Modal processing of Hartmann and Shack&#x2013;Hartmann patterns by means of a least squares fitting of the transverse aberrations: erratum

Geovanni Hernández-Gómez; Daniel Malacara-Doblado; Zacarías Malacara-Hernández; Daniel Malacara-Hernández

doi:10.1364/AO.54.001870

Abstract

In the previously published paper [Appl. Opt. 53, 7422 (2014) [CrossRef] ], we had an error in some equations as described below.

In our paper [1] we found an error in Eqs. (27)–(32); the correct expressions should be as follows, where ${TA}_{ρ}$ has been replaced at some places by $ρ$ :

Δ {TA}^{2} = {(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ})}^{2} + {(\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ})}^{2},

ε = \sum_{i = 1}^{N} [Δ {TA}^{2}] = \sum_{i = 1}^{N} [{(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ})}^{2} + {(\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ})}^{2}],

\frac{\partial ε}{\partial a_{k}} = \sum_{i = 1}^{N} [(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ}) \frac{\partial g_{k} (ρ, θ)}{\partial ρ} + \frac{1}{ρ} (\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ}) \frac{\partial g_{k} (ρ, θ)}{\partial θ}] = 0,

\sum_{i = 1}^{N} [\frac{\partial g_{k} (ρ, θ)}{\partial ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} + \frac{1}{ρ^{2}} \frac{\partial g_{k} (ρ, θ)}{\partial θ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ}] = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial ρ} \pm \frac{{TA}_{θ}}{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial θ}),

\sum_{j = 2}^{L} {\sum_{i = 1}^{N} (\frac{\partial g_{k} (ρ, θ)}{\partial ρ \partial a_{k}} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} + \frac{1}{ρ^{2}} \frac{\partial g_{k} (ρ, θ)}{\partial θ} \frac{\partial g_{j} (ρ, θ)}{\partial θ})} a_{j} = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial ρ} + \frac{{TA}_{θ}}{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial θ}),

\sum_{j = 2}^{L} {\sum_{i = 1}^{N} (n (k) n (j) ρ^{n (k) + n (j) - 2} {\begin{cases} \sin \\ \cos \end{cases}} l (k) θ \cdot {\begin{cases} \sin \\ \cos \end{cases}} l (j) θ + \frac{l (k) l (j)}{ρ^{2}} ρ^{n (k) + n (j)} {\begin{cases} \cos \\ - \sin \end{cases}} l (k) θ \cdot {\begin{cases} \cos \\ - \sin \end{cases}} l (j) θ)} a_{j} = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} n (k) ρ^{n (k) - 1} {\begin{cases} \sin \\ \cos \end{cases}} l (k) θ + \frac{{TA}_{θ}}{ρ} l (k) ρ^{n (k)} {\begin{cases} \cos \\ - \sin \end{cases}} l (k) θ) .

No other equation is affected, and any further results are not affected by this error.

Reference

1. G. Hernández-Gómez, D. Malacara-Doblado, Z. Malacara-Hernández, and D. Malacara-Hernández, “Modal processing of Hartmann and Shack–Hartmann patterns by means of a least squares fitting of the transverse aberrations,” Appl. Opt. 53, 7422–7434 (2014). [CrossRef]

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Equations (6)

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Δ {TA}^{2} = {(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ})}^{2} + {(\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ})}^{2},

ε = \sum_{i = 1}^{N} [Δ {TA}^{2}] = \sum_{i = 1}^{N} [{(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ})}^{2} + {(\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ})}^{2}],

\frac{\partial ε}{\partial a_{k}} = \sum_{i = 1}^{N} [(r_{w} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} - {TA}_{ρ}) \frac{\partial g_{k} (ρ, θ)}{\partial ρ} + \frac{1}{ρ} (\frac{r_{w}}{ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ} - {TA}_{θ}) \frac{\partial g_{k} (ρ, θ)}{\partial θ}] = 0,

\sum_{i = 1}^{N} [\frac{\partial g_{k} (ρ, θ)}{\partial ρ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} + \frac{1}{ρ^{2}} \frac{\partial g_{k} (ρ, θ)}{\partial θ} \sum_{j = 2}^{L} a_{j} \frac{\partial g_{j} (ρ, θ)}{\partial θ}] = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial ρ} \pm \frac{{TA}_{θ}}{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial θ}),

\sum_{j = 2}^{L} {\sum_{i = 1}^{N} (\frac{\partial g_{k} (ρ, θ)}{\partial ρ \partial a_{k}} \frac{\partial g_{j} (ρ, θ)}{\partial ρ} + \frac{1}{ρ^{2}} \frac{\partial g_{k} (ρ, θ)}{\partial θ} \frac{\partial g_{j} (ρ, θ)}{\partial θ})} a_{j} = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial ρ} + \frac{{TA}_{θ}}{ρ} \frac{\partial g_{k} (ρ, θ)}{\partial θ}),

\sum_{j = 2}^{L} {\sum_{i = 1}^{N} (n (k) n (j) ρ^{n (k) + n (j) - 2} {\begin{cases} \sin \\ \cos \end{cases}} l (k) θ \cdot {\begin{cases} \sin \\ \cos \end{cases}} l (j) θ + \frac{l (k) l (j)}{ρ^{2}} ρ^{n (k) + n (j)} {\begin{cases} \cos \\ - \sin \end{cases}} l (k) θ \cdot {\begin{cases} \cos \\ - \sin \end{cases}} l (j) θ)} a_{j} = \frac{1}{r_{w}} \sum_{i = 1}^{N} ({TA}_{ρ} n (k) ρ^{n (k) - 1} {\begin{cases} \sin \\ \cos \end{cases}} l (k) θ + \frac{{TA}_{θ}}{ρ} l (k) ρ^{n (k)} {\begin{cases} \cos \\ - \sin \end{cases}} l (k) θ) .

Modal processing of Hartmann and Shack–Hartmann patterns by means of a least squares fitting of the transverse aberrations: erratum

Abstract

Reference

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Equations (6)

Applied Optics