Generalized beam quality factor of aberrated truncated Gaussian laser beams: erratum

Cosmas Mafusire; Andrew Forbes

doi:10.1364/JOSAA.33.001111

Abstract

A correction is given for Eqs. (10a) and (10b) [J. Opt. Soc. Am. A 28, 1372 (2011) [CrossRef] ].

There were typographical errors in Eqs. (10a) and (10b) of this paper [1]. In both equations, the term in the square brackets of the denominator has been correctly raised to power 3. At the same time, in Eq. (10b) the round bracket after the term $B_{31}^{2}$ is correctly replaced by a square one and the coefficient of $B_{31} B_{33}$ is corrected by making it negative. Therefore the correct equations are supposed to be as follows:

M_{x}^{4} = \frac{1}{π {[\exp (2 γ^{2}) - 1]}^{3} γ^{8}} {24 π^{3} [B_{22}^{2} [\exp (2 γ^{2}) - 1] {[\exp (2 γ^{2}) - 2 γ^{2} - 1]}^{2} γ^{4} + 3 (B_{31}^{2} [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 2 γ^{2} - 1] [\exp (2 γ^{2}) - 1 - 2 (γ^{4} + γ^{2})] γ^{2} + A_{31}^{2} [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [5 \exp (4 γ^{2}) + 2 γ^{2} (γ^{2} + 1) - 2 \exp (2 γ^{2}) (9 γ^{4} + γ^{2} + 5) + 5] γ^{2} + 2 (B_{31} B_{33} + A_{31} A_{33}) [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [\exp (2 γ^{2}) - 2 (γ^{4} + γ^{2}) - 1] γ^{2} + [\exp (2 γ^{2}) - 1] {(B_{33}^{2} + A_{33}^{2}) [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [\exp (2 γ^{2}) - 1 - 2 (γ^{4} + γ^{2})] γ^{2} + 20 A_{40}^{2} [2 γ^{2} (γ^{2} + 2) + \exp (4 γ^{2}) - 2 \exp (2 γ^{2}) (2 γ^{6} - γ^{4} + 2 γ^{2} + 1) + 1]})] + [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] {π [\exp (2 γ^{2}) - 1 - 2 γ^{2}] + 32} γ^{8}},

REFERENCE

1. C. Mafusire and A. Forbes, “Generalized beam quality factor of aberrated truncated Gaussian laser beams,” J. Opt. Soc. Am. A 28, 1372–1378 (2011). [CrossRef]

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M_{x}^{4} = \frac{1}{π {[\exp (2 γ^{2}) - 1]}^{3} γ^{8}} {24 π^{3} [B_{22}^{2} [\exp (2 γ^{2}) - 1] {[\exp (2 γ^{2}) - 2 γ^{2} - 1]}^{2} γ^{4} + 3 (B_{31}^{2} [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 2 γ^{2} - 1] [\exp (2 γ^{2}) - 1 - 2 (γ^{4} + γ^{2})] γ^{2} + A_{31}^{2} [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [5 \exp (4 γ^{2}) + 2 γ^{2} (γ^{2} + 1) - 2 \exp (2 γ^{2}) (9 γ^{4} + γ^{2} + 5) + 5] γ^{2} + 2 (B_{31} B_{33} + A_{31} A_{33}) [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [\exp (2 γ^{2}) - 2 (γ^{4} + γ^{2}) - 1] γ^{2} + [\exp (2 γ^{2}) - 1] {(B_{33}^{2} + A_{33}^{2}) [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [\exp (2 γ^{2}) - 1 - 2 (γ^{4} + γ^{2})] γ^{2} + 20 A_{40}^{2} [2 γ^{2} (γ^{2} + 2) + \exp (4 γ^{2}) - 2 \exp (2 γ^{2}) (2 γ^{6} - γ^{4} + 2 γ^{2} + 1) + 1]})] + [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] {π [\exp (2 γ^{2}) - 1 - 2 γ^{2}] + 32} γ^{8}},

M_{y}^{4} = \frac{1}{π {[\exp (2 γ^{2}) - 1]}^{3} γ^{8}} {24 π^{3} [B_{22}^{2} [\exp (2 γ^{2}) - 1] {[\exp (2 γ^{2}) - 1 - 2 γ^{2}]}^{2} γ^{4} + 3 (B_{31}^{2} [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [5 \exp (4 γ^{2}) + 2 γ^{2} (γ^{2} + 1) - 2 \exp (2 γ^{2}) (9 γ^{4} + γ^{2} + 5) + 5] γ^{2} - 2 B_{31} B_{33} [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [\exp (2 γ^{2}) - 1 - 2 (γ^{4} + γ^{2})] γ^{2} + [\exp (2 γ^{2}) - 1] (\exp (4 γ^{2}) {[B_{33}^{2} + {(A_{31} - A_{33})}^{2}] γ^{2} + 20 A_{40}^{2}} + γ^{2} {40 A_{40}^{2} (γ^{2} + 2) + [B_{33}^{2} + {(A_{31} - A_{33})}^{2}] [4 γ^{6} + 6 γ^{4} + 4 γ^{2} + 1]} - 2 \exp (2 γ^{2}) {[B_{33}^{2} + {(A_{31} - A_{33})}^{2}] {[γ^{3} + γ]}^{2} + 20 A_{40}^{2} (2 γ^{6} - γ^{4} + 2 γ^{2} + 1)] + 20 A_{40}^{2}})] + [\exp (2 γ^{2}) - 1] [\exp (2 γ^{2}) - 1 - 2 γ^{2}] [π (\exp (2 γ^{2}) - 1 - 2 γ^{2}) + 32] γ^{8}} .

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Journal of the Optical Society of America A