## Invisibility cloaks for irregular particles using coordinate transformations

Optics Express, Vol. 16, Issue 9, pp. 6134-6145 (2008)

http://dx.doi.org/10.1364/OE.16.006134

Acrobat PDF (1877 KB)

### Abstract

Invisibility cloaks for ellipsoids, rounded cuboids and rounded cylinders have been studied on the basis of the coordinate transformation approach. The resultant material property tensors for irregular cloaks are more complicated in comparison with those for the spherical invisibility cloak. A generalized Discrete Dipole Approximation (DDA) formalism has been used to simulate the scattered field distribution in the vicinity of the aforementioned irregular cloaks illuminated by an incident plane wave. Simulated scattering efficiencies are on the order of 10^{-5}, and the simulated electric-field distribution outside of a cloak is the same as that of the incident radiation.

© 2008 Optical Society of America

## 1. Introduction

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

5. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

6. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. **8**, 118 (2006). [CrossRef]

7. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express **15**, 14772–14782 (2007). [CrossRef] [PubMed]

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

5. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

6. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. **8**, 118 (2006). [CrossRef]

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

**8**, 247 (2006). [CrossRef]

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

**8**, 247 (2006). [CrossRef]

**8**, 247 (2006). [CrossRef]

7. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express **15**, 14772–14782 (2007). [CrossRef] [PubMed]

**312**, 1780 (2006). [CrossRef] [PubMed]

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

**8**, 247 (2006). [CrossRef]

8. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977 (2006). [CrossRef] [PubMed]

**312**, 1780 (2006). [CrossRef] [PubMed]

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

10. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. **99**, 063903 (2007). [CrossRef] [PubMed]

11. B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B **76**, 121101 (2007). [CrossRef]

8. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

12. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

13. H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A **77**, 013825 (2008). [CrossRef]

14. D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. **92**, 013505 (2008). [CrossRef]

15. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express **16**, 2068–2079 (2008). [CrossRef] [PubMed]

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977 (2006). [CrossRef] [PubMed]

16. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. **1**, 224–227 (2007). [CrossRef]

17. B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. **100**, 063904 (2008). [CrossRef] [PubMed]

18. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. **186**, 705–714 (1973). [CrossRef]

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

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

*μ*≠1 [15

15. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express **16**, 2068–2079 (2008). [CrossRef] [PubMed]

## 2. Transformation equations

*x*, with

^{i}*i*=1,2,3. Under a coordinate transformation that maps point

*x*to point

^{i}*T*is transformed as follows:

^{ij}*ε*and the permeability tensor

^{ij}*μ*are transformed as

^{ij}*et al.*[2

**312**, 1780 (2006). [CrossRef] [PubMed]

*et al.*[4

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

*cloaking region*, and leaves region (III), the

*cloaked region*, radiation-free. This implies that there is no interaction between anything in the cloaked region and sources in the outside domain. Outside of the cloaking region, the radiation fields remain unchanged, as if neither the cloaking material nor any cloaked object exists. In principle, the regions (I), (I′), and (III) can be of arbitrary shapes, although the transformation matrix Eq.(2) associated with irregular shapes may be quite complicated.

## 3. Coordinate transformations for ellipsoidal cloaks

**312**, 1780 (2006). [CrossRef] [PubMed]

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

*x*-,

*y*-, and

*z*-directions with lengths

*α*

_{1}

*b*,

*α*

_{2}

*b*, and

*α*

_{3}

*b*, respectively, the outer boundary is described by

*α*

_{1},

*α*

_{2}, and

*α*

_{3}are positive numbers describing the aspect ratio of the ellipsoid, and do not transform as a vector. Inside this boundary, we start with the coordinate transformation determined by Eq.(14) in Ref. [4

**14**, 9794–9804 (2006). [CrossRef] [PubMed]

*r*as a scaled distance

*r*guarantees that

*r*=

*b*for any point

*x*on the outer boundary. Therefore,

^{i}*ε*is a small positive number. In the limit when

*ε*approaches 0, this surface becomes the origin. A little algebra will show that the transformed surface becomes

*α*

_{1}

*b*,

*α*

_{2}

*b*, and

*α*

_{3}

*b*into an ellipsoidal shell with the same lengths of the outer semi-axes and lengths of the inner semi-axes

*α*

_{1}

*a*,

*α*

_{2}

*a*, and

*α*

_{3}

*a*. Outside this region, we assume the identity transformation. In the following discussions, all equations apply only to the internal region.

*x*=2

*πa*

_{eff}/

*λ*where

*a*

_{eff}=(3

*V*/4

*π*)

^{1/3}with

*V*the volume of the irregular region, and

*λ*is the wavelength of some incident radiation. The ellipsoid shown in Fig. 2 has a size parameter of

*x*=8, with

*α*

_{1}=

*α*

_{2}=1, and

*α*

_{3}=2, or

*b*=2

^{1/3}8

*λ*/(2

*π*). This ellipsoid is practically a spheroid with its axis of symmetry lying along the

*z*direction.

*z*-direction. A coordinate transformation Eq.(5) with

*a*=0.5

*b*transforms this closed region into the cloaking region shown in Fig. 2(b). The grid lines in the region transform accordingly, and the transformed grid lines, which are geodesics in the transformed coordinates, can be interpreted as light rays and wave fronts in the presence of the cloak. As expected, the radiation field never penetrates into the cloaked region. Evidently, the light rays and wave fronts beyond the outer boundary remain the same, as an identity transformation is applied in this region.

*θ*to the

*z*-direction. We just need to do the coordinate transformation in the same Cartesian coordinates spanned by the three semi-axes of the ellipsoid as can be seen in Fig. 3, where we show the transformation applied to the same ellipsoid, but subject to an incident radiance in the

*x*-

*z*plane making an angle of

*θ*=30° to the

*z*-axis.

*ε*and

^{ij}*μ*in the cloaking region can be determined by substituting Eqs.(5) and (6) into Eq.(2), leading to a transformation matrix as follows:

^{ij}*i*and

*k*. For the vacuum,

*ε*=

^{ij}*μ*=

^{ij}*δ*. Therefore, a combination of Eq.(1) and Eq.(9) gives the material properties in the cloaking region as follows:

^{ij}*x*→

^{i}*x̃*,

^{i}*r*→

*r̃*,

*r*′→

*r*, with

*r*given by Eq.(6),

*r̃*and

*x̃*given by

^{i}*α*=1, Eq.(10) reduces to the simple results described by the Eq.(20) in Ref. [4

_{i}**14**, 9794–9804 (2006). [CrossRef] [PubMed]

## 4. Coordinate transformations for rounded-cuboids

*et al.*[12

12. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

*superellipsoid*[22, 23

23. T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shapes particles,” Part. Part. Syst. Charact. **19**, 256–268 (2002). [CrossRef]

23. T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shapes particles,” Part. Part. Syst. Charact. **19**, 256–268 (2002). [CrossRef]

*n*. We call this shape an

*order-n-cuboid*for convenience. As can be seen in Fig. 4(a), an order-

*n*-cuboid approaches a cube as the exponent

*n*increases. In Fig. 4(b) we show the scattering patterns associated with an order-10-cuboidal scattering particle with

*α*

_{1}=

*α*

_{2}=

*α*

_{3}and with a cubical scattering particle. As can be seen in this figure, an order-

*n*-cuboid with

*n*=10 is already a good approximation of a cuboid as far as light scattering is concerned.

*n*-cuboidal cloak, we can still use the coordinate transformation Eq.(5), and redefine the scaled magnitude

*r*again as

*b*replaced by

*a*.

## 5. Coordinate transformations for rounded cylinders

*e*=1 and

*m*=2/

*n*. Therefore, Eq.(12) becomes

*order-n-cylinder*. As Fig. 6(a) shows, an order-

*n*-cylinder approaches a cylinder as the exponent

*n*increases. Figure 6(b) implies that an order-

*n*-cylinder with

*n*=10 gives a good approximation to a cylinder as far as light scattering is concerned.

*n*-cylindrical cloak by redefining the scaled magnitude

*r*as

*b*replaced by

*a*. The corresponding transformation matrix will then be

*n*-cylindrical cloak is a hybrid of the corresponding tensors for an ellipsoidal cloak and for an order-

*n*-cuboidal cloak, given by

*z*=0, Eq.(17) and Eq.(4) are identical, and Eq.(18) and Eq.(6) are identical; When

*x*=0 or

*y*=0, Eq.(17) and Eq.(13) are identical, and Eq.(18) and Eq.(14) are identical. Namely, for an order-

*n*-cylindrical cloak, the wave fronts and light rays behave the same as those associated with an ellipsoidal cloak in the

*x*-

*y*plane (similar to the situation shown in Fig. 2(b) and Fig. 3(b)), and the same as those associated with an order-

*n*-cuboidal cloak in the

*x*-

*z*plane and the

*y*-

*z*plane (same as the situation shown in Fig. 5).

## 6. DDA simulations

15. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express **16**, 2068–2079 (2008). [CrossRef] [PubMed]

*x*-

*z*plane. The DDA simulations in the

*x*-

*z*plane for a rounded cylindrical cloak look similar to Fig. 9. It can be noticed that the plane-wave feature outside of the cloak is perfectly kept. The field in the cloaking region is compressed with patterns that are consistent with the predictions of the coordinate transformation approach as can be seen in Fig. 2(b), Fig. 3(b), and Fig. 5. The field in the cloaked region is close to 0 with a leakage of about 10% of the radiating field, or about 1% of the radiative energy, into the cloaked region due to the discretization of material properties in the DDA calculations.

*x*=8, cloak parameter

*a*=0.5

*b*, and two particle orientations, all simulated scattering efficiencies

*Q*

_{sca}=

*C*

_{sca}/(

*πa*

^{2}

_{eff}) are on the order of 10

^{-3}, which is 3 orders lower than that for a regular dielectric particle. The simulated scattering efficiencies are on the order of 10

^{-5}for smaller cloaks (

*x*=5) applied to smaller cloaked regions (

*a*=0.3

*b*).

## 7. Conclusions

*n*-cuboidal cloaks, and 3-D rounded-cylindrical cloaks approximated by order-

*n*-cylindrical cloaks. Numerical calculations using a generalized Discrete Dipole Approximation method were carried out to simulate the light scattering of plane-wave incident radiation associated with such cloaking objects. The simulated electric-field distributions in the vicinity of the cloak and scattering efficiencies suggest that these cloaks do not change the electric field outside at all. Therefore, outside observers cannot detect any object embedded within the cloaked region. Moreover, simulated scattering efficiencies of as low as 10

^{-5}have been observed.

## Acknowledgments

## References and links

1. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

3. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. |

4. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

5. | U. Leonhardt, “Optical conformal mapping,” Science |

6. | U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. |

7. | D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express |

8. | S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E |

9. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

10. | H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. |

11. | B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B |

12. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. |

13. | H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A |

14. | D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. |

15. | Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express |

16. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. |

17. | B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. |

18. | E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. |

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

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

21. | J. D. Jackson, |

22. | I. D. Faux and M. J. Pratt, |

23. | T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shapes particles,” Part. Part. Syst. Charact. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(290.0290) Scattering : Scattering

(160.3918) Materials : Metamaterials

(280.1350) Remote sensing and sensors : Backscattering

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 20, 2008

Revised Manuscript: April 4, 2008

Manuscript Accepted: April 4, 2008

Published: April 16, 2008

**Citation**

Yu You, George W. Kattawar, Peng-Wang Zhai, and Ping Yang, "Invisibility cloaks for irregular particles using coordinate transformations," Opt. Express **16**, 6134-6145 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6134

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### References

- A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell??s equations," J. Mod. Opt. 43, 773-793 (1996). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006). [CrossRef]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006). [CrossRef]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express 15, 14772-14782 (2007). [CrossRef] [PubMed]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977 (2006). [CrossRef] [PubMed]
- H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interaction with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
- B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007). [CrossRef]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell??s equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008). [CrossRef]
- H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen, and J. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2008). [CrossRef]
- D.-H. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]
- Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, "Zero-backscatter cloak for aspherical particles using a generalized DDA formalism," Opt. Express 16, 2068-2079 (2008). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, "Optical cloaking with metamaterials," Nat. Photon. 1, 224-227 (2007). [CrossRef]
- B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008). [CrossRef] [PubMed]
- E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973). [CrossRef]
- B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975).
- I. D. Faux and M. J. Pratt, Computational geometry for design and manufacture (Wiley, Chichester 1979).
- T. Wriedt, "Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superellipsoids and realistically shapes particles," Part. Part. Syst. Charact. 19, 256-268 (2002). [CrossRef]

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