Boundary element method for surface nonlinear optics of nanoparticles: erratum

Jouni Mäkitalo; Saku Suuriniemi; Martti Kauranen

doi:10.1364/OE.21.010205

Abstract

We report a correction to the numerical procedure, in which the source vector lacked a factor 1/2 and the integration in Eq. (19) was incorrect. The errors are inconsequential for the main results.

We wish to point out small errors in the mathematical details of our paper (Ref. [1]). The errors are related to the technical details and are inconsequential for the main results.

The operators 𝒦_l involve an integration over a strong singularity, so that the integral exists by means of the Cauchy principal value:

\begin{array}{l} 𝒦_{l} f (r) & = \int_{\partial V_{l}} [\nabla^{'} G_{l} (r, r^{'})] \times f (r^{'}) d S^{'} \\ = \frac{1}{2} f (r) \times n_{l} + 𝒦_{l}^{'} f (r), \end{array}

where

𝒦_{l}^{'} f (r) = lim_{a \to 0} \int_{\partial V_{l} - D (r, a)} [\nabla^{'} G_{l} (r, r^{'})] \times f (r^{'}) d S^{'}

where D(r, a) is a disk of radius a around r and it is assumed that the boundary ∂V_l is smooth. Thus the proper way to write Eqs. (7) and (8) of [1] would be

\begin{array}{l} {(𝒟_{1} J_{1}^{S} + 𝒦_{1}^{'} M_{1}^{S} - 𝒟_{2} J_{2}^{S} - 𝒦_{2}^{'} M_{2}^{S})}_{tan} = - \frac{1}{2 ε^{'}} \nabla_{S} P_{n}^{S}, \\ {(- 𝒦_{1}^{'} J_{1}^{S} + \frac{1}{η_{1}^{2}} 𝒟_{1} M_{1}^{S} + 𝒦_{2}^{'} J_{2}^{S} - \frac{1}{η_{2}^{2}} 𝒟_{2} M_{2}^{S})}_{tan} = - i \frac{1}{2} Ω P^{S} \times n . \end{array}

Although the Eqs. (7) and (8) are correct, the method of moments procedure should be applied to the operators involving the principal values. This was not noticed in the implementation of the numerical procedure. Consequently, the first two elements of the source vector b lacked a factor 1/2. This factor does not appear in the PMCHWT formulation for linear scattering due to the continuity of E _tan and H _tan.

Furthermore, there is a mistake in the source integral (19). For if $P_{n}^{S}$ is constant in each triangle, the integral still vanishes for RWG functions. This may be overcome by first expanding $\nabla_{S} P_{n}^{S}$ in RWG basis as

\nabla_{S} P_{n}^{S} = \sum_{l = 1}^{N} p_{l} f_{l} .

where p_l can be obtained in the same way as

b_{m}^{n 1}

by integration by parts. Then the coefficients

b_{m}^{n 2}

can be computed directly as

b_{m}^{n 2} = \frac{1}{ε^{'}} \sum_{l} p_{l} \int_{S_{m} \cap S_{l}} f_{m} \cdot n \times f_{l} d S,

where l runs over indices for which S_m ∩ S_l ≠ ∅. In [2], an alternative distributional approach / was presented for evaluating the coefficients

b_{m}^{n 1}

and

b_{m}^{n 2}

.

Finally, we point out a curious observation, that these two mistakes very accurately cancelled each other out in the numerical results, as is apparent from the comparison to the multipole solution. In general this cancellation is not to be expected. Through use of the corrected expressions we verified the validity of the results in Fig. 5.

References and links

1. J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19, 23386–23399 (2011) [CrossRef] [PubMed]

2. C. Forestiere, A. Capretti, and G. Miano, “Surface Integral Method for the Second Harmonic Generation in Metal Nanoparticles,” arXiv:1301.1880 [physics.optics] (2013)

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Equations (5)

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\begin{array}{l} 𝒦_{l} f (r) & = \int_{\partial V_{l}} [\nabla^{'} G_{l} (r, r^{'})] \times f (r^{'}) d S^{'} \\ = \frac{1}{2} f (r) \times n_{l} + 𝒦_{l}^{'} f (r), \end{array}

𝒦_{l}^{'} f (r) = lim_{a \to 0} \int_{\partial V_{l} - D (r, a)} [\nabla^{'} G_{l} (r, r^{'})] \times f (r^{'}) d S^{'}

\begin{array}{l} {(𝒟_{1} J_{1}^{S} + 𝒦_{1}^{'} M_{1}^{S} - 𝒟_{2} J_{2}^{S} - 𝒦_{2}^{'} M_{2}^{S})}_{tan} = - \frac{1}{2 ε^{'}} \nabla_{S} P_{n}^{S}, \\ {(- 𝒦_{1}^{'} J_{1}^{S} + \frac{1}{η_{1}^{2}} 𝒟_{1} M_{1}^{S} + 𝒦_{2}^{'} J_{2}^{S} - \frac{1}{η_{2}^{2}} 𝒟_{2} M_{2}^{S})}_{tan} = - i \frac{1}{2} Ω P^{S} \times n . \end{array}

\nabla_{S} P_{n}^{S} = \sum_{l = 1}^{N} p_{l} f_{l} .

b_{m}^{n 2} = \frac{1}{ε^{'}} \sum_{l} p_{l} \int_{S_{m} \cap S_{l}} f_{m} \cdot n \times f_{l} d S,

Abstract

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Equations (5)

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