## Abstract

We provide new expressions for equations that were incorrectly presented in a recent paper [Opt. Express **18**, 656 (2010)].

©2010 Optical Society of America

In our recent paper [1], we described in Eq. (3) a defocus phase function *S*(*ρ*) that included two constants *M* and *N*. The expressions for the constants, as presented in Eqs. (4) and (5) were incorrect. The correct expressions are:

(1)$$M=\frac{2}{3}\frac{{n}_{2}^{2}}{{\mathrm{NA}}^{2}}[1-{(1-\frac{{\mathrm{NA}}^{2}}{{n}_{2}^{2}})}^{\frac{3}{2}}],$$ (2)$$N={(1-{M}^{2}-\frac{1}{2}\frac{{\mathrm{NA}}^{2}}{{n}_{2}^{2}})}^{\frac{1}{2}}.$$ The modified spherical aberration function $\widehat{\Phi}$
_{SA} is then obtained as:

(3)$${\hat{\Phi}}_{\mathrm{SA}}\left(\rho \right)={\Phi}_{\mathrm{SA}}\left(\rho \right)-xS\left(\rho \right)$$ where *x* is chosen to minimise the RMS value of $\widehat{\Phi}$
_{SA}. We note that the other results presented in the paper were not affected by this editing error.

## References and links

**1. **A. Jesacher, G. D. Marshall, T. Wilson, and M. J. Booth, “Adaptive optics for direct laser writing with plasma emission aberration sensing,” Opt. Express **18**, 656–661 (2010). [CrossRef] [PubMed]

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### Equations (3)

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(1)
$$M=\frac{2}{3}\frac{{n}_{2}^{2}}{{\mathrm{NA}}^{2}}[1-{(1-\frac{{\mathrm{NA}}^{2}}{{n}_{2}^{2}})}^{\frac{3}{2}}],$$
(2)
$$N={(1-{M}^{2}-\frac{1}{2}\frac{{\mathrm{NA}}^{2}}{{n}_{2}^{2}})}^{\frac{1}{2}}.$$
(3)
$${\hat{\Phi}}_{\mathrm{SA}}\left(\rho \right)={\Phi}_{\mathrm{SA}}\left(\rho \right)-xS\left(\rho \right)$$