm{.}$$
(5)
$${R}^{2}\left({x}^{\prime}\right)={L}_{f}R\left(x\right){R}^{\prime}\left(x\right)\mathrm{.}$$
(6)
$$R\left({x}^{\prime}\right)=R\left(x\right)\mathrm{exp}\left[i(ax+b)\right];$$
(7)
$$\frac{{R}^{\prime}\left(x\right)}{R\left(x\right)}=\frac{1}{{L}_{f}}\text{\hspace{0.17em}}\mathrm{exp}\left[2i\left(ax+b\right)\right]\mathrm{.}$$
(8)
$$R\left(x\right){|}_{x={L}_{t}/2}={R}_{0},$$
(9)
$${R}^{\prime}\left(x\right){|}_{x={L}_{t}/2}=0,$$
(10)
$${R}^{\prime}\left(x\right){|}_{x=0}=0,$$
(11)
$$2\pi {\int}_{0}^{{L}_{t}/2}{R}^{2}\left(x\right)\mathrm{d}x=\pi {{R}_{0}}^{2}{L}_{f0}\mathrm{.}$$
(12)
$$R\left(x\right)=c\text{\hspace{0.17em}}\mathrm{exp}\left\{\frac{-i}{2ad\left({L}_{s}\right){L}_{f0}}\text{\hspace{0.17em}}\mathrm{exp}\left[2i(ax+b)\right]\right\};$$
(13)
$$R\left(x\right)=c\text{\hspace{0.17em}}\mathrm{exp}\left[\frac{1}{2ad\left({L}_{s}\right){L}_{f0}}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}2(ax+b)\right]\mathrm{.}$$
(14)
$$c={R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left[\frac{-1}{2ad\left({L}_{s}\right){L}_{f0}}\text{\hspace{0.17em}}\mathrm{sin}(a{L}_{t}+2b)\right],$$
(15)
$$a=\frac{\pi}{{L}_{t}},b=-\frac{\pi}{4}+n\pi ,n=0,\text{\xb1}1,\text{\xb1}2\text{, \hspace{0.17em}}\dots \text{\hspace{0.17em}}\mathrm{.}$$
(16)
$$R\left(x\right)={R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left\{\frac{-{L}_{t}}{2\pi d\left({L}_{s}\right){L}_{f0}}\text{\hspace{0.17em}}[1-\mathrm{sin}(\frac{2\pi x}{{L}_{t}}-\frac{\pi}{2})]\right\}\mathrm{.}$$