## Computation of Mie derivatives |

Applied Optics, Vol. 52, Issue 20, pp. 4997-5006 (2013)

http://dx.doi.org/10.1364/AO.52.004997

Enhanced HTML Acrobat PDF (526 KB)

### Abstract

Analytical expressions are obtained for the derivatives of Mie scattering coefficients with respect to the electrical radius of the spherical scattering particle, and to the relative permittivity and permeability of both the particle and the surrounding medium. Their corresponding approximate expressions are developed to avoid numerical overflow based on the logarithmic derivative of Riccati–Bessel functions. The analytical expressions have been verified by comparing their results with those calculated by analytical expressions developed by Mathematica. Compared with the numerical derivative, the analytical expressions and approximate expressions show a higher accuracy and are 2.0 and 2.8 times, respectively, faster in the case of a single magnetodielectric sphere. Generally, for spheres with an electrical radius in a large range, the approximate expressions can yield acceptable accuracy and computation time up to a high order. This work can be used in the design of nonmetallic metamaterials, and in the retrieval of aerosol properties from remote sensing data. An example calculation is given for the design of an optical, all-dielectric, mu-negative metamaterial consisting of a simple cubic array of tellurium nanoparticles.

© 2013 Optical Society of America

**OCIS Codes**

(290.4020) Scattering : Mie theory

(160.3918) Materials : Metamaterials

**ToC Category:**

Scattering

**History**

Original Manuscript: April 25, 2013

Revised Manuscript: June 5, 2013

Manuscript Accepted: June 12, 2013

Published: July 10, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Yang Li and Nicola Bowler, "Computation of Mie derivatives," Appl. Opt. **52**, 4997-5006 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ao-52-20-4997

Sort: Year | Journal | Reset

### References

- J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012). [CrossRef]
- R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007). [CrossRef]
- Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009). [CrossRef]
- Y. Li and N. Bowler, “Rational design of double-negative metamaterials consisting of 3D arrays of two different nonmetallic spheres arranged on a simple tetragonal lattice,” in 2011 IEEE International Symposium on Antennas and Propagation (2011), pp. 1494–1497.
- Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011). [CrossRef]
- Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012). [CrossRef]
- I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006). [CrossRef]
- I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009). [CrossRef]
- K. L. Kumley and E. F. Kuester, “Effect of scatterer size variations on the reflection and transmission properties of a metafilm,” presented at National Radio Science Meeting, Boulder, CO (2012).
- E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).
- Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013). [CrossRef]
- G. Thomas, S. Bass, R. Grainger, and A. Lambert, “Retrieval of aerosol refractive index from extinction spectra with a damped harmonic-oscillator band model,” Appl. Opt. 44, 1332–1341 (2005). [CrossRef]
- O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005). [CrossRef]
- F. Xu and A. B. Davis, “Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method,” Opt. Lett. 36, 4464–4466 (2011). [CrossRef]
- R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012). [CrossRef]
- R. Grainger, J. Lucas, G. Thomas, and G. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004). [CrossRef]
- G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 330, 377–445 (1908). [CrossRef]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).
- H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
- B. Verner, “Note on the recurrence between Mie’s coefficients,” J. Opt. Soc. Am. 66, 1424–1425 (1976). [CrossRef]
- H. Du, “Mie-scattering calculation,” Appl. Opt. 43, 1951–1956 (2004). [CrossRef]
- L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).
- J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
- J. V. Dave, “Scattering of electromagnetic radiation by a large absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969). [CrossRef]
- W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980). [CrossRef]
- C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003). [CrossRef]
- P. E. Falloon, “Theory and computation of spheroidal harmonics with general arguments,” Master of Science, The University of Western Australia (2001).
- C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.