Convergence of the discrete dipole approximation. I. Theoretical analysis: erratum

Maxim A. Yurkin; Valeri P. Maltsev; Alfons G. Hoekstra

doi:10.1364/JOSAA.32.002407

Abstract

We report and address the errors in the analysis of the weighted discretization in Section 2.F of our published paper [J. Opt. Soc. Am. A 23, 2578 (2006) [CrossRef] ].

The following errors in the analysis of the weighted discretization in [1] resulted from the incorrect interpretation of this formulation of the discrete dipole approximation, originally proposed by Piller [2]. Moreover, this interpretation was not fully expressed in [1], which also cast ambiguity on the definition of the formulation. So we start with explicit statement of a correct interpretation, as corresponding to a numerical scheme,

E_{i} = E_{i}^{inc} + \sum_{j \neq i} {\bar{G}}_{i j}^{(0)} V_{j} {\bar{χ}}_{j}^{e} E_{j} + ({\bar{M}}_{i}^{e} {\bar{χ}}_{i}^{e} - {\bar{L}}_{i} χ_{i}) E_{i},

which is a direct cast of Eq. (13) in [2] into the notation of [1] and can be considered an extension of Eq. (10) in the latter.

The first error was in the multiplier of ${\bar{G}}^{s}$ in the second integral in Eq. (92) in [1], which also contained a typographical error (“ $r$ ” instead of “ $r_{i}$ ”). The correct expression is

{\bar{M}}_{i}^{e} {\bar{χ}}_{i}^{e} E_{i} = (\int_{V_{i}^{p}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) χ_{i}^{p} + \int_{V_{i}^{s}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) χ_{i}^{s} {\bar{T}}_{i} - {\bar{G}}^{s} (r_{i}, r^{'}) χ_{i}^{p})) E_{i},

where the left-hand side is updated according to the notation of Eq. (E1) to explicitly indicate that it is an approximation to the rigorous

M (V_{i}, r_{i})

given by Eq. (4) in [1]. The corrected Eq. (92) is equivalent to Eq. (9) in [2].

The second error was in the third line of Eq. (96) in [1] in a multiplier of $\bar{L}$ , cf. Eq. (E1). Correcting both errors also results in vanishing of the last two lines of Eq. (96). The final corrected expression is

h_{i i}^{s h} = (M (V_{i}, r_{i}) - \bar{L} (\partial V_{i}, r_{i}) P_{i}^{p}) - (\int_{V_{i}^{p}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) P_{i}^{p} + \int_{V_{i}^{s}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) P_{i}^{s} - \bar{L} (\partial V_{i}, r_{i}) P_{i}^{p}) = \int_{V_{i}^{p}} d^{3} r^{'} \bar{G} (r_{i}, r^{'}) (P (r^{'}) - P_{i}^{p}) + \int_{V_{i}^{s}} d^{3} r^{'} \bar{G} (r_{i}, r^{'}) (P (r^{'}) - P_{i}^{s}) .

Also, there was a typographic error in the expression in the beginning of a text line immediately before Eq. (96)—it should read

h_{ii}^{sh}

.

Then, it follows that the phrase “and the third one is transformed to $\bar{L}$ the same way as in expression (80)” after Eq. (96) should be removed, and Eq. (97) should read

| h_{ii}^{sh} | \leq c_{86} d .

The whole paragraph after Eq. (97) should be removed, i.e., the original weighted discretization effectively reduces shape errors and requires no further improvements. That is the main conclusion of this erratum.

Finally, Eq. (98) should read

{‖ h^{d} ‖}_{1} \leq \sum_{j \in \partial V} (\sum_{l = 1}^{K_{\max}} c_{84} n_{s} (l) l^{- 4} + c_{88} d) \leq c_{89} N d,

i.e., the term

c_{87}

is removed, but the total order of errors is unchanged.

There was also a sign error in Eq. (6); the correct one is

\bar{L} (δ V_{0}, r) = \oint_{δ V_{0}} d^{2} r^{'} \frac{{\hat{n}}^{'} \hat{R}}{R^{3}} .

All other parts of [1], including abstract and conclusions, remain unchanged and are not affected by this erratum.

Acknowledgment

We thank Olli S. Vartia and Ari Seppälä for a discussion that led us to discovery of the described errors.

REFERENCES

1. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23, 2578–2591 (2006). [CrossRef]

2. N. B. Piller, “Influence of the edge meshes on the accuracy of the coupled-dipole approximation,” Opt. Lett. 22, 1674–1676 (1997). [CrossRef]

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Equations (6)

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E_{i} = E_{i}^{inc} + \sum_{j \neq i} {\bar{G}}_{i j}^{(0)} V_{j} {\bar{χ}}_{j}^{e} E_{j} + ({\bar{M}}_{i}^{e} {\bar{χ}}_{i}^{e} - {\bar{L}}_{i} χ_{i}) E_{i},

{\bar{M}}_{i}^{e} {\bar{χ}}_{i}^{e} E_{i} = (\int_{V_{i}^{p}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) χ_{i}^{p} + \int_{V_{i}^{s}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) χ_{i}^{s} {\bar{T}}_{i} - {\bar{G}}^{s} (r_{i}, r^{'}) χ_{i}^{p})) E_{i},

h_{i i}^{s h} = (M (V_{i}, r_{i}) - \bar{L} (\partial V_{i}, r_{i}) P_{i}^{p}) - (\int_{V_{i}^{p}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) P_{i}^{p} + \int_{V_{i}^{s}} d^{3} r^{'} (\bar{G} (r_{i}, r^{'}) - {\bar{G}}^{s} (r_{i}, r^{'})) P_{i}^{s} - \bar{L} (\partial V_{i}, r_{i}) P_{i}^{p}) = \int_{V_{i}^{p}} d^{3} r^{'} \bar{G} (r_{i}, r^{'}) (P (r^{'}) - P_{i}^{p}) + \int_{V_{i}^{s}} d^{3} r^{'} \bar{G} (r_{i}, r^{'}) (P (r^{'}) - P_{i}^{s}) .

| h_{ii}^{sh} | \leq c_{86} d .

{‖ h^{d} ‖}_{1} \leq \sum_{j \in \partial V} (\sum_{l = 1}^{K_{\max}} c_{84} n_{s} (l) l^{- 4} + c_{88} d) \leq c_{89} N d,

\bar{L} (δ V_{0}, r) = \oint_{δ V_{0}} d^{2} r^{'} \frac{{\hat{n}}^{'} \hat{R}}{R^{3}} .

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Journal of the Optical Society of America A