Discrete dipole approximation

Last updated
In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles. Shape DDA1.svg
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.

Contents



Magnitude of the electric field strength E = | E - | {\displaystyle E=|{\vec {E}}|} (colored) and the Poynting vector (black arrows) in the near field of the vertically oriented dipole in the image plane. Blue/red colors indicate an electric field oriented downwards/upwards. DipoleRadiation.gif
Magnitude of the electric field strength (colored) and the Poynting vector (black arrows) in the near field of the vertically oriented dipole in the image plane. Blue/red colors indicate an electric field oriented downwards/upwards.

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 cubical 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.

There are several reviews of DDA method. [7] [6] [8] [9]

Extensions

The method was improved by Draine, Flatau, and Goodman, who applied the fast Fourier transform to solve fast convolution problems arising in the discrete dipole approximation (DDA). This allowed for the calculation of scattering by large targets. They distributed an open-source code DDSCAT. [7] [10] There are now several DDA implementations, [6] extensions to periodic targets, [11] and particles placed on or near a plane substrate. [12] [13] Comparisons with exact techniques have also been published. [14] Other aspects, such as the validity criteria of the discrete dipole approximation, were published. [15] The DDA was also extended to employ rectangular or cuboid dipoles, [16] which are more efficient for highly oblate or prolate particles.

Fast Fourier Transform for fast convolution calculations

The Fast Fourier Transform (FFT) method was introduced in 1991 by Goodman, Draine, and Flatau [17] for the discrete dipole approximation. They utilized a 3D FFT GPFA written by Clive Temperton. The interaction matrix was extended to twice its original size to incorporate negative lags by mirroring and reversing the interaction matrix. Several variants have been developed since then. Barrowes, Teixeira, and Kong [18] in 2001 developed a code that uses block reordering, zero padding, and a reconstruction algorithm, claiming minimal memory usage. McDonald, Golden, and Jennings [19] in 2009 used a 1D FFT code and extended the interaction matrix in the x, y, and z directions of the FFT calculations, suggesting memory savings due to this approach. This variant was also implemented in the MATLAB 2021 code by Shabaninezhad and Ramakrishna [20] . Other techniques to accelerate convolutions have been suggested in a general context [21] [22] along with faster evaluations of Fast Fourier Transforms arising in DDA problem solvers.

Conjugate gradient iteration schemes and preconditioning

Some of the early calculations of the polarization vector were based on direct inversion [3] and the implementation of the conjugate gradient method by Petravic and Kuo-Petravic. [23] Subsequently, many other conjugate gradient methods have been tested. [24] Advances in the preconditioning of linear systems of equations arising in the DDA setup have also been reported. [25]


Discrete dipole approximation codes

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). There are some published comparisons of existing DDA codes. [14]

General-purpose open-source DDA codes

These codes typically use regular grids (cubical 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]

NameAuthorsReferencesLanguageUpdatedFeatures
DDSCAT Draine and Flatau [7] Fortran2019 (v. 7.3.3)Can also handle periodic particles and efficiently calculate near fields. Uses OpenMP acceleration.
DDscat.C++ Choliy [26] C++2017 (v. 7.3.1)Version of DDSCAT translated to C++ with some further improvements.
ADDA Yurkin, Hoekstra, and contributors [27] [28] C2020 (v. 1.4.0)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] [29] C2009 (v. 0.4.1)Uses both OpenMP and MPI parallelization. Focuses on computational efficiency.
DDA-GPU Kieß [30] C++2016Runs on GPU (OpenCL). Algorithms are partly based on ADDA.
VIE-FFT Sha [31] C/C++2019Also calculates near fields and material absorption. Named differently, but the algorithms are very similar to the ones used in the mainstream DDA.
VoxScatter Groth, Polimeridis, and White [32] Matlab2019Uses circulant preconditioner for accelerating iterative solvers
IF-DDA Chaumet, Sentenac, and Sentenac [33] Fortran, GUI in C++ with Qt2021 (v. 0.9.19)Idiot-friendly DDA. Uses OpenMP and HDF5. Has a separate version (IF-DDAM) for multi-layered substrate.
MPDDA Shabaninezhad, Awan, and Ramakrishna [20] Matlab2021 (v. 1.0)Runs on GPU (using Matlab capabilities)

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.


NameAuthorsReferencesLanguageUpdatedFeatures
DDSURF, DDSUB, DDFILMSchmehl, Nebeker, and Zhang [12] [34] [35] Fortran2008Rigorous handling of semi-infinite substrate and finite films (with arbitrary particle placement), but only 2D FFT acceleration is used.
DDMMMackowski [36] Fortran2002Calculates T-matrix, which can then be used to efficiently calculate orientation-averaged scattering properties.
CDAMcMahon [37] Matlab2006
DDA-SI Loke [38] Matlab2014 (v. 0.2)Rigorous handling of substrate, but no FFT acceleration is used.
PyDDA DmitrievPython2015Reimplementation of DDA-SI
e-DDA Vaschillo and Bigelow [39] Fortran2019 (v. 2.0)Simulates electron-energy loss spectroscopy and cathodoluminescence. Built upon DDSCAT 7.1.
DDEELS Geuquet, Guillaume and Henrard [40] Fortran2013 (v. 2.1)Simulates electron-energy loss spectroscopy and cathodoluminescence. Handles substrate through image approximation, but no FFT acceleration is used.
T-DDAEdalatpour [41] Fortran2015Simulates near-field radiative heat transfer. The computational bottleneck is direct matrix inversion (no FFT acceleration is used). Uses OpenMP and MPI parallelization.
CDDARosales, Albella, González, Gutiérrez, and Moreno [42] 2021Applies to chiral systems (solves coupled equations for electric and magnetic fields)
PyDScat Yibin Jiang, Abhishek Sharma and Leroy Cronin [43] Python2023Simulates nanostructures undergoing structural transformation with GPU acceleration.

See also

Related Research Articles

<span class="mw-page-title-main">Rayleigh scattering</span> Light scattering by small particles

Rayleigh scattering, named after the 19th-century British physicist Lord Rayleigh, is the predominantly elastic scattering of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering medium, the amount of scattering is inversely proportional to the fourth power of the wavelength, e.g., a blue color is scattered much more than a red color as light propagates through air.

<span class="mw-page-title-main">Mie scattering</span> Scattering of an electromagnetic plane wave by a sphere

In electromagnetism, the Mie solution to Maxwell's equations describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the form of an infinite series of spherical multipole partial waves. It is named after German physicist Gustav Mie.

<span class="mw-page-title-main">Optical tweezers</span> Scientific instruments

Optical tweezers are scientific instruments that use a highly focused laser beam to hold and move microscopic and sub-microscopic objects like atoms, nanoparticles and droplets, in a manner similar to tweezers. If the object is held in air or vacuum without additional support, it can be called optical levitation.

<span class="mw-page-title-main">Zernike polynomials</span> Polynomial sequence

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging.

<span class="mw-page-title-main">Computational electromagnetics</span> Branch of physics

Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.

An atmospheric radiative transfer model, code, or simulator calculates radiative transfer of electromagnetic radiation through a planetary atmosphere.

Light scattering by particles is the process by which small particles scatter light causing optical phenomena such as the blue color of the sky, and halos.

Gans theory or Mie-Gans theory is the extension of Mie theory for the case of spheroidal particles. It gives the scattering characteristics of both oblate and prolate spheroidal particles much smaller than the excitation wavelength. Since it is a solution of the Maxwell equations it should technically not be called a theory.

<span class="mw-page-title-main">Bruce T. Draine</span> American astrophysicist

Bruce Thomas Draine is an American astrophysicist. He is Professor of Astrophysical Sciences at Princeton University.

GEISA - GEISA is a computer-accessible spectroscopic database, designed to facilitate accurate forward radiative transfer calculations using a line-by-line and layer-by-layer approach. It was started in 1974, at Laboratoire de Météorologie Dynamique (LMD) in France. GEISA is maintained by the ARA group at LMD for its scientific part and by the ETHER group at IPSL for its technical part. Currently, GEISA is involved in activities related to the assessment of the capabilities of IASI through the GEISA/IASI database derived from GEISA.

Multiangle light scattering (MALS) describes a technique for measuring the light scattered by a sample into a plurality of angles. It is used for determining both the absolute molar mass and the average size of molecules in solution, by detecting how they scatter light. A collimated beam from a laser source is most often used, in which case the technique can be referred to as multiangle laser light scattering (MALLS). The insertion of the word laser was intended to reassure those used to making light scattering measurements with conventional light sources, such as Hg-arc lamps that low-angle measurements could now be made. Until the advent of lasers and their associated fine beams of narrow width, the width of conventional light beams used to make such measurements prevented data collection at smaller scattering angles. In recent years, since all commercial light scattering instrumentation use laser sources, this need to mention the light source has been dropped and the term MALS is used throughout.

Codes for electromagnetic scattering by spheres - this article list codes for electromagnetic scattering by a homogeneous sphere, layered sphere, and cluster of spheres.

<span class="mw-page-title-main">Rigorous coupled-wave analysis</span> Semi-analytic method of computational electromagnetism

Rigorous coupled-wave analysis (RCWA), also known as Fourier modal method (FMM), is a semi-analytical method in computational electromagnetics that is most typically applied to solve scattering from periodic dielectric structures. It is a Fourier-space method so devices and fields are represented as a sum of spatial harmonics.

Differential dynamic microscopy (DDM) is an optical technique that allows performing light scattering experiments by means of a simple optical microscope. DDM is suitable for typical soft materials such as for instance liquids or gels made of colloids, polymers and liquid crystals but also for biological materials like bacteria and cells.

The Optical Stretcher is a dual-beam optical trap that is used for trapping and deforming ("stretching") micrometer-sized soft matter particles, such as biological cells in suspension. The forces used for trapping and deforming objects arise from photon momentum transfer on the surface of the objects, making the Optical Stretcher - unlike atomic force microscopy or micropipette aspiration - a tool for contact-free rheology measurements.

In the small wavelength limit, the total scattering cross-section of an impenetrable sphere is twice its geometrical cross-sectional area.

Atmospheric lidar is a class of instruments that uses laser light to study atmospheric properties from the ground up to the top of the atmosphere. Such instruments have been used to study, among other, atmospheric gases, aerosols, clouds, and temperature.

Rayleigh–Gans approximation, also known as Rayleigh–Gans–Debye approximation and Rayleigh–Gans–Born approximation, is an approximate solution to light scattering by optically soft particles. Optical softness implies that the relative refractive index of particle is close to that of the surrounding medium. The approximation holds for particles of arbitrary shape that are relatively small but can be larger than Rayleigh scattering limits.

<span class="mw-page-title-main">Method of moments (electromagnetics)</span> Numerical method in computational electromagnetics

The method of moments (MoM), also known as the moment method and method of weighted residuals, is a numerical method in computational electromagnetics. It is used in computer programs that simulate the interaction of electromagnetic fields such as radio waves with matter, for example antenna simulation programs like NEC that calculate the radiation pattern of an antenna. Generally being a frequency-domain method, it involves the projection of an integral equation into a system of linear equations by the application of appropriate boundary conditions. This is done by using discrete meshes as in finite difference and finite element methods, often for the surface. The solutions are represented with the linear combination of pre-defined basis functions; generally, the coefficients of these basis functions are the sought unknowns. Green's functions and Galerkin method play a central role in the method of moments.

Akira Ishimaru is a Japanese-American electrical engineer and professor emeritus at Department of Electrical and Computer Engineering at University of Washington. He is best known for his contributions to the theory of wave scattering in random media.

References

  1. 1 2 Singham, Shermila B.; Salzman, Gary C. (1986). "Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation". J. Chem. Phys. 84 (5). AIP Publishing: 2658–2667. Bibcode:1986JChPh..84.2658S. doi:10.1063/1.450338.
  2. ↑ DeVoe, Howard (1964-07-15). "Optical Properties of Molecular Aggregates. I. Classical Model of Electronic Absorption and Refraction". J. Chem. Phys. 41 (2). AIP Publishing: 393–400. Bibcode:1964JChPh..41..393D. doi:10.1063/1.1725879.
  3. 1 2 E. M. Purcell; C. R. Pennypacker (1973). "Scattering and absorption of light by nonspherical dielectric grains". Astrophys. J. 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". Opt. Lett. 12 (1). The Optical Society: 10–12. Bibcode:1987OptL...12...10S. doi:10.1364/ol.12.000010. PMID   19738776.
  5. ↑ H. A. Lorentz, Theory of Electrons (Teubner, Leipzig, 1909)
  6. 1 2 3 4 M. A. Yurkin; A. G. Hoekstra (2007). "The discrete dipole approximation: an overview and recent developments" (PDF). J. Quant. Spectrosc. Radiat. Transfer. 106 (1–3): 558–589. arXiv: 0704.0038 . Bibcode:2007JQSRT.106..558Y. doi:10.1016/j.jqsrt.2007.01.034. S2CID   119572857.
  7. 1 2 3 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. ↑ Yurkin, Maxim A. (2023). "Discrete Dipole Approximation". Light, Plasmonics and Particles. Elsevier. pp. 167–198.
  9. ↑ Chaumet, Patrick Christian (2022). "The discrete dipole approximation: A review". Mathematics. 10 (17). MDPI: 3049. doi: 10.3390/math10173049 .
  10. ↑ B. T. Draine; P. J. Flatau (2008). "The discrete dipole approximation for periodic targets: theory and tests". J. Opt. Soc. Am. A. 25 (11): 2693–3303. arXiv: 0809.0338 . Bibcode:2008JOSAA..25.2693D. doi:10.1364/JOSAA.25.002693. PMID   18978846. S2CID   15747060.
  11. ↑ Chaumet, Patrick C.; Rahmani, Adel; Bryant, Garnett W. (2003-04-02). "Generalization of the coupled dipole method to periodic structures". Phys. Rev. B. 67 (16). American Physical Society (APS): 165404. arXiv: physics/0305051 . Bibcode:2003PhRvB..67p5404C. doi:10.1103/physrevb.67.165404. S2CID   26726283.
  12. 1 2 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". J. Opt. Soc. Am. A. 14 (11). The Optical Society: 3026–3036. Bibcode:1997JOSAA..14.3026S. doi:10.1364/josaa.14.003026.
  13. ↑ 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.
  14. 1 2 Penttilä, Antti; Zubko, Evgenij; Lumme, Kari; Muinonen, Karri; Yurkin, Maxim A.; et al. (2007). "Comparison between discrete dipole implementations and exact techniques". J. Quant. Spectrosc. Radiat. Transfer. 106 (1–3). Elsevier BV: 417–436. Bibcode:2007JQSRT.106..417P. doi:10.1016/j.jqsrt.2007.01.026.
  15. ↑ Zubko, Evgenij; Petrov, Dmitry; Grynko, Yevgen; Shkuratov, Yuriy; Okamoto, Hajime; et al. (2010-03-04). "Validity criteria of the discrete dipole approximation". Appl. Opt. 49 (8). The Optical Society: 1267–1279. Bibcode:2010ApOpt..49.1267Z. doi:10.1364/ao.49.001267. hdl: 2115/50065 . PMID   20220882.
  16. ↑ D. A. Smunev; P. C. Chaumet; M. A. Yurkin (2015). "Rectangular dipoles in the discrete dipole approximation" (PDF). J. Quant. Spectrosc. Radiat. Transfer. 156: 67–79. Bibcode:2015JQSRT.156...67S. doi:10.1016/j.jqsrt.2015.01.019.
  17. ↑ Goodman, John J.; Draine, Bruce T.; Flatau, Piotr J. (1991). "Application of fast-Fourier-transform techniques to the discrete-dipole approximation". Optics Letters. 16 (15). Optica Publishing Group: 1198–1200.
  18. ↑ Barrowes, B. E.; Teixeira, F. L.; Kong, J. A. (2001). "Fast algorithm for matrix–vector multiply of asymmetric multilevel block‐Toeplitz matrices in 3‐D scattering". Microwave and Optical Technology Letters. 31 (1): 28–32.
  19. 1 2 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. S2CID   10285783.
  20. 1 2 M. Shabaninezhad; M. G. Awan; G. Ramakrishna (2021). "MATLAB package for discrete dipole approximation by graphics processing unit: Fast Fourier Transform and Biconjugate Gradient". J. Quant. Spectrosc. Radiat. Transfer. 262: 107501. Bibcode:2021JQSRT.26207501S. doi:10.1016/j.jqsrt.2020.107501. S2CID   233839571.
  21. ↑ Fu, Daniel Y; Kumbong, Hermann; Nguyen, Eric; Ré, Christopher (2023). "FlashFFTConv: Efficient Convolutions for Long Sequences with Tensor Cores". arXiv: 2311.05908 [cs.LG].
  22. ↑ Bowman, John C.; Roberts, Malcolm (2011). "Efficient dealiased convolutions without padding". SIAM Journal on Scientific Computing. 33 (1). SIAM: 386–406. arXiv: 1008.1366 . Bibcode:2011SJSC...33..386B. doi:10.1137/100787933.
  23. ↑ Petravic, M.; Kuo-Petravic, G. (1979). "An ILUCG algorithm which minimizes in the euclidean norm". Journal of Computational Physics. 32 (2): 263–269.
  24. ↑ Chaumet, Patrick C. (2024). "A comparative study of efficient iterative solvers for the discrete dipole approximation". Journal of Quantitative Spectroscopy and Radiative Transfer. 312. Elsevier. Bibcode:2024JQSRT.31208816C. doi:10.1016/j.jqsrt.2023.108816. S2CID   264805146.
  25. ↑ Chaumet, Patrick C.; Maire, Guillaume; Sentenac, Anne (2023). "Accelerating the discrete dipole approximation by initializing with a scalar solution and using a circulant preconditioning". Journal of Quantitative Spectroscopy and Radiative Transfer. 298. Elsevier. Bibcode:2023JQSRT.29808505C. doi:10.1016/j.jqsrt.2023.108505.
  26. ↑ 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.
  27. ↑ 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. S2CID   119574693.
  28. ↑ 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.
  29. ↑ J. McDonald (2007). OpenDDA - a novel high-performance computational framework for the discrete dipole approximation (PDF) (PhD). Galway: National University of Ireland.
  30. ↑ 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 .
  31. ↑ 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.
  32. ↑ S. P. Groth; A.G. Polimeridis; J.K. White (2020). "Accelerating the discrete dipole approximation via circulant preconditioning". J. Quant. Spectrosc. Radiat. Transfer. 240: 106689. Bibcode:2020JQSRT.24006689G. doi:10.1016/j.jqsrt.2019.106689. S2CID   209969404.
  33. ↑ P. C. Chaumet; D. Sentenac; G. Maire; T. Zhang; A. Sentenac (2021). "IFDDA, an easy-to-use code for simulating the field scattered by 3D inhomogeneous objects in a stratified medium: tutorial". J. Opt. Soc. Am. A. 38 (12): 1841–1852. Bibcode:2021JOSAA..38.1841C. doi: 10.1364/JOSAA.432685 .
  34. ↑ 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.
  35. ↑ 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.
  36. ↑ 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.
  37. ↑ 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.
  38. ↑ 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.
  39. ↑ 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.
  40. ↑ 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.
  41. ↑ S. Edalatpour; M. Čuma; 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. S2CID   21556373.
  42. ↑ S. A. Rosales; P. Albella; F. González; Y. Gutierrez; F. Moreno (2021). "CDDA: extension and analysis of the discrete dipole approximation for chiral systems". Opt. Express. 29 (19): 30020–30034. Bibcode:2021OExpr..2930020R. doi: 10.1364/OE.434061 . hdl: 10902/24774 . PMID   34614734.
  43. ↑ Jiang, Yibin; Sharma, Abhishek; Cronin, Leroy (2023). "An Accelerated Method for Investigating Spectral Properties of Dynamically Evolving Nanostructures". The Journal of Physical Chemistry Letters. 14 (16): 3929–3938. doi:10.1021/acs.jpclett.3c00395. PMC   10150391 . PMID   37078273.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.