## Convergence of the discrete dipole approximation. I. Theoretical analysis

JOSA A, Vol. 23, Issue 10, pp. 2578-2591 (2006)

http://dx.doi.org/10.1364/JOSAA.23.002578

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### Abstract

We perform a rigorous theoretical convergence analysis of the discrete dipole approximation (DDA). We prove that errors in any measured quantity are bounded by a sum of a linear term and a quadratic term in the size of a dipole d when the latter is in the range of DDA applicability. Moreover, the linear term is significantly smaller for cubically than for noncubically shaped scatterers. Therefore, for small d, errors for cubically shaped particles are much smaller than for noncubically shaped ones. The relative importance of the linear term decreases with increasing size; hence convergence of DDA for large enough scatterers is quadratic in the common range of d. Extensive numerical simulations are carried out for a wide range of d. Finally, we discuss a number of new developments in DDA and their consequences for convergence.

© 2006 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(260.2110) Physical optics : Electromagnetic optics

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: January 23, 2006

Manuscript Accepted: April 18, 2006

**Citation**

Maxim A. Yurkin, Valeri P. Maltsev, and Alfons G. Hoekstra, "Convergence of the discrete dipole approximation. I. Theoretical analysis," J. Opt. Soc. Am. A **23**, 2578-2591 (2006)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-10-2578

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### References

- E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973). [CrossRef]
- B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
- B. T. Draine, "The discrete dipole approximation for light scattering by irregular targets," in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.
- B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT 6.1," http://xxx.arxiv.org/abs/astro-ph/0409262 (2004).
- J. J. Goodman, B. T. Draine, and P. J. Flatau, "Application of fast-Fourier-transform techniques to the discrete-dipole approximation," Opt. Lett. 16, 1198-1200 (1991).
- G. C. Hsiao and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propag. 45, 316-328 (1997). [CrossRef]
- K. F. Warnick and W. C. Chew, "Error analysis of the moment method," IEEE Antennas Propag. Mag. 46, 38-53 (2004).
- B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- J. I. Hage, J. M. Greenberg, and R. T. Wang, "Scattering from arbitrarily shaped particles: theory and experiment," Appl. Opt. 30, 1141-1152 (1991).
- F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803-814 (1993). [CrossRef]
- N. B. Piller, "Influence of the edge meshes on the accuracy of the coupled-dipole approximation," Opt. Lett. 22, 1674-1676 (1997).
- N. B. Piller and O. J. F. Martin, "Increasing the performance of the coupled-dipole approximation: a spectral approach," IEEE Trans. Antennas Propag. 46, 1126-1137 (1998). [CrossRef]
- N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999). [CrossRef]
- A. G. Hoekstra, J. Rahola, and P. M. A. Sloot, "Accuracy of internal fields in volume integral equation simulations of light scattering," Appl. Opt. 37, 8482-8497 (1998).
- S. D. Druger and B. V. Bronk, "Internal and scattered electric fields in the discrete dipole approximation," J. Opt. Soc. Am. B 16, 2239-2246 (1999).
- Y. L. Xu and B. A. S. Gustafson, "Comparison between multisphere light-scattering calculations: rigorous solution and discrete-dipole approximation," Astrophys. J. 513, 894-909 (1999). [CrossRef]
- M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
- M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, "Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy," J. Opt. Soc. Am. A 23, 2592-2601 (2006).
- A. Lakhtakia, "Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic-fields," Int. J. Mod. Phys. C 3, 583-603 (1992). [CrossRef]
- F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transf. 79, 775-824 (2003). [CrossRef]
- C. P. Davis and K. F. Warnick, "On the physical interpretation of the Sobolev norm in error estimation," Appl. Comput. Electromagn. Soc. J. 20, 144-150 (2005).
- A. D. Yanghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).
- G. H. Goedecke and S. G. O'Brien, "Scattering by irregular inhomogeneous particles via the digitized Green's function algorithm," Appl. Opt. 27, 2431-2438 (1988).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- J. Rahola, "On the eigenvalues of the volume integral operator of electromagnetic scattering," SIAM (Soc. Ind. Appl. Math) J. Sci. Comput. (USA) 21, 1740-1754 (2000). [CrossRef]
- A. Lakhtakia and G. W. Mulholland, "On two numerical techniques for light scattering by dielectric agglomerated structures," J. Res. Natl. Inst. Stand. Technol. 98, 699-716 (1993).
- J. I. Hage and J. M. Greenberg, "A model for the optical properties of porous grains," Astrophys. J. 361, 251-259 (1990). [CrossRef]
- C. E. Dungey and C. F. Bohren, "Light scattering by nonspherical particles: a refinement to the coupled-dipole method," J. Opt. Soc. Am. A 8, 81-87 (1991).
- H. Okamoto, "Light scattering by clusters: the A1-term method," Opt. Rev. 2, 407-412 (1995).
- B. T. Draine and J. J. Goodman, "Beyond Clausius-Miossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993). [CrossRef]
- J. I. Peltoniemi, "Variational volume integral equation method for electromagnetic scattering by irregular grains," J. Quant. Spectrosc. Radiat. Transf. 55, 637-647 (1996). [CrossRef]
- D. Gutkowicz-Krusin and B. T. Draine, "Propagation of electromagnetic waves on a rectangular lattice of polarizable points," http://xxx.arxiv.org/abs/astro-ph/0403082 (2004).
- A. Rahmani, P. C. Chaumet, and G. W. Bryant, "Coupled dipole method with an exact long-wavelength limit and improved accuracy at finite frequencies," Opt. Lett. 27, 2118-2120 (2002).
- A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004). [CrossRef]
- P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004). [CrossRef]
- K. F. Evans and G. L. Stephens, "Microwave radiative transfer through clouds composed of realistically shaped ice crystals. Part 1. Single scattering properties," J. Atmos. Sci. 52, 2041-2057 (1995).
- "Amsterdam DDA," http://www.science.uva.nl/research/scs/Software/adda.
- A. G. Hoekstra, M. D. Grimminck, and P. M. A. Sloot, "Large scale simulations of elastic light scattering by a fast discrete dipole approximation," Int. J. Mod. Phys. C 9, 87-102 (1998). [CrossRef]
- M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chemyshev, A. G. Hoekstra, and V. P. Maltsev, "Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and discrete dipole approximation," Appl. Opt. 44, 5249-5256 (2005). [CrossRef]
- "Description of the national computer cluster Lisa," http://www.sara.nl/userinfo/lisa/description/mdex.html(2005).

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