A laser beam quality definition based on induced temperature rise: erratum

Harold C. Miller

doi:10.1364/OE.21.005635

Abstract

An erratum is presented to correct an error in an equation used to determine laser beam quality from power-in-the-bucket curves.

In the recent paper on laser beam quality [1], the mathematical form of Eq. (19) in section 5 is incorrect as published. Equation (19) came from a straightforward substitution using Eq. (5), Eq. (17) and Eq. (18) in the reference. The resulting equation for the beam quality (BQ) is,

B Q = \frac{\int_{0}^{\infty} {(\frac{d P I B}{d θ} (θ))}_{i d e a l} \frac{d θ}{θ}}{\int_{0}^{\infty} {(\frac{d P I B}{d θ} (θ))}_{m e a s u r e d} \frac{d θ}{θ}} .

PIB(θ) is the power-in-the-bucket function and θ is the radial coordinate (expressed as a diffraction angle) in the far-field plane. One can perform a change-of-variables such that the integration variable is now the PIB. Since PIB(∞) = P, the total power in the beam, the limits of integration also change and Eq. (19) should correctly read,

B Q = \frac{\int_{0}^{P} \frac{d P I B}{θ {(P I B)}_{i d e a l}}}{\int_{0}^{P} \frac{d P I B}{θ {(P I B)}_{m e a s u r e d}}} .

The impact of the shapes of the measured and ideal far-field intensity distributions are now captured by the inverse function, θ(PIB), which can be read-off a measured or calculated PIB(θ) curve. The intention was to clearly show how the beam quality depends on the distribution of power in the far-field. Equation (1) above may be more useful than Eq. (2) from a numerical analysis perspective, although either equation can be used. The author regrets any confusion that may have been caused by the oversight.

Reference

1. H. C. Miller, “A laser beam quality definition based on induced temperature rise,” Opt. Express 20(27), 28819–28828 (2012). [CrossRef] [PubMed]

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B Q = \frac{\int_{0}^{\infty} {(\frac{d P I B}{d θ} (θ))}_{i d e a l} \frac{d θ}{θ}}{\int_{0}^{\infty} {(\frac{d P I B}{d θ} (θ))}_{m e a s u r e d} \frac{d θ}{θ}} .

B Q = \frac{\int_{0}^{P} \frac{d P I B}{θ {(P I B)}_{i d e a l}}}{\int_{0}^{P} \frac{d P I B}{θ {(P I B)}_{m e a s u r e d}}} .

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