Modeling the evolution of spatial beam parameters in parabolic index fibers: erratum

Roey Zuitlin; Yariv Shamir; Yoav Sintov; Mark Shtaif

doi:10.1364/OL.38.001067

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.

Equation (1) in [1] defines the $x$ component of the beam parameter product and of $M^{2}$ . That definition is only valid when the beam’s effective radius of curvature is infinite at the fiber’s output facet. The correct definition, which takes into account the beam’s effective radius, is [2]

M^{2} = \sqrt{4 B σ_{x}^{2} (z) + A^{2}},

where

A = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y (x - {〈 x 〉}_{(z)}) \times [E (x, y, z) \cdot \frac{\partial E^{*}}{\partial x} (x, y, z) - c.c.],

B = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y {| \frac{\partial E}{\partial x} (x, y, z) |}^{2} + \frac{1}{4} {\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y {[E (x, y, z) \cdot \frac{\partial E^{*}}{\partial x} (x, y, z) - c.c.]}}^{2} .

Performing the same procedure as in [1] and taking advantage of the property

H_{m}^{'} (x) = 2 m H_{m - 1} (x)

produces the following expression for

M^{2}

in a parabolic index (PI) fiber:

{(M_{x}^{2})}^{2} = 4 [σ_{x 0}^{2} σ_{k_{x 0}}^{2} - {(C + x_{0} k_{x 0})}^{2}],

where the parameters

A

,

B

,

C

,

x_{0}

,

k_{x 0}

,

σ_{x 0}^{2}

,

σ_{k_{x 0}}^{2}

, and

Δ β

are defined in [1]. The fact that

M^{2}

remains constant in the process of propagation is consistent with the fact that a PI fiber can be represented in terms of an ABCD matrix, and hence it should preserve

M^{2}

. The second term on the right-handside of Eq. (5) accounts for the fact that the waist of the incident beam does not necessarily coincide with the fiber’s input facet (where

σ_{x 0}^{2}

and

σ_{k_{x 0}}^{2}

are evaluated). When PI fibers are used to deliver high-power beams, a focusing effect of the beam to a small width may occur inside the fiber, leading to destructive power densities [3]. Extracting

A

and

B

from the parameters of the incident beam, as was done after Eq. (14) of [1], we express the beam’s root-mean-square width inside the fiber as

σ_{x}^{2} (z) = \frac{w^{4} σ_{k_{x 0}}^{2}}{4} \sin^{2} (Δ β z) + σ_{x 0}^{2} \cos^{2} (Δ β z) - \frac{w^{2}}{2} (C + x_{0} k_{x 0}) \sin (2 Δ β z) .

As we discussed previously, the third term on the right-hand side of Eq. (6) can be eliminated by properly focusing the incident beam. The first and second terms exhibit sinusoidal fluctuations between the values

(1 / 4) w^{4} σ_{k_{x 0}}^{2}

and

σ_{x 0}^{2}

, and the beam’s minimal width is the smaller of these two values. A plausible criterion for an optimized PI-fiber-based delivery system is to require that

\frac{w^{4} σ_{k_{x 0}}^{2}}{4} = σ_{x 0}^{2},

in which case the width of the beam remains constant during propagation in the fiber. This is exactly the same criterion that was defined in Eq. (16) of [1].

To conclude, we pointed out an inaccuracy in the definition of $M^{2}$ used in [1]. The analysis reported in [1] was shown to be valuable for determining the evolution of the beamwidth in the fiber and for establishing a criterion for which the beamwidth remains constant. This criterion helps in avoiding destructive processes resulting from the buildup of high power densities in the process of propagation.

References

1. R. Zuitlin, Y. Shamir, Y. Sintov, and M. Shtaif, Opt. Lett. 37, 3636 (2012). [CrossRef]

2. H. Yoda, P. Polynkin, and M. Mansuripur, J. Lightwave Technol. 24, 1350 (2006). [CrossRef]

3. S. G. Krivoshlykov and W. Neuberger, “Power laser delivery fiber system with enhanced damage threshold,” U.S. patent 5,557,701 (September 17, 1996).

Previous Article Next Article

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

M^{2} = \sqrt{4 B σ_{x}^{2} (z) + A^{2}},

A = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y (x - {〈 x 〉}_{(z)}) \times [E (x, y, z) \cdot \frac{\partial E^{*}}{\partial x} (x, y, z) - c.c.],

B = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y {| \frac{\partial E}{\partial x} (x, y, z) |}^{2} + \frac{1}{4} {\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y {[E (x, y, z) \cdot \frac{\partial E^{*}}{\partial x} (x, y, z) - c.c.]}}^{2} .

H_{m}^{'} (x) = 2 m H_{m - 1} (x)

{(M_{x}^{2})}^{2} = 4 [σ_{x 0}^{2} σ_{k_{x 0}}^{2} - {(C + x_{0} k_{x 0})}^{2}],

σ_{x}^{2} (z) = \frac{w^{4} σ_{k_{x 0}}^{2}}{4} \sin^{2} (Δ β z) + σ_{x 0}^{2} \cos^{2} (Δ β z) - \frac{w^{2}}{2} (C + x_{0} k_{x 0}) \sin (2 Δ β z) .

\frac{w^{4} σ_{k_{x 0}}^{2}}{4} = σ_{x 0}^{2},

Abstract

References

Cited By

Equations (7)

Optics Letters