188:
reversing the interaction matrix. Several variants have been developed since then. Barrowes, Teixeira, and Kong in 2001 developed a code that uses block reordering, zero padding, and a reconstruction algorithm, claiming minimal memory usage. McDonald, Golden, and
Jennings 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. Other techniques to accelerate convolutions have been suggested in a general context along with faster evaluations of Fast Fourier Transforms arising in DDA problem solvers.
772:
784:
156:
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
152:(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.
73:
17:
179:
near a plane substrate. Comparisons with exact techniques have also been published. Other aspects, such as the validity criteria of the discrete dipole approximation, were published. The DDA was also extended to employ rectangular or cuboid dipoles, which are more efficient for highly oblate or prolate particles.
134:
The basic idea of the DDA was introduced in 1964 by DeVoe 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,
155:
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
196:
Some of the early calculations of the polarization vector were based on direct inversion and the implementation of the conjugate gradient method by
Petravic and Kuo-Petravic. Subsequently, many other conjugate gradient methods have been tested. Advances in the preconditioning of linear systems of
178:
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. There are now several DDA implementations, extensions to periodic targets, and particles placed on or
187:
The Fast
Fourier Transform (FFT) method was introduced in 1991 by Goodman, Draine, and Flatau 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
139:
and
Pennypacker 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
163:
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.
39:
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
771:
234:
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.
147:
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
2613:
1270:
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".
213:
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
1769:
Chaumet, Patrick C.; Maire, Guillaume; Sentenac, Anne (2023). "Accelerating the discrete dipole approximation by initializing with a scalar solution and using a circulant preconditioning".
783:
1586:
M. Shabaninezhad; M. G. Awan; G. Ramakrishna (2021). "MATLAB package for discrete dipole approximation by graphics processing unit: Fast
Fourier Transform and Biconjugate Gradient".
1841:
Moncada-Villa, E.; Cuevas, J. C. (2022). "Thermal discrete dipole approximation for near-field radiative heat transfer in many-body systems with arbitrary nonreciprocal bodies".
1343:
Penttilä, Antti; Zubko, Evgenij; Lumme, Kari; Muinonen, Karri; Yurkin, Maxim A.; et al. (2007). "Comparison between discrete dipole implementations and exact techniques".
119:
1307:
222:(extinction, absorption, and scattering), internal fields and angle-resolved scattered fields (phase function). There are some published comparisons of existing DDA codes.
2361:
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".
1006:
718:
Simulates near-field radiative heat transfer. The computational bottleneck is direct matrix inversion (no FFT acceleration is used). Uses OpenMP and MPI parallelization.
1378:
Zubko, Evgenij; Petrov, Dmitry; Grynko, Yevgen; Shkuratov, Yuriy; Okamoto, Hajime; et al. (2010-03-04). "Validity criteria of the discrete dipole approximation".
160:. This highlights that the approximation of point dipoles is equivalent to that of discretizing the integral equation, and thus decreases with decreasing dipole size.
1970:
1500:
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".
1910:
1432:
339:
Implements fast and rigorous consideration of a plane substrate, and allows rectangular-cuboid voxels for highly oblate or prolate particles. Can also calculate
125:(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.
946:
Singham, Shermila Brito; Bohren, Craig F. (1987-01-01). "Light scattering by an arbitrary particle: a physical reformulation of the coupled dipole method".
205:
Thermal discrete dipole approximation is an extension of the original DDA to simulations of near-field heat transfer between 3D arbitrarily-shaped objects.
1473:
Goodman, John J.; Draine, Bruce T.; Flatau, Piotr J. (1991). "Application of fast-Fourier-transform techniques to the discrete-dipole approximation".
838:
Singham, Shermila B.; Salzman, Gary C. (1986). "Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation".
2307:
777:
Scattering by periodic structures such as slabs, gratings, of periodic cubes placed on a surface, can be solved in the discrete dipole approximation.
699:
Simulates electron-energy loss spectroscopy and cathodoluminescence. Handles substrate through image approximation, but no FFT acceleration is used.
1629:
Fu, Daniel Y; Kumbong, Hermann; Nguyen, Eric; Ré, Christopher (2023). "FlashFFTConv: Efficient
Convolutions for Long Sequences with Tensor Cores".
2618:
2326:
V. L. Y. Loke; P. M. Mengüç; Timo A. Nieminen (2011). "Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB".
1530:
J. McDonald; A. Golden; G. Jennings (2009). "OpenDDA: a novel high-performance computational framework for the discrete dipole approximation".
1804:
Edalatpour, S.; Čuma, M.; Trueax, T.; Backman, R.; Francoeur, M. (2015). "Convergence analysis of the thermal discrete dipole approximation".
512:
acceleration is absent or reduced, the code focuses on specific applications not allowing easy calculation of standard scattering quantities.
508:
These list include codes that do not qualify for the previous section. The reasons may include the following: source code is not available,
1214:
Chaumet, Patrick C.; Rahmani, Adel; Bryant, Garnett W. (2003-04-02). "Generalization of the coupled dipole method to periodic structures".
815:
810:
2396:
N. Geuquet; L. Henrard (2010). "EELS and optical response of a noble metal nanoparticle in the frame of a discrete dipole approximation".
2423:
S. Edalatpour; M. Čuma; T. Trueax; R. Backman; M. Francoeur (2015). "Convergence analysis of the thermal discrete dipole approximation".
873:
DeVoe, Howard (1964-07-15). "Optical
Properties of Molecular Aggregates. I. Classical Model of Electronic Absorption and Refraction".
2121:
S. P. Groth; A.G. Polimeridis; J.K. White (2020). "Accelerating the discrete dipole approximation via circulant preconditioning".
157:
2598:
800:
2166:"IFDDA, an easy-to-use code for simulating the field scattered by 3D inhomogeneous objects in a stratified medium: tutorial"
2220:
E. Bae; H. Zhang; E. D. Hirleman (2008). "Application of the discrete dipole approximation for dipoles embedded in film".
1726:
Chaumet, Patrick C. (2024). "A comparative study of efficient iterative solvers for the discrete dipole approximation".
426:
and material absorption. Named differently, but the algorithms are very similar to the ones used in the mainstream DDA.
20:
In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles.
2263:
D. W. Mackowski (2002). "Discrete dipole moment method for calculation of the T matrix for nonspherical particles".
1911:"The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength"
149:
911:
E. M. Purcell; C. R. Pennypacker (1973). "Scattering and absorption of light by nonspherical dielectric grains".
423:
348:
344:
282:
231:
2603:
2012:
1153:
B. T. Draine; P. J. Flatau (2008). "The discrete dipole approximation for periodic targets: theory and tests".
555:
Rigorous handling of semi-infinite substrate and finite films (with arbitrary particle placement), but only 2D
219:
2207:
Modeling of light scattering from features above and below surfaces using the discrete-dipole approximation
2031:
M. Zimmermann; A. Tausendfreund; S. Patzelt; G. Goch; S. Kieß; M. Z. Shaikh; M. Gregoire; S. Simon (2012).
556:
509:
175:
472:
Idiot-friendly DDA. Uses OpenMP and HDF5. Has a separate version (IF-DDAM) for multi-layered substrate.
2497:
2442:
2335:
2272:
2229:
2177:
2130:
2085:
2044:
1985:
1935:
1891:
1778:
1735:
1669:
1595:
1549:
1447:
1387:
1352:
1280:
1233:
1172:
1079:
1031:
955:
920:
882:
847:
136:
2545:"An Accelerated Method for Investigating Spectral Properties of Dynamically Evolving Nanostructures"
789:
Scattering by infinite object (such as cylinder) can be solved in the discrete dipole approximation.
79:
2608:
2309:
Effects of geometrical order on the linear and nonlinear optical properties of metal nanoparticles
2466:
2432:
2146:
1951:
1925:
1850:
1813:
1751:
1659:
1630:
1611:
1565:
1539:
1249:
1223:
1196:
1162:
1047:
1021:
2014:
OpenDDA - a novel high-performance computational framework for the discrete dipole approximation
1070:
Draine, B.T.; P.J. Flatau (1994). "Discrete dipole approximation for scattering calculations".
140:
electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation.
2574:
2525:
2458:
2378:
2288:
2245:
2103:
1650:
Bowman, John C.; Roberts, Malcolm (2011). "Efficient dealiased convolutions without padding".
1413:
1188:
971:
582:, which can then be used to efficiently calculate orientation-averaged scattering properties.
2564:
2556:
2515:
2505:
2450:
2405:
2370:
2343:
2280:
2237:
2185:
2138:
2093:
2052:
1993:
1943:
1860:
1823:
1786:
1743:
1708:
1696:
1677:
1603:
1557:
1509:
1482:
1455:
1403:
1395:
1360:
1322:
1288:
1241:
1180:
1133:
1087:
1039:
963:
928:
890:
855:
674:
Simulates electron-energy loss spectroscopy and cathodoluminescence. Built upon DDSCAT 7.1.
215:
579:
171:
144:
122:
2569:
2544:
2501:
2446:
2339:
2276:
2233:
2181:
2134:
2089:
2048:
1989:
1939:
1895:
1782:
1739:
1673:
1599:
1553:
1451:
1391:
1356:
1284:
1237:
1176:
1083:
1035:
959:
924:
886:
851:
1879:
340:
41:
2486:"CDDA: extension and analysis of the discrete dipole approximation for chiral systems"
1308:"Rigorous and fast discrete dipole approximation for particles near a plane interface"
736:
Applies to chiral systems (solves coupled equations for electric and magnetic fields)
679:
651:
2592:
2150:
1955:
1755:
1712:
1615:
1051:
757:
Simulates nanostructures undergoing structural transformation with GPU acceleration.
48:
2470:
1880:"The discrete dipole approximation code DDscat.C++: features, limitations and plans"
1569:
1253:
1200:
2409:
452:
60:
1971:"The discrete-dipole-approximation code ADDA: capabilities and known limitations"
2560:
2347:
2142:
1997:
1947:
1864:
1790:
1747:
1607:
1459:
1364:
1043:
2454:
1827:
1245:
380:
Uses both OpenMP and MPI parallelization. Focuses on computational efficiency.
805:
294:
72:
36:
1561:
1326:
477:
431:
2284:
2241:
1292:
1184:
1091:
2578:
2529:
2462:
2382:
2292:
2249:
2107:
1417:
1192:
975:
360:
2190:
2165:
2098:
2073:
1486:
1399:
1138:
1121:
967:
230:
These codes typically use regular grids (cubical or rectangular cuboid),
2484:
S. A. Rosales; P. Albella; F. González; Y. Gutierrez; F. Moreno (2021).
1228:
1007:"The discrete dipole approximation: an overview and recent developments"
2520:
630:
406:
385:
56:
16:
2510:
2485:
2374:
2057:
2032:
1681:
1408:
894:
1513:
859:
741:
352:
319:
286:
44:
2164:
P. C. Chaumet; D. Sentenac; G. Maire; T. Zhang; A. Sentenac (2021).
605:
314:
Version of DDSCAT translated to C++ with some further improvements.
2437:
2074:"Optical design of organic solar cell with hybrid plasmonic system"
1855:
1818:
1635:
932:
1930:
1664:
1544:
1167:
1026:
71:
52:
15:
625:
Rigorous handling of substrate, but no FFT acceleration is used.
447:
Uses circulant preconditioner for accelerating iterative solvers
47:. This technique is used in a variety of applications including
143:
Nature provides the physical inspiration for the DDA - in 1909
281:
Can also handle periodic particles and efficiently calculate
261:
2072:
W. E. I. Sha; W. C. H. Choy; Y. P. Chen; W. C. Chew (2011).
197:
equations arising in the DDA setup have also been reported.
1771:
1728:
988:
H. A. Lorentz, Theory of Electrons (Teubner, Leipzig, 1909)
401:
Runs on GPU (OpenCL). Algorithms are partly based on ADDA.
2033:"In-process measuring procedure for sub-100 nm structures"
1697:"An ILUCG algorithm which minimizes in the euclidean norm"
1433:"Rectangular dipoles in the discrete dipole approximation"
1105:
Yurkin, Maxim A. (2023). "Discrete Dipole Approximation".
192:
Conjugate gradient iteration schemes and preconditioning
183:
Fast Fourier Transform for fast convolution calculations
2614:
Scattering, absorption and radiative transfer (optics)
2543:
Jiang, Yibin; Sharma, Abhishek; Cronin, Leroy (2023).
82:
1909:
M. A. Yurkin; V. P. Maltsev; A. G. Hoekstra (2007).
1431:D. A. Smunev; P. C. Chaumet; M. A. Yurkin (2015).
726:Rosales, Albella, González, Gutiérrez, and Moreno
113:
2315:(PhD). Nashville, TN, USA: Vanderbilt University.
2209:(PhD). Tempe, AZ, USA: Arizona State University.
1525:
1523:
1338:
1336:
1222:(16). American Physical Society (APS): 165404.
158:volume integral equation for the electric field
2020:(PhD). Galway: National University of Ireland.
1265:
1263:
746:Yibin Jiang, Abhishek Sharma and Leroy Cronin
1581:
1579:
1122:"The discrete dipole approximation: A review"
1000:
998:
996:
994:
906:
904:
833:
831:
8:
1065:
1063:
1061:
1481:(15). Optica Publishing Group: 1198–1200.
351:(MPI) parallelization and can run on GPU (
167:There are several reviews of DDA method.
2568:
2549:The Journal of Physical Chemistry Letters
2519:
2509:
2436:
2189:
2097:
2056:
1929:
1854:
1817:
1663:
1634:
1543:
1407:
1227:
1166:
1137:
1025:
106:
95:
94:
89:
81:
76:Magnitude of the electric field strength
1502:Microwave and Optical Technology Letters
517:
497:Runs on GPU (using Matlab capabilities)
347:calculation is not very efficient. Uses
237:
1695:Petravic, M.; Kuo-Petravic, G. (1979).
827:
767:
174:, Flatau, and Goodman, who applied the
1279:(11). The Optical Society: 3026–3036.
2328:J. Quant. Spectrosc. Radiat. Transfer
2123:J. Quant. Spectrosc. Radiat. Transfer
1978:J. Quant. Spectrosc. Radiat. Transfer
1969:M. A. Yurkin; A. G. Hoekstra (2011).
1918:J. Quant. Spectrosc. Radiat. Transfer
1588:J. Quant. Spectrosc. Radiat. Transfer
1440:J. Quant. Spectrosc. Radiat. Transfer
1386:(8). The Optical Society: 1267–1279.
1345:J. Quant. Spectrosc. Radiat. Transfer
1014:J. Quant. Spectrosc. Radiat. Transfer
1005:M. A. Yurkin; A. G. Hoekstra (2007).
482:Shabaninezhad, Awan, and Ramakrishna
226:General-purpose open-source DDA codes
201:Thermal discrete dipole approximation
7:
1652:SIAM Journal on Scientific Computing
816:Method of moments (electromagnetics)
811:Finite-difference time-domain method
1315:The Journal of Physical Chemistry C
1306:M. A. Yurkin; M. Huntemann (2015).
1120:Chaumet, Patrick Christian (2022).
324:Yurkin, Hoekstra, and contributors
209:Discrete dipole approximation codes
14:
954:(1). The Optical Society: 10–12.
341:emission (decay-rate) enhancement
1701:Journal of Computational Physics
846:(5). AIP Publishing: 2658–2667.
782:
770:
457:Chaumet, Sentenac, and Sentenac
1107:Light, Plasmonics and Particles
684:Geuquet, Guillaume and Henrard
2619:Computational electromagnetics
2410:10.1016/j.ultramic.2010.01.013
881:(2). AIP Publishing: 393–400.
801:Computational electromagnetics
436:Groth, Polimeridis, and White
114:{\displaystyle E=|{\vec {E}}|}
107:
100:
90:
1:
1532:Int. J. High Perf. Comp. Appl
1351:(1–3). Elsevier BV: 417–436.
1109:. Elsevier. pp. 167–198.
25:Discrete dipole approximation
1713:10.1016/0021-9991(79)90133-5
544:Schmehl, Nebeker, and Zhang
462:Fortran, GUI in C++ with Qt
35:, is a method for computing
33:coupled dipole approximation
2561:10.1021/acs.jpclett.3c00395
2348:10.1016/j.jqsrt.2011.03.012
2143:10.1016/j.jqsrt.2019.106689
1998:10.1016/j.jqsrt.2011.01.031
1948:10.1016/j.jqsrt.2007.01.033
1865:10.1103/PhysRevB.106.235430
1791:10.1016/j.jqsrt.2023.108505
1748:10.1016/j.jqsrt.2023.108816
1608:10.1016/j.jqsrt.2020.107501
1460:10.1016/j.jqsrt.2015.01.019
1365:10.1016/j.jqsrt.2007.01.026
1044:10.1016/j.jqsrt.2007.01.034
646:Reimplementation of DDA-SI
170:The method was improved by
2635:
2455:10.1103/PhysRevE.91.063307
1828:10.1103/PhysRevE.91.063307
1246:10.1103/physrevb.67.165404
150:Clausius-Mossotti relation
349:Message Passing Interface
232:conjugate gradient method
135:was proposed in 1973 by
1562:10.1177/1094342008097914
1327:10.1021/acs.jpcc.5b09271
2285:10.1364/JOSAA.19.000881
2242:10.1364/JOSAA.25.001728
1884:Adv. Astron. Space Phys
1293:10.1364/josaa.14.003026
1185:10.1364/JOSAA.25.002693
1092:10.1364/JOSAA.11.001491
220:integral cross-sections
2306:M. D. McMahon (2006).
2205:B. M. Nebeker (1998).
659:Vaschillo and Bigelow
559:acceleration is used.
541:DDSURF, DDSUB, DDFILM
176:fast Fourier transform
126:
115:
21:
2599:Computational science
1878:V. Y. Choliy (2013).
504:Specialized DDA codes
116:
75:
19:
2191:10.1364/JOSAA.432685
2099:10.1364/OE.19.015908
2011:J. McDonald (2007).
1658:(1). SIAM: 386–406.
1487:10.1364/OL.16.001198
1400:10.1364/ao.49.001267
1139:10.3390/math10173049
968:10.1364/ol.12.000010
80:
2502:2021OExpr..2930020R
2496:(19): 30020–30034.
2447:2015PhRvE..91f3307E
2340:2011JQSRT.112.1711L
2277:2002JOSAA..19..881M
2234:2008JOSAA..25.1728B
2182:2021JOSAA..38.1841C
2135:2020JQSRT.24006689G
2090:2011OExpr..1915908S
2084:(17): 15908–15918.
2049:2012JLasA..24d2010Z
1990:2011JQSRT.112.2234Y
1940:2007JQSRT.106..546Y
1896:2013AASP....3...66C
1783:2023JQSRT.29808505C
1740:2024JQSRT.31208816C
1674:2011SJSC...33..386B
1600:2021JQSRT.26207501S
1554:2009arXiv0908.0863M
1452:2015JQSRT.156...67S
1392:2010ApOpt..49.1267Z
1357:2007JQSRT.106..417P
1321:(52): 29088–29094.
1285:1997JOSAA..14.3026S
1238:2003PhRvB..67p5404C
1177:2008JOSAA..25.2693D
1084:1994JOSAA..11.1491D
1036:2007JQSRT.106..558Y
960:1987OptL...12...10S
925:1973ApJ...186..705P
887:1964JChPh..41..393D
852:1986JChPh..84.2658S
343:of point emitters.
2265:J. Opt. Soc. Am. A
2222:J. Opt. Soc. Am. A
2170:J. Opt. Soc. Am. A
1273:J. Opt. Soc. Am. A
1155:J. Opt. Soc. Am. A
1132:(17). MDPI: 3049.
1072:J. Opt. Soc. Am. A
266:Draine and Flatau
127:
121:(colored) and the
111:
22:
2555:(16): 3929–3938.
2511:10.1364/OE.434061
2375:10.1021/nn302980u
2334:(11): 1711–1725.
2176:(12): 1841–1852.
2058:10.2351/1.4719936
1984:(13): 2234–2247.
1843:Physical Review B
1806:Physical Review E
1682:10.1137/100787933
1161:(11): 2693–3303.
895:10.1063/1.1725879
764:Gallery of shapes
761:
760:
501:
500:
103:
31:), also known as
2626:
2583:
2582:
2572:
2540:
2534:
2533:
2523:
2513:
2481:
2475:
2474:
2440:
2420:
2414:
2413:
2404:(8): 1075–1080.
2393:
2387:
2386:
2369:(8): 7497–7504.
2358:
2352:
2351:
2323:
2317:
2316:
2314:
2303:
2297:
2296:
2260:
2254:
2253:
2228:(7): 1728–1736.
2217:
2211:
2210:
2202:
2196:
2195:
2193:
2161:
2155:
2154:
2118:
2112:
2111:
2101:
2069:
2063:
2062:
2060:
2028:
2022:
2021:
2019:
2008:
2002:
2001:
1975:
1966:
1960:
1959:
1933:
1924:(1–3): 546–557.
1915:
1906:
1900:
1899:
1875:
1869:
1868:
1858:
1838:
1832:
1831:
1821:
1801:
1795:
1794:
1766:
1760:
1759:
1723:
1717:
1716:
1692:
1686:
1685:
1667:
1647:
1641:
1640:
1638:
1626:
1620:
1619:
1583:
1574:
1573:
1547:
1527:
1518:
1517:
1514:10.1002/mop.1348
1497:
1491:
1490:
1470:
1464:
1463:
1437:
1428:
1422:
1421:
1411:
1375:
1369:
1368:
1340:
1331:
1330:
1312:
1303:
1297:
1296:
1267:
1258:
1257:
1231:
1211:
1205:
1204:
1170:
1150:
1144:
1143:
1141:
1117:
1111:
1110:
1102:
1096:
1095:
1078:(4): 1491–1499.
1067:
1056:
1055:
1029:
1020:(1–3): 558–589.
1011:
1002:
989:
986:
980:
979:
943:
937:
936:
908:
899:
898:
870:
864:
863:
860:10.1063/1.450338
835:
786:
774:
695:
670:
621:
518:
493:
468:
422:Also calculates
376:
335:
310:
277:
238:
216:Mueller matrices
120:
118:
117:
112:
110:
105:
104:
96:
93:
2634:
2633:
2629:
2628:
2627:
2625:
2624:
2623:
2604:Electrodynamics
2589:
2588:
2587:
2586:
2542:
2541:
2537:
2483:
2482:
2478:
2422:
2421:
2417:
2398:Ultramicroscopy
2395:
2394:
2390:
2360:
2359:
2355:
2325:
2324:
2320:
2312:
2305:
2304:
2300:
2262:
2261:
2257:
2219:
2218:
2214:
2204:
2203:
2199:
2163:
2162:
2158:
2120:
2119:
2115:
2071:
2070:
2066:
2030:
2029:
2025:
2017:
2010:
2009:
2005:
1973:
1968:
1967:
1963:
1913:
1908:
1907:
1903:
1877:
1876:
1872:
1840:
1839:
1835:
1803:
1802:
1798:
1768:
1767:
1763:
1725:
1724:
1720:
1694:
1693:
1689:
1649:
1648:
1644:
1628:
1627:
1623:
1585:
1584:
1577:
1529:
1528:
1521:
1499:
1498:
1494:
1472:
1471:
1467:
1435:
1430:
1429:
1425:
1377:
1376:
1372:
1342:
1341:
1334:
1310:
1305:
1304:
1300:
1269:
1268:
1261:
1229:physics/0305051
1213:
1212:
1208:
1152:
1151:
1147:
1119:
1118:
1114:
1104:
1103:
1099:
1069:
1068:
1059:
1009:
1004:
1003:
992:
987:
983:
945:
944:
940:
910:
909:
902:
872:
871:
867:
837:
836:
829:
824:
797:
790:
787:
778:
775:
766:
693:
668:
619:
515:
506:
491:
466:
374:
333:
308:
275:
228:
211:
203:
194:
185:
132:
123:Poynting vector
78:
77:
69:
66:
12:
11:
5:
2632:
2630:
2622:
2621:
2616:
2611:
2606:
2601:
2591:
2590:
2585:
2584:
2535:
2476:
2415:
2388:
2353:
2318:
2298:
2271:(5): 881–893.
2255:
2212:
2197:
2156:
2113:
2064:
2023:
2003:
1961:
1901:
1870:
1849:(23): 235430.
1833:
1796:
1761:
1718:
1707:(2): 263–269.
1687:
1642:
1621:
1575:
1519:
1492:
1475:Optics Letters
1465:
1423:
1370:
1332:
1298:
1259:
1206:
1145:
1112:
1097:
1057:
990:
981:
938:
933:10.1086/152538
900:
865:
826:
825:
823:
820:
819:
818:
813:
808:
803:
796:
793:
792:
791:
788:
781:
779:
776:
769:
765:
762:
759:
758:
755:
752:
749:
747:
744:
738:
737:
734:
731:
729:
727:
724:
720:
719:
716:
713:
710:
708:
705:
701:
700:
697:
690:
687:
685:
682:
676:
675:
672:
665:
662:
660:
657:
648:
647:
644:
641:
638:
636:
633:
627:
626:
623:
616:
613:
611:
608:
602:
601:
599:
596:
593:
591:
588:
584:
583:
576:
573:
570:
568:
565:
561:
560:
553:
550:
547:
545:
542:
538:
537:
534:
531:
528:
525:
522:
505:
502:
499:
498:
495:
488:
485:
483:
480:
474:
473:
470:
463:
460:
458:
455:
449:
448:
445:
442:
439:
437:
434:
428:
427:
420:
417:
414:
412:
409:
403:
402:
399:
396:
393:
391:
388:
382:
381:
378:
371:
368:
366:
363:
357:
356:
337:
330:
327:
325:
322:
316:
315:
312:
305:
302:
300:
297:
291:
290:
289:acceleration.
279:
272:
269:
267:
264:
258:
257:
254:
251:
248:
245:
242:
227:
224:
210:
207:
202:
199:
193:
190:
184:
181:
131:
130:Basic concepts
128:
109:
102:
99:
92:
88:
85:
13:
10:
9:
6:
4:
3:
2:
2631:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2596:
2594:
2580:
2576:
2571:
2566:
2562:
2558:
2554:
2550:
2546:
2539:
2536:
2531:
2527:
2522:
2517:
2512:
2507:
2503:
2499:
2495:
2491:
2487:
2480:
2477:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2439:
2434:
2431:(6): 063307.
2430:
2426:
2419:
2416:
2411:
2407:
2403:
2399:
2392:
2389:
2384:
2380:
2376:
2372:
2368:
2364:
2357:
2354:
2349:
2345:
2341:
2337:
2333:
2329:
2322:
2319:
2311:
2310:
2302:
2299:
2294:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2259:
2256:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2216:
2213:
2208:
2201:
2198:
2192:
2187:
2183:
2179:
2175:
2171:
2167:
2160:
2157:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2117:
2114:
2109:
2105:
2100:
2095:
2091:
2087:
2083:
2079:
2075:
2068:
2065:
2059:
2054:
2050:
2046:
2043:(4): 042010.
2042:
2038:
2037:J. Laser Appl
2034:
2027:
2024:
2016:
2015:
2007:
2004:
1999:
1995:
1991:
1987:
1983:
1979:
1972:
1965:
1962:
1957:
1953:
1949:
1945:
1941:
1937:
1932:
1927:
1923:
1919:
1912:
1905:
1902:
1897:
1893:
1889:
1885:
1881:
1874:
1871:
1866:
1862:
1857:
1852:
1848:
1844:
1837:
1834:
1829:
1825:
1820:
1815:
1812:(6): 063307.
1811:
1807:
1800:
1797:
1792:
1788:
1784:
1780:
1776:
1772:
1765:
1762:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1722:
1719:
1714:
1710:
1706:
1702:
1698:
1691:
1688:
1683:
1679:
1675:
1671:
1666:
1661:
1657:
1653:
1646:
1643:
1637:
1632:
1625:
1622:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1582:
1580:
1576:
1571:
1567:
1563:
1559:
1555:
1551:
1546:
1541:
1537:
1533:
1526:
1524:
1520:
1515:
1511:
1507:
1503:
1496:
1493:
1488:
1484:
1480:
1476:
1469:
1466:
1461:
1457:
1453:
1449:
1445:
1441:
1434:
1427:
1424:
1419:
1415:
1410:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1374:
1371:
1366:
1362:
1358:
1354:
1350:
1346:
1339:
1337:
1333:
1328:
1324:
1320:
1316:
1309:
1302:
1299:
1294:
1290:
1286:
1282:
1278:
1274:
1266:
1264:
1260:
1255:
1251:
1247:
1243:
1239:
1235:
1230:
1225:
1221:
1217:
1210:
1207:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1169:
1164:
1160:
1156:
1149:
1146:
1140:
1135:
1131:
1127:
1123:
1116:
1113:
1108:
1101:
1098:
1093:
1089:
1085:
1081:
1077:
1073:
1066:
1064:
1062:
1058:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1023:
1019:
1015:
1008:
1001:
999:
997:
995:
991:
985:
982:
977:
973:
969:
965:
961:
957:
953:
949:
942:
939:
934:
930:
926:
922:
918:
914:
907:
905:
901:
896:
892:
888:
884:
880:
876:
875:J. Chem. Phys
869:
866:
861:
857:
853:
849:
845:
841:
840:J. Chem. Phys
834:
832:
828:
821:
817:
814:
812:
809:
807:
804:
802:
799:
798:
794:
785:
780:
773:
768:
763:
756:
753:
750:
748:
745:
743:
740:
739:
735:
732:
730:
728:
725:
722:
721:
717:
714:
711:
709:
706:
703:
702:
698:
691:
688:
686:
683:
681:
678:
677:
673:
666:
663:
661:
658:
656:
654:
650:
649:
645:
642:
639:
637:
634:
632:
629:
628:
624:
617:
614:
612:
609:
607:
604:
603:
600:
597:
594:
592:
589:
586:
585:
581:
577:
574:
571:
569:
566:
563:
562:
558:
554:
551:
548:
546:
543:
540:
539:
535:
532:
529:
526:
523:
520:
519:
516:
513:
511:
503:
496:
489:
486:
484:
481:
479:
476:
475:
471:
464:
461:
459:
456:
454:
451:
450:
446:
443:
440:
438:
435:
433:
430:
429:
425:
421:
418:
415:
413:
410:
408:
405:
404:
400:
397:
394:
392:
389:
387:
384:
383:
379:
372:
369:
367:
364:
362:
359:
358:
354:
350:
346:
342:
338:
331:
328:
326:
323:
321:
318:
317:
313:
306:
303:
301:
298:
296:
293:
292:
288:
284:
280:
273:
270:
268:
265:
263:
260:
259:
255:
252:
249:
246:
243:
240:
239:
236:
233:
225:
223:
221:
217:
208:
206:
200:
198:
191:
189:
182:
180:
177:
173:
168:
165:
161:
159:
153:
151:
146:
141:
138:
129:
124:
97:
86:
83:
74:
70:
67:
64:
62:
58:
54:
50:
49:nanophotonics
46:
43:
38:
34:
30:
26:
18:
2552:
2548:
2538:
2493:
2490:Opt. Express
2489:
2479:
2428:
2425:Phys. Rev. E
2424:
2418:
2401:
2397:
2391:
2366:
2362:
2356:
2331:
2327:
2321:
2308:
2301:
2268:
2264:
2258:
2225:
2221:
2215:
2206:
2200:
2173:
2169:
2159:
2126:
2122:
2116:
2081:
2078:Opt. Express
2077:
2067:
2040:
2036:
2026:
2013:
2006:
1981:
1977:
1964:
1921:
1917:
1904:
1887:
1883:
1873:
1846:
1842:
1836:
1809:
1805:
1799:
1777:. Elsevier.
1774:
1770:
1764:
1734:. Elsevier.
1731:
1727:
1721:
1704:
1700:
1690:
1655:
1651:
1645:
1624:
1591:
1587:
1538:(1): 42–61.
1535:
1531:
1508:(1): 28–32.
1505:
1501:
1495:
1478:
1474:
1468:
1443:
1439:
1426:
1383:
1379:
1373:
1348:
1344:
1318:
1314:
1301:
1276:
1272:
1219:
1216:Phys. Rev. B
1215:
1209:
1158:
1154:
1148:
1129:
1125:
1115:
1106:
1100:
1075:
1071:
1017:
1013:
984:
951:
947:
941:
916:
913:Astrophys. J
912:
878:
874:
868:
843:
839:
652:
514:
507:
229:
212:
204:
195:
186:
169:
166:
162:
154:
142:
133:
68:
65:
61:astrophysics
59:physics and
55:scattering,
32:
28:
24:
23:
2521:10902/24774
1126:Mathematics
707:Edalatpour
578:Calculates
432:VoxScatter
424:near fields
345:Near-fields
283:near fields
42:polarizable
2609:Scattering
2593:Categories
2438:1502.02186
2129:: 106689.
1856:2206.14921
1819:1502.02186
1636:2311.05908
1594:: 107501.
1409:2115/50065
822:References
806:Mie theory
567:Mackowski
527:References
295:DDscat.C++
247:References
37:scattering
2151:209969404
1956:119574693
1931:0704.0037
1890:: 66–70.
1756:264805146
1665:1008.1366
1616:233839571
1545:0908.0863
1446:: 67–79.
1380:Appl. Opt
1168:0809.0338
1052:119572857
1027:0704.0038
948:Opt. Lett
635:Dmitriev
536:Features
365:McDonald
256:Features
101:→
2579:37078273
2570:10150391
2530:34614734
2471:21556373
2463:26172822
2383:22849410
2363:ACS Nano
2293:11999964
2250:18594631
2108:21934954
1570:10285783
1418:20220882
1254:26726283
1201:15747060
1193:18978846
976:19738776
795:See also
712:Fortran
692:2013 (v.
689:Fortran
667:2019 (v.
664:Fortran
618:2014 (v.
590:McMahon
580:T-matrix
572:Fortran
549:Fortran
530:Language
490:2021 (v.
469:0.9.19)
465:2021 (v.
373:2009 (v.
332:2020 (v.
307:2017 (v.
274:2019 (v.
271:Fortran
250:Language
2498:Bibcode
2443:Bibcode
2336:Bibcode
2273:Bibcode
2230:Bibcode
2178:Bibcode
2131:Bibcode
2086:Bibcode
2045:Bibcode
1986:Bibcode
1936:Bibcode
1892:Bibcode
1779:Bibcode
1736:Bibcode
1670:Bibcode
1596:Bibcode
1550:Bibcode
1448:Bibcode
1388:Bibcode
1353:Bibcode
1281:Bibcode
1234:Bibcode
1173:Bibcode
1080:Bibcode
1032:Bibcode
956:Bibcode
921:Bibcode
919:: 705.
883:Bibcode
848:Bibcode
751:Python
742:PyDScat
640:Python
615:Matlab
595:Matlab
533:Updated
524:Authors
487:Matlab
441:Matlab
407:VIE-FFT
386:DDA-GPU
377:0.4.1)
361:OpenDDA
336:1.4.0)
311:7.3.1)
299:Choliy
285:. Uses
278:7.3.3)
253:Updated
244:Authors
145:Lorentz
137:Purcell
57:aerosol
45:dipoles
2577:
2567:
2528:
2469:
2461:
2381:
2291:
2248:
2149:
2106:
1954:
1754:
1614:
1568:
1416:
1252:
1199:
1191:
1050:
974:
704:T-DDA
694:
680:DDEELS
669:
620:
606:DDA-SI
492:
467:
453:IF-DDA
416:C/C++
375:
353:OpenCL
334:
309:
287:OpenMP
276:
262:DDSCAT
172:Draine
2467:S2CID
2433:arXiv
2313:(PDF)
2147:S2CID
2018:(PDF)
1974:(PDF)
1952:S2CID
1926:arXiv
1914:(PDF)
1851:arXiv
1814:arXiv
1752:S2CID
1660:arXiv
1631:arXiv
1612:S2CID
1566:S2CID
1540:arXiv
1436:(PDF)
1311:(PDF)
1250:S2CID
1224:arXiv
1197:S2CID
1163:arXiv
1048:S2CID
1022:arXiv
1010:(PDF)
754:2023
733:2021
723:CDDA
715:2015
696:2.1)
671:2.0)
643:2015
631:PyDDA
622:0.2)
610:Loke
598:2006
575:2002
564:DDMM
552:2008
494:1.0)
478:MPDDA
444:2019
419:2019
398:2016
390:Kieß
53:radar
2575:PMID
2526:PMID
2459:PMID
2379:PMID
2289:PMID
2246:PMID
2104:PMID
1414:PMID
1189:PMID
972:PMID
655:-DDA
587:CDA
521:Name
411:Sha
395:C++
320:ADDA
304:C++
241:Name
2565:PMC
2557:doi
2516:hdl
2506:doi
2451:doi
2406:doi
2402:110
2371:doi
2344:doi
2332:112
2281:doi
2238:doi
2186:doi
2139:doi
2127:240
2094:doi
2053:doi
1994:doi
1982:112
1944:doi
1922:106
1861:doi
1847:106
1824:doi
1787:doi
1775:298
1744:doi
1732:312
1709:doi
1678:doi
1604:doi
1592:262
1558:doi
1510:doi
1483:doi
1456:doi
1444:156
1404:hdl
1396:doi
1361:doi
1349:106
1323:doi
1319:119
1289:doi
1242:doi
1181:doi
1134:doi
1088:doi
1040:doi
1018:106
964:doi
929:doi
917:186
891:doi
856:doi
557:FFT
510:FFT
355:).
29:DDA
2595::
2573:.
2563:.
2553:14
2551:.
2547:.
2524:.
2514:.
2504:.
2494:29
2492:.
2488:.
2465:.
2457:.
2449:.
2441:.
2429:91
2427:.
2400:.
2377:.
2365:.
2342:.
2330:.
2287:.
2279:.
2269:19
2267:.
2244:.
2236:.
2226:25
2224:.
2184:.
2174:38
2172:.
2168:.
2145:.
2137:.
2125:.
2102:.
2092:.
2082:19
2080:.
2076:.
2051:.
2041:24
2039:.
2035:.
1992:.
1980:.
1976:.
1950:.
1942:.
1934:.
1920:.
1916:.
1886:.
1882:.
1859:.
1845:.
1822:.
1810:91
1808:.
1785:.
1773:.
1750:.
1742:.
1730:.
1705:32
1703:.
1699:.
1676:.
1668:.
1656:33
1654:.
1610:.
1602:.
1590:.
1578:^
1564:.
1556:.
1548:.
1536:23
1534:.
1522:^
1506:31
1504:.
1479:16
1477:.
1454:.
1442:.
1438:.
1412:.
1402:.
1394:.
1384:49
1382:.
1359:.
1347:.
1335:^
1317:.
1313:.
1287:.
1277:14
1275:.
1262:^
1248:.
1240:.
1232:.
1220:67
1218:.
1195:.
1187:.
1179:.
1171:.
1159:25
1157:.
1130:10
1128:.
1124:.
1086:.
1076:11
1074:.
1060:^
1046:.
1038:.
1030:.
1016:.
1012:.
993:^
970:.
962:.
952:12
950:.
927:.
915:.
903:^
889:.
879:41
877:.
854:.
844:84
842:.
830:^
370:C
329:C
218:,
63:.
51:,
2581:.
2559::
2532:.
2518::
2508::
2500::
2473:.
2453::
2445::
2435::
2412:.
2408::
2385:.
2373::
2367:6
2350:.
2346::
2338::
2295:.
2283::
2275::
2252:.
2240::
2232::
2194:.
2188::
2180::
2153:.
2141::
2133::
2110:.
2096::
2088::
2061:.
2055::
2047::
2000:.
1996::
1988::
1958:.
1946::
1938::
1928::
1898:.
1894::
1888:3
1867:.
1863::
1853::
1830:.
1826::
1816::
1793:.
1789::
1781::
1758:.
1746::
1738::
1715:.
1711::
1684:.
1680::
1672::
1662::
1639:.
1633::
1618:.
1606::
1598::
1572:.
1560::
1552::
1542::
1516:.
1512::
1489:.
1485::
1462:.
1458::
1450::
1420:.
1406::
1398::
1390::
1367:.
1363::
1355::
1329:.
1325::
1295:.
1291::
1283::
1256:.
1244::
1236::
1226::
1203:.
1183::
1175::
1165::
1142:.
1136::
1094:.
1090::
1082::
1054:.
1042::
1034::
1024::
978:.
966::
958::
935:.
931::
923::
897:.
893::
885::
862:.
858::
850::
653:e
108:|
98:E
91:|
87:=
84:E
27:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
