## Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers

Optics Express, Vol. 15, Issue 26, pp. 17902-17911 (2007)

http://dx.doi.org/10.1364/OE.15.017902

Acrobat PDF (258 KB)

### Abstract

We compare the discrete dipole approximation (DDA) and the finite difference time domain (FDTD) method for simulating light scattering of spheres in a range of size parameters x up to 80 and refractive indices *m* up to 2. Using parallel implementations of both methods, we require them to reach a certain accuracy goal for scattering quantities and then compare their performance. We show that relative performance sharply depends on *m*. The DDA is faster for smaller *m*, while the FDTD for larger values of *m*. The break-even point lies at *m*=1.4. We also compare the performance of both methods for a few particular biological cells, resulting in the same conclusions as for optically soft spheres.

© 2007 Optical Society of America

## 1. Introduction

1. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer **106**, 558–589 (2007). [CrossRef]

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

4. P. Yang and K. N. LiouM. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications,, eds. (Academic Press, New York, 2000), pp. 173–221.

5. Y. You, G. W. Kattawar, C. H. Li, and P. Yang, “Internal dipole radiation as a tool for particle identification,” Appl. Opt. **45**, 9115–9124 (2006). [CrossRef] [PubMed]

6. J. P. He, A. Karlsson, J. Swartling, and S. Andersson-Engels, “Light scattering by multiple red blood cells,” J. Opt. Soc. Am. A **21**, 1953–1961 (2004). [CrossRef]

7. T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer **60**, 411–423 (1998). [CrossRef]

*x*and refractive index

*m*, which includes almost the whole range of biological cells (

*x*up to 80). Second, we set the accuracy to be reached by both methods, which makes the performance results more informative.

8. I. V. Kolesnikova, S. V. Potapov, M. A. Yurkin, A. G. Hoekstra, V. P. Maltsev, and K. A. Semyanov, “Determination of volume, shape and refractive index of individual blood platelets,” J. Quant. Spectrosc. Radiat. Transfer **102**, 37–45 (2006). [CrossRef]

## 2. DDA and FDTD implementations

13. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transfer **106**, 546–557 (2007). [CrossRef]

14. “Amsterdam DDA,” http://www.science.uva.nl/research/scs/Software/adda (2007).

15. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer **106**, 417–436 (2007). [CrossRef]

1. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer **106**, 558–589 (2007). [CrossRef]

16. N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: A spectral approach,” IEEE Trans. Ant. Propag. **46**, 1126–1137 (1998). [CrossRef]

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

19. N. B. Piller, “Influence of the edge meshes on the accuracy of the coupled-dipole approximation,” Opt. Lett. **22**, 1674–1676 (1997). [CrossRef]

20. 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*very close to 1 (see Section 4). Hence decreasing shape errors will not drastically improve the overall performance of the DDA in this study. We do not analyze the impact of specific formulations because 1) they would not significantly change the final conclusion; 2) we want this comparison to be useful mostly to researchers who would rather use the publicly available code than implement the latest theoretical developments themselves.

10. R. S. Brock, X. Hu, P. Yang, and J. Q. Lu, “Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells,” Opt. Express **13**, 5279–5292 (2005). [CrossRef] [PubMed]

4. P. Yang and K. N. LiouM. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications,, eds. (Academic Press, New York, 2000), pp. 173–221.

21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comp. Phys. **114**, 185–200 (1994). [CrossRef]

## 3. Test objects

*x*(defined as

*ka*, where

*a*is the sphere radius and

*k*is the wavenumber of the incident light) and refractive index

*m*=

*m*

^{′}+

*im*

^{″}. Here

*m*

^{″}is fixed at 1.5×10

^{-5}. This imaginary part of

*m*does not significantly influence the final simulation results; however, it may decrease the simulation time for the FDTD, at least for

*m*′=1.02, as indicated by previous preliminary studies (data not shown). The lower limit for x is 10 and the upper limit depends on

*m*

^{′}(to keep the computational times manageable). It decreases from 80 to 20 for

*m*

^{′}increasing from 1.02 to 2. The exact set of

*x*,

*m*

^{′}pairs is shown in Table 1. For each sphere we compute the extinction efficiency

*Q*

_{ext}, asymmetry parameter <cos

*θ*>, and Mueller matrix in one scattering plane (polar angle

*θ*changes from 0° to 180° in steps of 0.25°). From the whole Mueller matrix we analyze only the

*S*

_{11}element and the linear polarization

*P*=-

*S*

_{21}/

*S*

_{11}. We do not put any constraints on the number of dipoles to optimize sphericity of the dipole representation of the sphere [22

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

13. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transfer **106**, 546–557 (2007). [CrossRef]

*Q*

_{ext}less than 1%, and the root mean square (RMS) RE of

*S*

_{11}over the whole range of

*θ*less than 25%. All simulations were performed on the Lemieux cluster [23

23. “Lemieux,” http://www.psc.edu/machines/tcs/(2006).

*<*cosθ>.

*S*

_{11}is calculated in the

*yz*-plane for the same range of

*θ*, and

*Q*

_{ext}is calculated for incident light that is linearly polarized along the y-axis.

*z*-axis. The coefficients are set to their typical values:

*P*=-14.3 µm

^{2},

*Q*=38.9 µm

^{2},

*R*=-4.57 µm

^{4},

*S*=-0.193, which corresponds to an RBC with diameter 7.65 µm and maximum thickness 2.44 µm (see Fig. 1). Equation (1) is proposed in [24] as an extension to the discocyte shape described by Kuchel and Fackerell [25

25. P. W. Kuchel and E. D. Fackerell, “Parametric-equation representation of biconcave erythrocytes,” Bull. Math. Biol. **61**, 209–220 (1999). [CrossRef]

*yz*-plane and

^{-5}i, which corresponds to the average hemoglobin concentration [24]. Note that

*m*

^{′}of the RBC is different from that of all other scatterers studied in this manuscript.

26. R. Hurwitz, J. Hozier, T. LeBien, J. Minowada, K. Gajl-Peczalska, I. Kubonishi, and J. Kersey, “Characterization of a leukemic cell line of the pre-B phenotype,” Int. J. Cancer **23**, 174–180 (1979). [CrossRef] [PubMed]

11. R. S. Brock, X. Hu, D. A. Weidner, J. R. Mourant, and J. Q. Lu, “Effect of detailed cell structure on light scattering distribution: FDTD study of a B-cell with 3D structure constructed from confocal images,” J. Quant. Spectrosc. Radiat. Transfer **102**, 25–36 (2006). [CrossRef]

11. R. S. Brock, X. Hu, D. A. Weidner, J. R. Mourant, and J. Q. Lu, “Effect of detailed cell structure on light scattering distribution: FDTD study of a B-cell with 3D structure constructed from confocal images,” J. Quant. Spectrosc. Radiat. Transfer **102**, 25–36 (2006). [CrossRef]

*m*

^{′}=1.023 and 1.071 respectively (as was done in previous studies of the BCP [11

11. R. S. Brock, X. Hu, D. A. Weidner, J. R. Mourant, and J. Q. Lu, “Effect of detailed cell structure on light scattering distribution: FDTD study of a B-cell with 3D structure constructed from confocal images,” J. Quant. Spectrosc. Radiat. Transfer **102**, 25–36 (2006). [CrossRef]

*m*

^{′}is the same as for spheres. Its orientation is the default one, so that

*z*-axis is normal to the layers used for 3D reconstruction.

*x*=68.1). Refractive indices are the same as for the BCP. We tune the discretization for the coated sphere to reach the same accuracy as for spheres. For biological cells we use a single discretization, because of lack of a rigorous exact solution. For the BCP the discretization is similar to those used for the coated sphere and for the RBC — similar to those used for

*m*

^{′}=1.08 spheres (see Table 3).

28. “Description of the national compute cluster Lisa,” http://www.sara.nl/userinfo/lisa/description/(2005).

## 4. Results for spheres

*x*and dpl and determines the memory consumption. Values of dpl cannot be directly compared between both methods because the typical values for the DDA [1

1. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer **106**, 558–589 (2007). [CrossRef]

4. P. Yang and K. N. LiouM. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications,, eds. (Academic Press, New York, 2000), pp. 173–221.

*x*just because the number of grid cells scale cubically with

*x*, if dpl is kept constant. Apart from that, the behavior of the methods is quite different. Dpl required by the DDA to reach the prescribed accuracy do not systematically depend on

*x*, except for

*m*

^{′}=1.7 and 2. However, dpl does depend on

*m*

^{′}— it increases both when

*m*

^{′}increases over 1.4 and approaches unity. The number of iterations for the DDA is relatively small and only moderately increases with

*x*for

*m*′=1.02 and 1.08. However, for larger

*m*

^{′}it rapidly increases both with

*m*

^{′}and

*x*. For

*m*

^{′}=1.7 and 2 this combines with increasing dpl leading to the sharp increase in computational time.

*x*and

*m*

^{′}studied. On the contrary, the number of time steps increase systematically with both

*x*and

*m*

^{′}, which is expected. The dependences of the FDTD performance on

*x*and

*m*

^{′}are less interdependent than that of the DDA. Comparing the overall performance of the two methods, one can see that for small

*m*

^{′}and large

*x*the DDA is an order of magnitude faster than the FDTD, and for large

*m*

^{′}vice versa. The boundary value of

*m*

^{′}is about 1.4, for which both methods are comparable. They are also comparable for small values of both

*m*

^{′}and

*x*. Memory requirements of the two methods are generally similar. However, they naturally correlate with computational time — in most cases the faster method is also less memory consuming.

*m*

^{′}, at least for large scatterers. Although enhancements mentioned in Section 2 definitely improve DDA performance in this regime, they do not solve the main problem of poor convergence of the iterative solver. However, this conclusion does not apply to particles with large

*m*

^{″}, which require a separate study.

*S*

_{11}is equal to 25%, we show

*S*

_{11}results for three sample spheres in Fig. 3. Three subfigures are for the same

*x*=20 and three different m′: 1.02, 1.4, and 2. Each of them shows the exact Mie solution and simulation results of the DDA and the FDTD. One can see that visual agreement is very good, probably more than enough for most applications.

*m*

^{′}is close to unity may seem counterintuitive. However, this is explained by the relative nature of the accuracy criterion and the large dynamical range of

*S*

_{11}(

*θ*) for optically soft spheres. This function has very sharp minima, the position of which depends on the exact shape of the particle. For example, consider a particular case of

*m*

^{′}=1.02,

*x*=20. The exact Mie solution for this case is shown in Fig. 3(a), dpl=20 is required for the DDA to reach good accuracy. If one uses dpl=10 (similar to those required for

*m*

^{′}=1.08) the relative errors are relatively large: their angle dependence is shown in Fig. 4 and the RMS value is 0.73. Using the methodology described elsewhere [29

29. 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). [CrossRef]

## 5. Sample applications to biological cells

29. 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). [CrossRef]

*Q*

_{ext})=2.6×10

^{-4}, RMSRE(

*S*

_{11})=0.12, RMSE(

*P*)=0.052.

*S*

_{11}results of both methods together with the reference results for the RBC and the BCP are shown in Fig. 5. In Fig. 5(b) we also included the Mie solution for the coated sphere model. One can see that it is a bad approximation of the realistic BCP shape.

*x*and

*m*(faster by about 50 times). One can also see that the DDA provides accurate results for realistic cell shape with dpl ≈10, at least for these two particular examples.

## 6. Conclusion

*x*up to 80 and

*m*

^{′}up to 2, using state-of-the-art parallel implementations of both methods, was performed requiring a certain accuracy of the simulated scattering quantities. The DDA is more than an order of magnitude faster for

*m*

^{′}≤1.2 and

*x*>30, while for

*m*

^{′}≥1.7 the FDTD is faster by the same extent.

*m*

^{′}=1.4 is a boundary value, for which both methods perform comparably. The DDA errors of

*S*

_{11}(

*θ*) for

*m*

^{′}=1.02 are mostly due to the shape errors, which are expected to be smaller for rough and/or inhomogeneous particles. Simulations for a few sample biological cells lead to the same conclusions.

*m*. However, this remains an open research question. On contrary, changing the details of the DDA model does not seem to be beneficial in the considered range of

*x*and

*m*. For the FDTD, a “safe” set of PML parameters was chosen; fine tuning these parameters could lead to a thinner PML and increase performance especially for the larger problem sizes. Also the FDTD code is designed to use memory conservatively; relaxing the memory restrictions would allow faster simulation times at the expense of additional memory use. However, all these improvements are not expected to cover an order of magnitude difference in the performance of two methods in the near future.

30. I. Ayranci, R. Vaillon, and N. Selcuk, “Performance of discrete dipole approximation for prediction of amplitude and phase of electromagnetic scattering by particles,” J. Quant. Spectrosc. Radiat. Transfer **103**, 83–101 (2007). [CrossRef]

## Acknowledgements

## References and links

1. | M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer |

2. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

3. | A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed., (Artech House, Boston, 2005). |

4. | P. Yang and K. N. LiouM. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications,, eds. (Academic Press, New York, 2000), pp. 173–221. |

5. | Y. You, G. W. Kattawar, C. H. Li, and P. Yang, “Internal dipole radiation as a tool for particle identification,” Appl. Opt. |

6. | J. P. He, A. Karlsson, J. Swartling, and S. Andersson-Engels, “Light scattering by multiple red blood cells,” J. Opt. Soc. Am. A |

7. | T. Wriedt and U. Comberg, “Comparison of computational scattering methods,” J. Quant. Spectrosc. Radiat. Transfer |

8. | I. V. Kolesnikova, S. V. Potapov, M. A. Yurkin, A. G. Hoekstra, V. P. Maltsev, and K. A. Semyanov, “Determination of volume, shape and refractive index of individual blood platelets,” J. Quant. Spectrosc. Radiat. Transfer |

9. | 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 with scanning flow cytometry and discrete dipole approximation,” Appl. Opt. |

10. | R. S. Brock, X. Hu, P. Yang, and J. Q. Lu, “Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells,” Opt. Express |

11. | R. S. Brock, X. Hu, D. A. Weidner, J. R. Mourant, and J. Q. Lu, “Effect of detailed cell structure on light scattering distribution: FDTD study of a B-cell with 3D structure constructed from confocal images,” J. Quant. Spectrosc. Radiat. Transfer |

12. | R. S. Brock and J. Q. Lu, “Numerical dispersion correction in a parallel FDTD code for the modeling of light scattering by biologic cells,” to be submitted to Appl. Opt. |

13. | M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transfer |

14. | “Amsterdam DDA,” http://www.science.uva.nl/research/scs/Software/adda (2007). |

15. | A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer |

16. | N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: A spectral approach,” IEEE Trans. Ant. Propag. |

17. | 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. |

18. | P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E |

19. | N. B. Piller, “Influence of the edge meshes on the accuracy of the coupled-dipole approximation,” Opt. Lett. |

20. | M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A |

21. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comp. Phys. |

22. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. |

23. | “Lemieux,” http://www.psc.edu/machines/tcs/(2006). |

24. | P. A. Avrorov, M. A. Yurkin, K. A. Semyanov, A. G. Hoekstra, P. A. Tarasov, and V. P. Maltsev, “Characterization of mature red blood cells with scanning flow cytometry,” in preparation. |

25. | P. W. Kuchel and E. D. Fackerell, “Parametric-equation representation of biconcave erythrocytes,” Bull. Math. Biol. |

26. | R. Hurwitz, J. Hozier, T. LeBien, J. Minowada, K. Gajl-Peczalska, I. Kubonishi, and J. Kersey, “Characterization of a leukemic cell line of the pre-B phenotype,” Int. J. Cancer |

27. | R. S. Brock, H. Ding, D. A. Weidner, T. J. McConnel, X. Hu, J. R. Mourant, and J. Q. Lu, “Modeling of the internal optical structure of the nuclei of B-cells,” in Frontiers in Optics (Optical Society of America, 2006), p. FTuE2. |

28. | “Description of the national compute cluster Lisa,” http://www.sara.nl/userinfo/lisa/description/(2005). |

29. | 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 |

30. | I. Ayranci, R. Vaillon, and N. Selcuk, “Performance of discrete dipole approximation for prediction of amplitude and phase of electromagnetic scattering by particles,” J. Quant. Spectrosc. Radiat. Transfer |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(170.1530) Medical optics and biotechnology : Cell analysis

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: October 10, 2007

Revised Manuscript: November 29, 2007

Manuscript Accepted: December 5, 2007

Published: December 17, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Maxim A. Yurkin, Alfons G. Hoekstra, R. S. Brock, and Jun Q. Lu, "Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers," Opt. Express **15**, 17902-17911 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-17902

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### References

- M. A. Yurkin and A. G. Hoekstra, "The discrete dipole approximation: an overview and recent developments," J. Quant. Spectrosc. Radiat. Transfer 106,558-589 (2007). [CrossRef]
- B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11,1491-1499 (1994). [CrossRef]
- A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed., (Artech House, Boston, 2005).
- P. Yang and K. N. Liou, "Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles," in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic Press, New York, 2000), pp. 173-221.
- Y. You, G. W. Kattawar, C. H. Li, and P. Yang, "Internal dipole radiation as a tool for particle identification," Appl. Opt. 45,9115-9124 (2006). [CrossRef] [PubMed]
- J. P. He, A. Karlsson, J. Swartling, and S. Andersson-Engels, "Light scattering by multiple red blood cells," J. Opt. Soc. Am. A 21,1953-1961 (2004). [CrossRef]
- T. Wriedt and U. Comberg, "Comparison of computational scattering methods," J. Quant. Spectrosc. Radiat. Transfer 60,411-423 (1998). [CrossRef]
- I. V. Kolesnikova, S. V. Potapov, M. A. Yurkin, A. G. Hoekstra, V. P. Maltsev, and K. A. Semyanov, "Determination of volume, shape and refractive index of individual blood platelets," J. Quant. Spectrosc. Radiat. Transfer 102,37-45 (2006). [CrossRef]
- 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 with scanning flow cytometry and discrete dipole approximation," Appl. Opt. 44,5249-5256 (2005). [CrossRef] [PubMed]
- R. S. Brock, X. Hu, P. Yang, and J. Q. Lu, "Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells," Opt. Express 13,5279-5292 (2005). [CrossRef] [PubMed]
- R. S. Brock, X. Hu, D. A. Weidner, J. R. Mourant, and J. Q. Lu, "Effect of detailed cell structure on light scattering distribution: FDTD study of a B-cell with 3D structure constructed from confocal images," J. Quant. Spectrosc. Radiat. Transfer 102,25-36 (2006). [CrossRef]
- R. S. Brock and J. Q. Lu, "Numerical dispersion correction in a parallel FDTD code for the modeling of light scattering by biologic cells," to be submitted to Appl. Opt.
- M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, "The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength," J. Quant. Spectrosc. Radiat. Transfer 106,546-557 (2007). [CrossRef]
- "Amsterdam DDA," http://www.science.uva.nl/research/scs/Software/adda (2007).
- A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, "Comparison between discrete dipole implementations and exact techniques," J. Quant. Spectrosc. Radiat. Transfer 106,417-436 (2007). [CrossRef]
- N. B. Piller and O. J. F. Martin, "Increasing the performance of the coupled-dipole approximation: A spectral approach," IEEE Trans. Ant. Propag. 46,1126-1137 (1998). [CrossRef]
- 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). [CrossRef]
- P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70,036606 (2004). [CrossRef]
- N. B. Piller, "Influence of the edge meshes on the accuracy of the coupled-dipole approximation," Opt. Lett. 22,1674-1676 (1997). [CrossRef]
- 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]
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic-waves," J. Comp. Phys. 114,185-200 (1994). [CrossRef]
- B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988). [CrossRef]
- "Lemieux," http://www.psc.edu/machines/tcs/ (2006).
- P. A. Avrorov, M. A. Yurkin, K. A. Semyanov, A. G. Hoekstra, P. A. Tarasov, and V. P. Maltsev, "Characterization of mature red blood cells with scanning flow cytometry," in preparation.
- P. W. Kuchel and E. D. Fackerell, "Parametric-equation representation of biconcave erythrocytes," Bull. Math. Biol. 61,209-220 (1999). [CrossRef]
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