  In this paper, we construct a special kind of breather solution of the
nonlinear Schr\"{o}dinger (NLS) equation, the so-called breather-positon ({\it
b-positon} for short), which can be obtained by taking the limit $\lambda_{j}$
$\rightarrow$ $\lambda_{1}$ of the Lax pair eigenvalues in the order-$n$
periodic solution which is generated by the $n$-fold Darboux transformation
from a special "seed" solution--plane wave. Further, an order-$n$ {\it
b-positon} gives an order-$n$ rogue wave under a limit $\lambda_1\rightarrow
\lambda_0$. Here $\lambda_0$ is a special eigenvalue in a breather of the NLS
equation such that its period goes to infinity. Several analytical plots of
order-2 breather confirm visually this double degeneration. The last limit in
this double degeneration can be realized approximately in an optical fiber
governed by the NLS equation, in which an injected initial ideal pulse is
created by a frequency comb system and a programable optical filter (wave
shaper) according to the profile of an analytical form of the {\it b-positon}
at a certain position $z_0$. We also suggest a new way to observe higher-order
rogue waves generation in an optical fiber, namely, measure the patterns at the
central region of the higher-order {\it b-positon} generated by above ideal
initial pulses when $\lambda_1$ is very close to the $\lambda_0$. The excellent
agreement between the numerical solutions generated from initial ideal inputs
with a low signal noise ratio and analytical solutions of order-2 {\it
b-positon}, supports strongly this way in a realistic optical fiber system. Our
results also show the validity of the generating mechanism of a higher-order
rogue waves from a multi-breathers through the double degeneration.
