Discrete Dipole Approximation
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Discrete Dipole Approximation
In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles.

Discrete dipole approximation (DDA), also known as coupled dipole approximation,[1] is a method for computing scattering of radiation by particles of arbitrary shape and by periodic structures. Given a target of arbitrary geometry, one seeks to calculate its scattering and absorption properties by an approximation of the continuum target by a finite array of small polarizable dipoles. This technique is used in a variety of applications including nanophotonics, radar scattering, aerosol physics and astrophysics.

## Basic concepts

The basic idea of the DDA was introduced in 1964 by DeVoe[2] who applied it to study the optical properties of molecular aggregates; retardation effects were not included, so DeVoe's treatment was limited to aggregates that were small compared with the wavelength. The DDA, including retardation effects, was proposed in 1973 by Purcell and Pennypacker[3] who used it to study interstellar dust grains. Simply stated, the DDA is an approximation of the continuum target by a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles interact with one another via their electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation.[1][4]

Nature provides the physical inspiration for the DDA - in 1909 Lorentz[5] showed that the dielectric properties of a substance could be directly related to the polarizabilities of the individual atoms of which it was composed, with a particularly simple and exact relationship, the Clausius-Mossotti relation (or Lorentz-Lorenz), when the atoms are located on a cubic lattice. We may expect that, just as a continuum representation of a solid is appropriate on length scales that are large compared with the interatomic spacing, an array of polarizable points can accurately approximate the response of a continuum target on length scales that are large compared with the interdipole separation.

For a finite array of point dipoles the scattering problem may be solved exactly, so the only approximation that is present in the DDA is the replacement of the continuum target by an array of N-point dipoles. The replacement requires specification of both the geometry (location of the dipoles) and the dipole polarizabilities. For monochromatic incident waves the self-consistent solution for the oscillating dipole moments may be found; from these the absorption and scattering cross sections are computed. If DDA solutions are obtained for two independent polarizations of the incident wave, then the complete amplitude scattering matrix can be determined. Alternatively, the DDA can be derived from volume integral equation for the electric field.[6] This highlights that the approximation of point dipoles is equivalent to that of discretizing the integral equation, and thus decreases with decreasing dipole size.

With the recognition that the polarizabilities may be tensors, the DDA can readily be applied to anisotropic materials. The extension of the DDA to treat materials with nonzero magnetic susceptibility is also straightforward, although for most applications magnetic effects are negligible.

## Extensions

The method was improved by Draine, Flatau, and Goodman who applied fast Fourier transform to calculate convolution problem arising in the DDA which allowed to calculate scattering by large targets. They distributed discrete dipole approximation open source code DDSCAT.[7][8] There are now several DDA implementations,[6] extensions to periodic targets[9] and particles placed on or near a plane substrate.[10][11] and comparisons with exact technique were published.[12] Other aspects such as the validity criteria of the discrete dipole approximation[13] was published. The DDA was also extended to employ rectangular or cuboid dipoles [14] which is more efficient for highly oblate or prolate particles.

## Discrete dipole approximation codes

There are reviews[7][6] as well as published comparison of existing codes.[12] Most of the codes apply to arbitrary-shaped inhomogeneous nonmagnetic particles and particle systems in free space or homogeneous dielectric host medium. The calculated quantities typically include the Mueller matrices, integral cross-sections (extinction, absorption, and scattering), internal fields and angle-resolved scattered fields (phase function).

### General-purpose open-source DDA codes

These codes typically use regular grids (cubic or rectangular cuboid), conjugate gradient method to solve large system of linear equations, and FFT-acceleration of the matrix-vector products which uses convolution theorem. Complexity of this approach is almost linear in number of dipoles for both time and memory.[6]

Name Authors References Language Updated Features
DDSCAT Draine and Flatau [7] Fortran 2019 (v.7.3.3) Can also handle periodic particles and efficiently calculate near fields. Uses OpenMP acceleration.
VoxScatter Samuel Groth, Polimeridis and White [15] Matlab 2020 Contains preconditioning acceleration
IF-DDA Chaumet, A. Sentenac, Henry, D. Sentenac FORTRAN and graphical user interface written in Matlab 2020 Idiot friendly discrete dipole approximation. Code available on github.
DDscat.C++ Choliy [16] C++ 2017 (v.7.3.1) Version of DDSCAT translated to C++ with some further improvements.
ADDA Yurkin, Hoekstra, and contributors [17][18] C 2018 (v.1.4.0-alpha) Implements fast and rigorous consideration of a plane substrate, and allows rectangular-cuboid voxels for highly oblate or prolate particles. Can also calculate emission (decay-rate) enhancement of point emitters.Near-fields calculation is not very efficient. Uses Message Passing Interface (MPI) parallelization and can run on GPU (OpenCL).
OpenDDA McDonald [19][20] C 2009 (v.0.4.1) Uses both OpenMP and MPI parallelization. Focuses on computational efficiency.
DDA-GPU Kieß [21] C++ 2016 Runs on GPU (OpenCL). Algorithms are partly based on ADDA.
VIE-FFT Sha [22] C/C++ 2019 Also calculates near fields and material absorption. Named differently, but the algorithms are very similar to the ones used in the mainstream DDA.

### Specialized DDA codes

These list include codes that do not qualify for the previous section. The reasons may include the following: source code is not available, FFT acceleration is absent or reduced, the code focuses on specific applications not allowing easy calculation of standard scattering quantities.

Name Authors References Language Updated Features
DDSURF, DDSUB, DDFILM Schmehl, Nebeker, and Zhang [10][23][24] Fortran 2008 Rigorous handling of semi-infinite substrate and finite films (with arbitrary particle placement), but only 2D FFT acceleration is used.
DDMM Mackowski [25] Fortran 2002 Calculates T-matrix, which can then be used to efficiently calculate orientation-averaged scattering properties.
CDA McMahon [26] Matlab 2006
DDA-SI Loke [27] Matlab 2014 (v.0.2) Rigorous handling of substrate, but no FFT acceleration is used.
PyDDA Python 2015 Reimplementation of DDA-SI
e-DDA Vaschillo and Bigelow [28] Fortran 2019 (v.2.0) Simulates electron-energy loss spectroscopy and cathodoluminescence. Built upon DDSCAT 7.1.
DDEELS Geuquet, Guillaume and Henrard [29] Fortran 2013 (v.2.1) Simulates electron-energy loss spectroscopy and cathodoluminescence. Handles substrate through image approximation, but no FFT acceleration is used.
T-DDA Edalatpour [30] Fortran 2015 Simulates near-field radiative heat transfer. The computational bottleneck is direct matrix inversion (no FFT acceleration is used). Uses OpenMP and MPI parallelization.

## References

1. ^ a b Singham, Shermila B.; Salzman, Gary C. (1986). "Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation". The Journal of Chemical Physics. AIP Publishing. 84 (5): 2658-2667. doi:10.1063/1.450338. ISSN 0021-9606.
2. ^ DeVoe, Howard (1964-07-15). "Optical Properties of Molecular Aggregates. I. Classical Model of Electronic Absorption and Refraction". The Journal of Chemical Physics. AIP Publishing. 41 (2): 393-400. doi:10.1063/1.1725879. ISSN 0021-9606.
3. ^ E. M. Purcell; C. R. Pennypacker (1973). "Scattering and absorption of light by nonspherical dielectric grains". Astrophysical Journal. 186: 705. Bibcode:1973ApJ...186..705P. doi:10.1086/152538.
4. ^ Singham, Shermila Brito; Bohren, Craig F. (1987-01-01). "Light scattering by an arbitrary particle: a physical reformulation of the coupled dipole method". Optics Letters. The Optical Society. 12 (1): 10-12. doi:10.1364/ol.12.000010. ISSN 0146-9592.
5. ^ H. A. Lorentz, Theory of Electrons (Teubner, Leipzig, 1909)
6. ^ a b c d M. A. Yurkin; A. G. Hoekstra (2007). "The discrete dipole approximation: an overview and recent developments". Journal of Quantitative Spectroscopy and Radiative Transfer. 106 (1-3): 558-589. arXiv:0704.0038. Bibcode:2007JQSRT.106..558Y. doi:10.1016/j.jqsrt.2007.01.034.
7. ^ a b c Draine, B.T.; P.J. Flatau (1994). "Discrete dipole approximation for scattering calculations". J. Opt. Soc. Am. A. 11 (4): 1491-1499. Bibcode:1994JOSAA..11.1491D. doi:10.1364/JOSAA.11.001491.
8. ^ B. T. Draine; P. J. Flatau (2008). "The discrete dipole approximation for periodic targets: theory and tests". J. Opt. Soc. Am. A. 25 (11): 2693. arXiv:0809.0338. Bibcode:2008JOSAA..25.2693D. doi:10.1364/JOSAA.25.002693.
9. ^ Chaumet, Patrick C.; Rahmani, Adel; Bryant, Garnett W. (2003-04-02). "Generalization of the coupled dipole method to periodic structures". Physical Review B. American Physical Society (APS). 67 (16): 165404. arXiv:physics/0305051. doi:10.1103/physrevb.67.165404. ISSN 0163-1829.
10. ^ a b Schmehl, Roland; Nebeker, Brent M.; Hirleman, E. Dan (1997-11-01). "Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique". Journal of the Optical Society of America A. The Optical Society. 14 (11): 3026-3036. doi:10.1364/josaa.14.003026. ISSN 1084-7529.
11. ^ M. A. Yurkin; M. Huntemann (2015). "Rigorous and fast discrete dipole approximation for particles near a plane interface" (PDF). The Journal of Physical Chemistry C. 119 (52): 29088-29094. doi:10.1021/acs.jpcc.5b09271.
12. ^ a b Penttilä, Antti; Zubko, Evgenij; Lumme, Kari; Muinonen, Karri; Yurkin, Maxim A.; et al. (2007). "Comparison between discrete dipole implementations and exact techniques". Journal of Quantitative Spectroscopy and Radiative Transfer. Elsevier BV. 106 (1-3): 417-436. doi:10.1016/j.jqsrt.2007.01.026. ISSN 0022-4073.
13. ^ Zubko, Evgenij; Petrov, Dmitry; Grynko, Yevgen; Shkuratov, Yuriy; Okamoto, Hajime; et al. (2010-03-04). "Validity criteria of the discrete dipole approximation". Applied Optics. The Optical Society. 49 (8): 1267-1279. doi:10.1364/ao.49.001267. hdl:2115/50065. ISSN 0003-6935.
14. ^ D. A. Smunev; P. C. Chaumet; M. A. Yurkin (2015). "Rectangular dipoles in the discrete dipole approximation" (PDF). Journal of Quantitative Spectroscopy and Radiative Transfer. 156: 67-79. Bibcode:2015JQSRT.156...67S. doi:10.1016/j.jqsrt.2015.01.019.
15. ^ Groth, Samuel P and Polimeridis, Athanasios G and White, Jacob K (2020). "Accelerating the discrete dipole approximation via circulant preconditioning". Journal of Quantitative Spectroscopy and Radiative Transfer. 240: 106689.CS1 maint: multiple names: authors list (link)
16. ^ V. Y. Choliy (2013). "The discrete dipole approximation code DDscat.C++: features, limitations and plans". Adv. Astron. Space Phys. 3: 66-70. Bibcode:2013AASP....3...66C.
17. ^ M. A. Yurkin; V. P. Maltsev; A. G. Hoekstra (2007). "The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength" (PDF). J. Quant. Spectrosc. Radiat. Transfer. 106 (1-3): 546-557. arXiv:0704.0037. Bibcode:2007JQSRT.106..546Y. doi:10.1016/j.jqsrt.2007.01.033.
18. ^ M. A. Yurkin; A. G. Hoekstra (2011). "The discrete-dipole-approximation code ADDA: capabilities and known limitations" (PDF). J. Quant. Spectrosc. Radiat. Transfer. 112 (13): 2234-2247. Bibcode:2011JQSRT.112.2234Y. doi:10.1016/j.jqsrt.2011.01.031.
19. ^ J. McDonald; A. Golden; G. Jennings (2009). "OpenDDA: a novel high-performance computational framework for the discrete dipole approximation". Int. J. High Perf. Comp. Appl. 23 (1): 42-61. arXiv:0908.0863. Bibcode:2009arXiv0908.0863M. doi:10.1177/1094342008097914.
20. ^ J. McDonald (2007). OpenDDA - a novel high-performance computational framework for the discrete dipole approximation (PDF) (PhD). Galway: National University of Ireland. Archived from the original (PDF) on 2011-07-27.
21. ^ M. Zimmermann; A. Tausendfreund; S. Patzelt; G. Goch; S. Kieß; M. Z. Shaikh; M. Gregoire; S. Simon (2012). "In-process measuring procedure for sub-100 nm structures". J. Laser Appl. 24 (4): 042010. Bibcode:2012JLasA..24d2010Z. doi:10.2351/1.4719936.
22. ^ W. E. I. Sha; W. C. H. Choy; Y. P. Chen; W. C. Chew (2011). "Optical design of organic solar cell with hybrid plasmonic system". Opt. Express. 19 (17): 15908-15918. Bibcode:2011OExpr..1915908S. doi:10.1364/OE.19.015908. PMID 21934954.
23. ^ B. M. Nebeker (1998). Modeling of light scattering from features above and below surfaces using the discrete-dipole approximation (PhD). Tempe, AZ, USA: Arizona State University.
24. ^ E. Bae; H. Zhang; E. D. Hirleman (2008). "Application of the discrete dipole approximation for dipoles embedded in film". J. Opt. Soc. Am. A. 25 (7): 1728-1736. Bibcode:2008JOSAA..25.1728B. doi:10.1364/JOSAA.25.001728. PMID 18594631.
25. ^ D. W. Mackowski (2002). "Discrete dipole moment method for calculation of the T matrix for nonspherical particles". J. Opt. Soc. Am. A. 19 (5): 881-893. Bibcode:2002JOSAA..19..881M. doi:10.1364/JOSAA.19.000881. PMID 11999964.
26. ^ M. D. McMahon (2006). Effects of geometrical order on the linear and nonlinear optical properties of metal nanoparticles (PDF) (PhD). Nashville, TN, USA: Vanderbilt University.
27. ^ V. L. Y. Loke; P. M. Mengüç; Timo A. Nieminen (2011). "Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB". J. Quant. Spectrosc. Radiat. Transfer. 112 (11): 1711-1725. Bibcode:2011JQSRT.112.1711L. doi:10.1016/j.jqsrt.2011.03.012.
28. ^ N. W. Bigelow; A. Vaschillo; V. Iberi; J. P. Camden; D. J. Masiello (2012). "Characterization of the electron- and photon-driven plasmonic excitations of metal nanorods". ACS Nano. 6 (8): 7497-7504. doi:10.1021/nn302980u. PMID 22849410.
29. ^ N. Geuquet; L. Henrard (2010). "EELS and optical response of a noble metal nanoparticle in the frame of a discrete dipole approximation". Ultramicroscopy. 110 (8): 1075-1080. doi:10.1016/j.ultramic.2010.01.013.
30. ^ S. Edalatpour; M. ?uma; T. Trueax; R. Backman; M. Francoeur (2015). "Convergence analysis of the thermal discrete dipole approximation". Phys. Rev. E. 91 (6): 063307. arXiv:1502.02186. Bibcode:2015PhRvE..91f3307E. doi:10.1103/PhysRevE.91.063307. PMID 26172822.