## Abstract

We describe experimental results and a theoretical analysis for
propagation in graded-index multimode fiber when diode laser light is
launched into the lowest-order propagation modes and the fiber
undergoes severe bending perturbations. Experimentally, near-field
modal interference images and transmission loss measurements were
obtained for different loop diameters. The data indicate that, when
the fundamental mode is excited, the light remains in lowest-order
modes even for small bend diameters. This is consistent with
analysis which predicts that, in a parabolic-index multimode fiber
subject to constant diameter bending, the light tends to oscillate
between lowest-order modes and remains trapped therein rather
than diffusing to high-order modes. Implications of these results
for an intrusion-resistant communication system with graded-index
multimode fiber are discussed.

© 2000 Optical Society of America

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### Equations (22)

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(1)
$$n={n}_{1}-\mathrm{\Delta}n\left({x}^{2}+{y}^{2}\right)/{a}^{2},$$
(2)
$$\mathrm{NA}={\left(2{n}_{1}\mathrm{\Delta}n\right)}^{1/2}.$$
(3)
$${u}_{\mathit{ij}}={u}_{i}\left(x\right){u}_{j}\left(y\right),$$
(4)
$${u}_{i}\left(x\right)={N}_{i}{H}_{i}\left(\mathrm{\alpha}x\right)exp\left(-{\mathrm{\alpha}}^{2}{x}^{2}/2\right),$$
(5)
$$\mathrm{\alpha}={\left(2\mathrm{\pi}\mathrm{NA}/\mathrm{\lambda}a\right)}^{1/2}.$$
(6)
$${\mathrm{\beta}}_{\mathit{ij}}={\mathrm{\beta}}_{c}-\left(i+j+1\right)\mathrm{\Delta}\mathrm{\beta},$$
(7)
$${\mathrm{\beta}}_{c}=2\mathrm{\pi}{n}_{1}/\mathrm{\lambda},$$
(8)
$$\mathrm{\Delta}\mathrm{\beta}={\mathrm{\alpha}}^{2}/{\mathrm{\beta}}_{c}.$$
(9)
$$i+j+1<\mathrm{\pi}\mathrm{NA}a/\mathrm{\lambda}.$$
(11)
$$u\left(x,y\right)=\sum _{i,j}{A}_{\mathit{ij}}{u}_{\mathit{ij}}\left(x,y\right),$$
(12)
$${u}_{\mathit{ij}}\left(x,y\right)exp\left(-i{\mathrm{\beta}}_{c}x\mathrm{\varphi}\right)=u\left(x,y\right)$$
(13)
$$x=\left({a}^{+}+a\right)/\left(\mathrm{\alpha}\sqrt{2}\right),$$
(14)
$${\int}_{-\infty}^{\infty}{u}_{n}{\mathit{au}}_{m}\mathrm{d}x=\sqrt{m}{\mathrm{\delta}}_{m,n+1},$$
(15)
$${\int}_{-\infty}^{\infty}{u}_{n}{a}^{+}{u}_{m}\mathrm{d}x=\sqrt{m+1}{\mathrm{\delta}}_{m,n-1}.$$
(16)
$${A}_{\left(i+1\right)j}=i{\mathrm{\beta}}_{c}\mathrm{\varphi}\sqrt{i+1}/\left(\sqrt{2}\mathrm{\alpha}\right),$$
(17)
$${A}_{\left(i-1\right)j}=i{\mathrm{\beta}}_{c}\mathrm{\varphi}\sqrt{i}/\left(\sqrt{2}\mathrm{\alpha}\right),$$
(18)
$${A}_{\mathit{lm}}=0\mathrm{if}m\ne j\mathrm{or}l\u3008i-1\mathrm{or}l\u3009i+1.$$
(19)
$$\frac{\mathrm{d}{A}_{\mathit{ij}}}{\mathrm{d}z}=i{\mathrm{\kappa}}_{i,\left(i+1\right)}{A}_{\left(i+1\right)j}exp\left(i\mathrm{\Delta}z\right)+i{\mathrm{\kappa}}_{i,\left(i-1\right)}{A}_{\left(i-1\right)j}exp\left(-i\mathrm{\Delta}z\right),$$
(20)
$${\mathrm{\kappa}}_{i,\left(i+1\right)}=\sqrt{\left(i+1\right)}{\mathrm{\kappa}}_{1},$$
(21)
$${\mathrm{\kappa}}_{i,\left(i-1\right)}=\sqrt{i}{\mathrm{\kappa}}_{1},$$
(22)
$${\mathrm{\kappa}}_{1}={\mathrm{\beta}}_{c}/\left(\sqrt{2}\mathrm{\alpha}R\right).$$