## Generalization of the optical theorem for light scattering from a particle at a planar interface |

JOSA A, Vol. 30, Issue 12, pp. 2519-2525 (2013)

http://dx.doi.org/10.1364/JOSAA.30.002519

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### Abstract

The optical theorem provides a powerful tool for calculating the extinction cross section of a particle from a solution to Maxwell’s equations, relating the cross section to the scattering amplitude in the forward direction. The theorem has been generalized by a number of other workers to consider a particle near an interface between media with different refractive indices. Here we present a derivation of the generalized optical theorem that is valid for a particle embedded in the interface, as well as an incident beam undergoing total internal reflection. We also obtain an additional useful physical result: we show that the far-field scattered field must be zero in the direction parallel to the interface. Our results enable the verification of computations of scattering by particles embedded in interfaces and may be relevant to experiments on colloidal particles at fluid interfaces.

© 2013 Optical Society of America

## 1. INTRODUCTION

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

6. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A **11**, 2851–2861 (1994). [CrossRef]

7. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A **25**, 1504–1513 (2008). [CrossRef]

8. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer **110**, 323–327 (2009). [CrossRef]

9. K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. **26**, 2995–2999 (1987). [CrossRef]

13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E **71**, 56610 (2005). [CrossRef]

14. D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. **11**, 138–142 (2012). [CrossRef]

*in situ*determination of the contact angles of submicrometer particles. The first step toward such a solution is the generalization of the optical theorem for a particle embedded at an interface.

*et al.*[16

16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. **1**, 131–135 (2004). [CrossRef]

## 2. DEFINITIONS AND FORMALISM

### A. Geometry

### B. Incident, Reflected, and Transmitted Fields

*incoming field*.

### C. Scattered Fields

## 3. CALCULATING THE SCATTERED POWER

### A. Form of the Key Terms

### B. Key Integral

### C. Generalized Optical Theorem

*et al.*[16

16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. **1**, 131–135 (2004). [CrossRef]

## 4. TOTAL INTERNAL REFLECTION

*et al.*[12

12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E **70**, 36611 (2004). [CrossRef]

## 5. TWO OTHER ARGUMENTS FOR ZERO FAR-FIELD SCATTERING ALONG THE INTERFACE

12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E **70**, 36611 (2004). [CrossRef]

13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E **71**, 56610 (2005). [CrossRef]

*will not*recover the case of a particle in a homogeneous medium. Even with a vanishingly small index mismatch, it is impossible to match fields in the far field, and so the Fresnel reflection coefficient must go to

18. W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. **67**, 1615–1619 (1977). [CrossRef]

*et al.*, who derive a similar version of the optical theorem by separately treating waves reflected and transmitted by the interface. The most rigorous argument for zero far-field scattering in the plane of the interface comes from the impossibility of matching the asymptotic form for the scattered field at the interface.

## 6. CONCLUSION

## ACKNOWLEDGMENTS

## REFERENCES

1. | M. Born, E. Wolf, and A. B. Bhatia, |

2. | H. van de Hulst, |

3. | C. F. Bohren and D. R. Huffman, |

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

5. | V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. |

6. | D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A |

7. | M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A |

8. | M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer |

9. | K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. |

10. | L. Sung, G. W. Mulholland, and T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. |

11. | J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. |

12. | P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E |

13. | D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E |

14. | D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. |

15. | P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. |

16. | D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. |

17. | A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in |

18. | W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. |

**OCIS Codes**

(290.0290) Scattering : Scattering

(290.2200) Scattering : Extinction

(290.2558) Scattering : Forward scattering

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: August 14, 2013

Revised Manuscript: October 17, 2013

Manuscript Accepted: October 17, 2013

Published: November 12, 2013

**Virtual Issues**

Vol. 9, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Alex Small, Jerome Fung, and Vinothan N. Manoharan, "Generalization of the optical theorem for light scattering from a particle at a planar interface," J. Opt. Soc. Am. A **30**, 2519-2525 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-12-2519

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### References

- M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
- H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
- V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002). [CrossRef]
- D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]
- M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008). [CrossRef]
- M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009). [CrossRef]
- K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987). [CrossRef]
- L. Sung, G. W. Mulholland, and T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999). [CrossRef]
- J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. 41, 5405–5412 (2002). [CrossRef]
- P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004). [CrossRef]
- D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005). [CrossRef]
- D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012). [CrossRef]
- P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).
- D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004). [CrossRef]
- A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.
- W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]

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