## The expansion coefficients of arbitrary shaped beam in oblique illumination

Optics Express, Vol. 15, Issue 2, pp. 735-746 (2007)

http://dx.doi.org/10.1364/OE.15.000735

Acrobat PDF (196 KB)

### Abstract

In Generalized Lorenz-Mie theories, (GLMTs), the most difficult task concerns the description of the illuminating beam. We provide an approach for expansions of the incident arbitrary shaped beam in spherical and spheroidal coordinates in the general case of oblique illumination. The representations for shaped beam coefficients are derived by using addition theorem for spherical vector wave functions under coordinate rotations. For

© 2007 Optical Society of America

## 1. Introduction

1. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A **5**,1427–1443 (1988). [CrossRef]

*g*

^{m}

_{n}are obtained when the shaped beam propagates parallel to the coordinate axis [1–3

1. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A **5**,1427–1443 (1988). [CrossRef]

4. K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the
,” J. Opt. Soc. Am. A **14**,
3014–3025 (1997). [CrossRef]

5. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. **36**,5188–5198 (1997). [CrossRef] [PubMed]

_{00}mode laser beam by using spherical harmonic expansions of the scalar potential for the electric and magnetic fields and by matching the electromagnetic fields at the surface [7

7. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. **64**,1632–1639 (1988). [CrossRef]

8. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. **34**,5542–5551 (1995). [CrossRef] [PubMed]

9. E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propagat. **41**,259–303 (1993). [CrossRef]

11. H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, “The imaging properties of scattering particles in laser beams,” Meas. Sci. Technol. **10**,564–574 (1999). [CrossRef]

12. T. Evers, H. Dahl, and T. Wriedt, “Extension of the program 3D MMP with a fifth order Gaussian beam,” Electron. Lett. **32**,1356–1357 (1996). [CrossRef]

13. A. Doicu and T. Wriedt, “Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Modern. Opt. **44**,785–801 (1997). [CrossRef]

14. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. **40**,2501–2509 (2001). [CrossRef]

*g*,

^{ms}_{n}*g´*and

^{ms}_{n}*G*,

^{m}_{n}*G´*, corresponding to spherical and spheroidal coordinates respectively, are given. In section 3, for particle center located on Gaussian beam axis and plane wave, the simplified representations for beam shape coefficients are obtained. The convergence of the beam shape coefficients is discussed in section 4. Section 5 is a conclusion.

^{m}_{n}## 2. Expansion of incident shaped beam with respect to spherical and spheroidal coordinates

### 2.1. Shaped beam coefficients in spherical coordinates

*z*´ to the positive

*z*´ of

*O*´

*x*´

*y*´

*z*´, with the middle of beam waist located at origin

*O*´. The system

*Ox*´´

*y*´´

*z*´´ is parallel to

*O*´

*x*´

*y*´

*z*´ and the Cartesian coordinates of

*O*´ in

*Ox*´´

*y*´´

*z*´´ are

*x*

_{0},

*y*

_{0},

*z*

_{0}. The center of the scatterer is located at the point

*O*of the Cartesian coordinate system

*which is obtained by rotating the system*

*Oxyz**Ox*´´

*y*´´

*z*´´ through Euler 23]. For spheroidal particle, the major axe of the spheroid is along the z axis. In this paper, we assume that the time-dependent part of the electromagnetic fields is

*e*.

^{-iωt}*Ox*´´

*y*´´

*z*´´ [1

1. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A **5**,1427–1443 (1988). [CrossRef]

14. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. **40**,2501–2509 (2001). [CrossRef]

*z*´ axis. They can be 1–3

**5**,1427–1443 (1988). [CrossRef]

6. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. **36**,2971–2978 (1997). [CrossRef] [PubMed]

*θ*,

*ϕ*are the angle coordinates of a point in space defined in the usual manner with respect to the

*xyz*axes, and

*θ*´´,

*ϕ*´´ to the

*x*´´

*y*´´

*z*´´ axes. Here the associated Legendre functions

*P*(cos

^{m}_{n}*θ*) 27].

*m*,

*s*,

*n*) are given by

*m*,

*s*,

*n*) = ρ

_{1}(

*m*,

*s*,

*n*) + iρ

_{2}(

*m*,

*s*,

*n*). ρ

_{1}(

*m*,

*s*,

*n*) and ρ

_{2}(

*m*,

*s*,

*n*) are the real and imaginary parts of ρ(

*m*,

*s*,

*n*) respectively.

*m*,

*s*,

*n*)into Eq. (2) and having the real and imaginary parts on both sides of the equality be equal respectively, we can obtain

*Oxyz*and

*Ox*´´

*y*´´

*z*´´, by using Eq. (5) multiplied by the spherical Bessel functions

*j*(

_{n}*kr*) and the definitions of

**w**stands for the spherical vector wave functions

**m**or

**n**,

*Oxyz*, as follows (TE Mode):

### 2.2. Shaped beam coefficients in spheroidal coordinates

**W**stands for the spheroidal vector wave functions

**M**or

**N**.

*a*representing the expansion coefficients, we can obtain

_{mn}*n*-

*s*=

*r*. One replaces

*s*by

*m*,

*l*by

*n*and

*m*by

*s*, then Eq. (14) or the expansion of the incident beam in terms of spheroidal vector wave functions attached to an arbitrarily oriented spheroid can be written as follows (TE Mode):

17. Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B. **84**,485–492 (2006). [CrossRef]

*γ*,

_{mn}*δ*,

_{mn}*α*,

_{mn}*β*,

_{mn}## 3. Expressions of beam shape coefficients for particle center located on Gaussian beam axis or plane wave illumination

*x*

_{0}= 0,

*y*

_{0}= 0) or plane wave illumination (

*x*

_{0}= 0,

*y*

_{0}= 0,

*z*

_{0},

*w*

_{2}→ ∞, we have [1

**5**,1427–1443 (1988). [CrossRef]

6. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. **36**,2971–2978 (1997). [CrossRef] [PubMed]

*w*

_{0}→ ∞ in

*g*) as follows

_{n}## 4. Numerical results

*b*of

*a*/

*b*=2 , and the size parameter 2

*πa*/λ=5.

*n*of the beam shape coefficients

*w*

_{0}=, given

*α*=

*π*/4,

*β*=

*π*/6,

*γ*=0,

*x*

_{0}=0,

*y*

_{0}=0,

*z*

_{0}=0. Computational results indicate that

*n*and

*m*.

## 5. Conclusion

*L*of approximation in the description of the Gaussian beam, the field components in the spherical coordinates (

*r*,θ ´´,ϕ´´) with respect to

*Ox*´´

*y*´´

*z*´´ are obtained as follows [1

**5**,1427–1443 (1988). [CrossRef]

25. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A **19**,1177–1179 (1979). [CrossRef]

*w*

_{0}is the beam’s electric-field half-width in the focal plane.

*E*

_{0}and

*H*

_{0}are linked by

**E**by -

**H**,

**H**by

**E**, ε by μ, μ by ε , from Eqs. (A1)-(A6)order L of approximation can also be expressed by

**5**,1427–1443 (1988). [CrossRef]

6. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. **36**,2971–2978 (1997). [CrossRef] [PubMed]

*C*coefficients are normalized factors for negative values of the index

_{nm}*m*

## Acknowledgments

## References and links

1. | G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A |

2. | G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. |

3. | G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients m
,” J. Opt. Soc. Am. A |

4. | K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the
,” J. Opt. Soc. Am. A |

5. | Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. |

6. | A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. |

7. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. |

8. | J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. |

9. | E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propagat. |

10. | F. M. Schulz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic wave by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. |

11. | H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, “The imaging properties of scattering particles in laser beams,” Meas. Sci. Technol. |

12. | T. Evers, H. Dahl, and T. Wriedt, “Extension of the program 3D MMP with a fifth order Gaussian beam,” Electron. Lett. |

13. | A. Doicu and T. Wriedt, “Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Modern. Opt. |

14. | Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. |

15. | Y. Han and Z. Wu, “The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam,” IEEE Trans. Antennas Propagat. |

16. | H. Y. Zhang and Y. P. Han, “Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam,” IEEE Trans. Antennas Propagat. |

17. | Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B. |

18. | B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. |

19. | S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. |

20. | O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. |

21. | S. Asano and G. Yamamoto, “Light scattering by a spheroid particle,” Appl. Opt. |

22. | J. Dalmas and R. Deleuil, “Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr,” Q. Appl. Math. |

23. | A. R. Edmonds, |

24. | B. P. Sinha and R. H. Macphie, “Translational addition theorems for spheroidal scalar and vector wave functions,” Q. Appl. Math. |

25. | L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A |

26. | C. Flammer, |

27. | J. A. Stratton, |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(260.2110) Physical optics : Electromagnetic optics

(290.4020) Scattering : Mie theory

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: September 11, 2006

Revised Manuscript: January 1, 2007

Manuscript Accepted: January 4, 2007

Published: January 22, 2007

**Virtual Issues**

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

**Citation**

Yiping Han, Huayong Zhang, and Guoxia Han, "The expansion coefficients of arbitrary shaped beam in oblique illumination," Opt. Express **15**, 735-746 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-735

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### References

- G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988). [CrossRef]
- G. Gouesbet, G. Gréhan, and B. Maheu, "Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods," Appl. Opt. 27, 4874-4883 (1988). [CrossRef] [PubMed]
- G. Gouesbet, G. Gréhan, and B. Maheu, "Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory," J. Opt. Soc. Am. A 7, 998-1003 (1990). [CrossRef]
- K. F. Ren, G. Gréhan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997). [CrossRef]
- Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres," Appl. Opt. 36, 5188-5198 (1997). [CrossRef] [PubMed]
- A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997). [CrossRef] [PubMed]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988). [CrossRef]
- J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995). [CrossRef] [PubMed]
- E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993). [CrossRef]
- F. M. Schulz, K. Stamnes, and J. J. Stamnes, "Scattering of electromagnetic wave by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates," Appl. Opt. 37, 7875-7896 (1998). [CrossRef]
- H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999). [CrossRef]
- T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996). [CrossRef]
- A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997). [CrossRef]
- Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001). [CrossRef]
- Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001). [CrossRef]
- H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005). [CrossRef]
- Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006). [CrossRef]
- B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).
- S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
- O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
- S. Asano and G. Yamamoto, "Light scattering by a spheroid particle," Appl. Opt. 14, 29-49 (1975). [PubMed]
- J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).
- A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.
- B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).
- L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979). [CrossRef]
- C. Flammer, Spheroidal wave functions, (Stanford University Press, Stanford, California, 1957).
- J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).

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