## Abstract

In a recently published paper [Opt. Lett. **37**, 3636 (2012)], equations for the evolution of the beam quality in a parabolic index (PI) fiber were introduced. Use of those equations for the extraction of the ${M}^{2}$ parameter was erroneous, as an incorrect definition for ${M}^{2}$ was assumed. When defined correctly, ${M}^{2}$ in PI fibers is shown here to be constant. Nonetheless, the optimization of the power delivery properties of PI fibers is governed by the criterion introduced in the paper under discussion.

© 2013 Optical Society of America

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### Equations (7)

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(1)
$${M}^{2}=\sqrt{4\mathcal{B}{\sigma}_{x}^{2}(z)+{\mathcal{A}}^{2}},$$
(2)
$$\mathcal{A}={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}\mathrm{d}x\mathrm{d}y(x-{\u3008x\u3009}_{(z)})\phantom{\rule{0ex}{0ex}}\times [E(x,y,z)\xb7\frac{\partial {E}^{*}}{\partial x}(x,y,z)-\text{c.c.}],$$
(3)
$$\mathcal{B}={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}\mathrm{d}x\mathrm{d}y{|\frac{\partial E}{\partial x}(x,y,z)|}^{2}+\frac{1}{4}\{{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}\mathrm{d}x\mathrm{d}y\phantom{\rule{0ex}{0ex}}{[E(x,y,z)\xb7\frac{\partial {E}^{*}}{\partial x}(x,y,z)-\text{c.c.}]\}}^{2}.$$
(4)
$${H}_{m}^{\prime}(x)=2m{H}_{m-1}(x)$$
(5)
$${({M}_{x}^{2})}^{2}=4[{\sigma}_{x0}^{2}{\sigma}_{{k}_{x0}}^{2}-{(C+{x}_{0}{k}_{x0})}^{2}],$$
(6)
$${\sigma}_{x}^{2}(z)=\frac{{w}^{4}{\sigma}_{{k}_{x0}}^{2}}{4}\text{\hspace{0.17em}}{\mathrm{sin}}^{2}(\mathrm{\Delta}\beta z)+{\sigma}_{x0}^{2}\text{\hspace{0.17em}}{\mathrm{cos}}^{2}(\mathrm{\Delta}\beta z)\phantom{\rule{0ex}{0ex}}-\frac{{w}^{2}}{2}(C+{x}_{0}{k}_{x0})\mathrm{sin}(2\mathrm{\Delta}\beta z).$$
(7)
$$\frac{{w}^{4}{\sigma}_{{k}_{x0}}^{2}}{4}={\sigma}_{x0}^{2},$$