Abstract

We propose an extrapolation technique that allows accuracy improvement of discrete dipole approximation computations. The performance of this technique was studied empirically on the basis of extensive simulations for five test cases using many different discretizations. The quality of the extrapolation improves with refining discretization, reaching extraordinary performance especially for cubically shaped particles. A 2-order-of-magnitude decrease of error is demonstrated. We also propose estimates of the extrapolation error, which are proven to be reliable. Finally, we propose a simple method to directly separate shape and discretization errors and illustrate this for one test case.

© 2006 Optical Society of America

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  20. C. L. Liu and A. J. Illingworth, "Reply to comment by Lemke, Okamoto and Quante on 'Error analysis of backscatter from discrete dipole approximation for different ice particle shapes'," Atmos. Res. 50, 1-2 (1999).
  21. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, "Convergence of the discrete dipole approximation. I. Theoretical analysis," J. Opt. Soc. Am. A 23, 2578-2591 (2006).
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  26. M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, 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).
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  33. M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
    [CrossRef]
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    [CrossRef]

2006 (2)

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, "Convergence of the discrete dipole approximation. I. Theoretical analysis," J. Opt. Soc. Am. A 23, 2578-2591 (2006).
[CrossRef]

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

2005 (1)

2004 (2)

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[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).
[CrossRef]

2000 (1)

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]

1999 (4)

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).
[CrossRef]

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]

C. L. Liu and A. J. Illingworth, "Reply to comment by Lemke, Okamoto and Quante on 'Error analysis of backscatter from discrete dipole approximation for different ice particle shapes'," Atmos. Res. 50, 1-2 (1999).

N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999).
[CrossRef]

1998 (4)

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).
[CrossRef]

H. Lemke, H. Okamoto, and M. Quante, "Comment on error analysis of backscatter from discrete dipole approximation for different ice particle shapes [Liu, C.-L., Illingworth, A. J., 1997, Atmos. Res. 44, 231-241]," Atmos. Res. 49, 189-197 (1998).

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]

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]

1997 (3)

1996 (1)

J. Rahola, "Solution of dense systems of linear equations in the discrete-dipole approximation," SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 17, 78-89 (1996).
[CrossRef]

1995 (2)

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).
[CrossRef]

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

1994 (1)

1993 (2)

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]

B. T. Draine and J. J. Goodman, "Beyond Clausius-Mossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

1991 (2)

1988 (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bronk, B. V.

Chaumet, P. C.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Chernyshev, A. V.

Collinge, M. J.

Dominik, A.

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

Draine, B. T.

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).
[CrossRef]

B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
[CrossRef]

B. T. Draine and J. J. Goodman, "Beyond Clausius-Mossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

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).
[CrossRef] [PubMed]

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

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).

Druger, S. D.

Evans, K. F.

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).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1990).

Flatau, P. J.

Frigo, M.

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).

Goodman, J. J.

B. T. Draine and J. J. Goodman, "Beyond Clausius-Mossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

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).
[CrossRef] [PubMed]

Greenberg, J. M.

Grimminck, M. D.

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]

Gustafson, B. A. S.

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]

Hage, J. I.

Hoekstra, A. G.

Hovenier, J. W.

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Illingworth, A. J.

C. L. Liu and A. J. Illingworth, "Reply to comment by Lemke, Okamoto and Quante on 'Error analysis of backscatter from discrete dipole approximation for different ice particle shapes'," Atmos. Res. 50, 1-2 (1999).

C. L. Liu and A. J. Illingworth, "Error analysis of backscatter from discrete dipole approximation for different ice particle shapes," Atmos. Res. 44, 231-241 (1997).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).

Koter, A. de

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

Lemke, H.

H. Lemke, H. Okamoto, and M. Quante, "Comment on error analysis of backscatter from discrete dipole approximation for different ice particle shapes [Liu, C.-L., Illingworth, A. J., 1997, Atmos. Res. 44, 231-241]," Atmos. Res. 49, 189-197 (1998).

Liu, C. L.

C. L. Liu and A. J. Illingworth, "Reply to comment by Lemke, Okamoto and Quante on 'Error analysis of backscatter from discrete dipole approximation for different ice particle shapes'," Atmos. Res. 50, 1-2 (1999).

C. L. Liu and A. J. Illingworth, "Error analysis of backscatter from discrete dipole approximation for different ice particle shapes," Atmos. Res. 44, 231-241 (1997).
[CrossRef]

Macke, A.

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

Maltsev, V. P.

Martin, O. J. F.

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]

Martin, P. G.

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]

Min, M.

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

Okamoto, H.

H. Lemke, H. Okamoto, and M. Quante, "Comment on error analysis of backscatter from discrete dipole approximation for different ice particle shapes [Liu, C.-L., Illingworth, A. J., 1997, Atmos. Res. 44, 231-241]," Atmos. Res. 49, 189-197 (1998).

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Piller, N. B.

N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999).
[CrossRef]

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, "Influence of the edge meshes on the accuracy of the coupled-dipole approximation," Opt. Lett. 22, 1674-1676 (1997).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1990).

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Quante, M.

H. Lemke, H. Okamoto, and M. Quante, "Comment on error analysis of backscatter from discrete dipole approximation for different ice particle shapes [Liu, C.-L., Illingworth, A. J., 1997, Atmos. Res. 44, 231-241]," Atmos. Res. 49, 189-197 (1998).

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Rahola, J.

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. 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).
[CrossRef]

J. Rahola, "Solution of dense systems of linear equations in the discrete-dipole approximation," SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (USA) 17, 78-89 (1996).
[CrossRef]

Raschke, E.

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

Rouleau, F.

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]

Semyanov, K. A.

Sentenac, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Sloot, P. M. A.

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]

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).
[CrossRef]

Stephens, G. L.

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).
[CrossRef]

Tarasov, P. A.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1990).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1990).

Wang, R. T.

Xu, Y. L.

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]

Yurkin, M. A.

Appl. Opt. (3)

Astrophys. J. (5)

B. T. Draine and J. J. Goodman, "Beyond Clausius-Mossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

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]

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]

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Atmos. Res. (3)

C. L. Liu and A. J. Illingworth, "Error analysis of backscatter from discrete dipole approximation for different ice particle shapes," Atmos. Res. 44, 231-241 (1997).
[CrossRef]

H. Lemke, H. Okamoto, and M. Quante, "Comment on error analysis of backscatter from discrete dipole approximation for different ice particle shapes [Liu, C.-L., Illingworth, A. J., 1997, Atmos. Res. 44, 231-241]," Atmos. Res. 49, 189-197 (1998).

C. L. Liu and A. J. Illingworth, "Reply to comment by Lemke, Okamoto and Quante on 'Error analysis of backscatter from discrete dipole approximation for different ice particle shapes'," Atmos. Res. 50, 1-2 (1999).

Contrib. Atmos. Phys. (1)

H. Okamoto, A. Macke, M. Quante, and E. Raschke, "Modeling of backscattering by non-spherical ice particles for the interpretation of cloud radar signals at 94GHz. An error analysis," Contrib. Atmos. Phys. 68, 319-334 (1995).

IEEE Trans. Antennas Propag. (1)

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]

Int. J. Mod. Phys. C (1)

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]

J. Atmos. Sci. (1)

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).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

M. Min, J. W. Hovenier, A. Dominik, A. de Koter, and M. A. Yurkin, "Absorption and scattering properties of arbitrary shaped particles in the Rayleigh domain: a rapid computational method and a theoretical foundation for the statistical approach," J. Quant. Spectrosc. Radiat. Transf. 97, 161-180 (2006).
[CrossRef]

Opt. Commun. (1)

N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Cubical discretization of a sphere using 16 dipoles per diameter (total 2176 dipoles).

Fig. 2
Fig. 2

Signed relative errors of Q e x t versus y and their fits by quadratic functions for (a) the k D = 8 cube and discretized k D = 10 sphere, (b) three spheres. Five and nine best points are used for fits in (a) and (b), respectively.

Fig. 3
Fig. 3

Errors of S 11 ( θ ) in logarithmic scale for extrapolation using five values of y in the intervals (a) [ 0.047 , 0.094 ] , (b) [ 0.094 , 0.19 ] , and (c) [ 0.38 , 0.75 ] for the k D = 8 cube. Estimate of the extrapolation error is 10(SE).

Fig. 4
Fig. 4

Errors of S 11 ( θ ) in logarithmic scale for extrapolation using five values of y in the intervals (a) [ 0.058 , 0.12 ] , (b) [ 0.12 , 0.23 ] and using four values of y in the interval (c) [ 0.23 , 0.93 ] ) for the discretized k D = 10 sphere. Estimate of the extrapolation error is 10(SE).

Fig. 5
Fig. 5

Errors of S 11 ( θ ) in logarithmic scale for extrapolation using nine values of y in the intervals (a) [ 0.018 , 0.070 ] , (b) [ 0.14 , 0.55 ] for the k D = 3 sphere. Estimate of the extrapolation error is 2(SE).

Fig. 6
Fig. 6

Errors of S 11 ( θ ) in logarithmic scale for extrapolation using nine values of y in the intervals (a) [ 0.059 , 0.23 ] , (b) [ 0.12 , 0.47 ] for the k D = 10 sphere. Estimate of the extrapolation error is 2(SE).

Fig. 7
Fig. 7

Errors of S 11 ( θ ) in logarithmic scale for extrapolation using nine values of y in the interval [ 0.18 , 0.70 ] for the k D = 30 sphere.

Fig. 8
Fig. 8

Comparison of discretization and shape errors of S 11 ( θ ) for the k D = 10 sphere discretized using 16 dipoles per D ( y = 0.93 ) .

Tables (2)

Tables Icon

Table 1 Extrapolation Errors of Q e x t a

Tables Icon

Table 2 Comparison of Shape and Discretization Errors of Q e x t for the k D = 10 Sphere Discretized with y = 0.93 a

Equations (2)

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δ ϕ y ( a 2 ϕ b 2 ϕ ln y ) y 2 + ( a 1 ϕ b 1 ϕ ln y ) y ,
ϕ y = a 0 + a 1 y + a 2 y 2 + ζ y ,

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