## Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams

Optics Express, Vol. 16, Issue 8, pp. 5926-5933 (2008)

http://dx.doi.org/10.1364/OE.16.005926

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### Abstract

A closed form formula is found for the Mie scattering coefficents of incident complex focus beams (which are a nonparaxial generalization of Gaussian beams) with any numerical aperture. This formula takes the compact form of multipoles evaluated at a single complex point. Included are the cases of incident scalar fields as well as electromagnetic fields with many polarizations, such as linear, circular, azimuthal and radial. Examples of incident radially and azimuthally polarized beams are presented.

© 2008 Optical Society of America

## 1. Introduction

## 2. Characteristics of complex focus fields

9. Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. **10**, 719–730 (1967). [CrossRef]

10. F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A **366**, 155–171 (1979). [CrossRef]

11. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A **57**, 2971–2979 (1998). [CrossRef]

12. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

11. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A **57**, 2971–2979 (1998). [CrossRef]

*U*

_{0}is a constant and

**ρ**_{0}=

**r**

_{0}+i

**q**. These fields do not include evanescent waves, but do include a significant counter-propagating component for small

*kq*. For

*kq*≫1, where

*q*=|

**q**| CF fields tend to paraxial Gaussian beams focused at

**r**

_{0}, traveling in the direction of

**q**and with waist width proportional to

11. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A **57**, 2971–2979 (1998). [CrossRef]

*kq*≳3 (i.e., when the counter-propagating components are negligible).

**ρ**_{0}. The resulting CF fields are given by the expressions:

**p**is a unit vector in the direction of the electric or magnetic dipole moment, respectively. Beams with several polarizations of interest can be obtained by using these fields. For example, a linearly polarized (in the sense of Richards and Wolf [13

13. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A **253**, 358–379 (1959). [CrossRef]

*z*direction and whose polarization (before focusing) is in the

*y*direction, can be written as

**E**

^{(E)}

_{CF}(

**r**;i

*q*

**ẑ**,

**ŷ**)+

**E**

^{(B)}

_{CF}(

**r**;i

*q*

**ẑ**,

**x̂**) [12

12. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

**E**

^{(E)}

_{CF}(

**r**;i

*q*

**ẑ**,

**ẑ**) and

**E**

^{(B)}

_{CF}(

**r**;i

*q*

**ẑ**,

**ẑ**) [14

14. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. **24**, 1543–1545 (1999). [CrossRef]

15. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. **179**, 1–7 (2000). [CrossRef]

## 3. Mie scattering of scalar CF fields

**r**

*,*

_{a}**r**

*and*

_{b}**r**

*:*

_{c}*(*

_{lm}**r**) are defined as

*j*is the spherical Bessel function of the first kind of order

_{l}*l*, (

*r*,

*θ*,

_{r}*ϕ*) are the spherical coordinates of the position vector

_{r}**r**and

*Y*is a spherical harmonic, defined by

_{lm}**r**

*=*

_{a}**r**(the spatial variable) and

**r**

*=*

_{b}

**ρ**_{0}=

**r**

_{0}+i

**q**. Let us now consider a scattering sphere of radius

*R*and refractive index

*n*. It is then convenient for

**r**

*to coincide with the center of the scatterer, which, for simplicity, is chosen as the origin (*

_{c}**r**

*=*

_{c}**0**). The incident CF field is then given by the remarkably simple expression

*ρ*

_{0}sin

*θ*cos

_{ρ}*ϕ*,

_{ρ}*ρ*

_{0}sin

*θ*sin

_{ρ}*ϕ*,

_{ρ}*ρ*

_{0}cos

*θ*) indeed returns

_{ρ}

**ρ**_{0}. The compact form of the coefficients in the multipolar expansion in Eq. (6) is the key result of this manuscript; the coefficients are themselves multipoles evaluated at a single complex point.

*U*

_{(sc)}(

**r**) is then found to have the form

*is the multipole field containing only outgoing components, i.e.,*

_{lm}*h*

^{(1)}

*is a spherical Hankel function of the first kind, so that ∏*

_{l}*satisfies the Sommerfeld radiation condition. The coefficients*

_{lm}*c*are the standard scaling factors for the multipolar components, which result from imposing boundary conditions (i.e., continuity of the field and its derivative with respect to

_{l}*r*) at the surface of the spherical scatterer. For a scalar field, these coefficients are given by [17]

## 4. Mie scattering of electromagnetic fields

**Y**

*(*

_{lm}*θ*,

*ϕ*) and

**Z**

*(*

_{lm}*θ*,

*ϕ*) are the vector spherical harmonics [18, 19

19. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. **15**, 234–244 (1974). [CrossRef]

**L**given by

**ρ**_{0}, defined in Eqs. (2), can then be written, respectively, as the linear combinations

*γ*

^{(I, II)}

*are given in terms of multipole fields at a complex point:*

_{lm}**Λ**

^{(I)}and

**Λ**

^{(II)}upon scattering. Thus, the scattered fields take the forms given below

**∏**

^{(I)}

*and*

_{lm}**∏**

^{(II)}

*are defined by replacing the spherical Bessel functions in Eqs. (11) with spherical Hankel functions of the first kind. The coefficients in Eqs. (16) may be calculated by applying the standard boundary conditions at a dielectric interface. These coefficients are found to be [17]*

_{lm}## 5. Examples: Radially and Azimuthally polarized beams

*z*axis [14

14. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. **24**, 1543–1545 (1999). [CrossRef]

*l*=1 to

*l*

_{max}=12. Note that when the scatterer is centered on the beam’s axis, only the

*m*=0 terms contribute. (For an arbitrarily polarized CF beam incident upon an on-axis scatterer,

*m*must be summed only from -1 to 1.)

*kR*=5 and refractive index

*n*=2, where the focus of the beam is located at the upper edge of the sphere (i.e.,

*k*

**r**

_{0}=5

**x̂**) and

*k*

**q**=4

**ẑ**. Each figure includes (a) a depiction of the incident field’s magnitude over the

*x*-

*z*plane, so as to demonstrate the relative location of the scatterer to the beam, as well as (b) the magnitude of the total field (including scattered and inside the scattering sphere). Additionally, the coefficients for both

**Λ**

^{(I)}

*(*

_{lm}**r**) and

**Λ**

^{(II)}

*(*

_{lm}**r**) are depicted in (c) to demonstrate convergence for

*l*

_{max}=12. Finally, the radiant intensity (i.e., irradiance in the far field) of the scattered field is shown in (d), configured to show back scattering in the center of the diagram, with forward scattering at the edges. Both figures have accompanying multimedia files which show the development of the fields, coefficients and radiant intensity as the focus moves laterally to the center of the scatterer, then axially to the front of the scatterer, and finally as the beam becomes more collimated (4≤

*kq*≤9). The location of the focus and the value of the parameter

*q*are indicated in (a).

## 6. Concluding remarks

20. M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express **14**, 6894–6905 (2006). [CrossRef] [PubMed]

## Acknowledgments

## A. Derivation of Eq. (3)

**u**and

**u**′ are unit vectors integrated over the complete unit sphere, and

*δ*(

**u**,

**u**′) is a Dirac delta function over the unit sphere. This function can be expressed in terms of the spherical harmonics, since they are a complete, orthonormal basis:

*θ*and

*ϕ*are the polar and azimuthal angles associated with

**u**, and likewise for the prime variables. By inserting Eq. (19) into Eq. (18), and using the relation [19

19. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. **15**, 234–244 (1974). [CrossRef]

## B. Derivation of Eq. (15)

*δ⃡*

_{⊥}(

**u**,

**u**′) is a tensor Dirac delta function for transverse vector distributions over the sphere, which satisfies

**u**·

*δ⃡*

_{⊥}(

**u**,

**u**′)=

*δ⃡*

_{⊥}(

**u**,

**u**′)·

**u**=

**0**. This function can be composed in terms of vector spherical harmonics as

19. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. **15**, 234–244 (1974). [CrossRef]

## References and links

1. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

2. | D. Ganic, X. Gan, and M. Gu, “Optical trapping force with annular and doughnut laser beams based on vectorial diffraction,” Opt. Express |

3. | Z. J. Smith and A. J. Berger, “Integrated raman- and angular- scattering microscopy,” Opt. Lett. |

4. | G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) |

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

6. | G. Gouesbet, G. Gréhan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) |

7. | R. Kant, “Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams,” J. Mod. Opt. |

8. | A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express |

9. | Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. and Quant. Elec. |

10. | F. A. L. Cullen and P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. Royal Soc. Lond. A |

11. | C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A |

12. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

13. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. Roy. Soc. A |

14. | C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. |

15. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

16. | J. D. Jackson, |

17. | M. Kerker, |

18. | J. D. Jackson, |

19. | A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. |

20. | M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(260.0260) Physical optics : Physical optics

(290.4020) Scattering : Mie theory

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: January 30, 2008

Revised Manuscript: April 4, 2008

Manuscript Accepted: April 8, 2008

Published: April 11, 2008

**Virtual Issues**

Vol. 3, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Nicole J. Moore and Miguel A. Alonso, "Closed form formula for Mie scattering of nonparaxial analogues of Gaussian beams," Opt. Express **16**, 5926-5933 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5926

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### References

- K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2807 (2004). [CrossRef]
- D. Ganic, X. Gan, and M. Gu, "Optical trapping force with annular and doughnut laser beams based on vectorial diffraction," Opt. Express 13, 1260-1265 (2005). [CrossRef] [PubMed]
- Z. J. Smith and A. J. Berger, "Integrated raman- and angular- scattering microscopy," Opt. Lett. 33, 714-716 (2008). [CrossRef] [PubMed]
- G. Gouesbet and G. Grehan, "Sur la generalisation de la theorie de Lorenz-Mie," J. Opt. (Paris) 13, 97-103 (1982). [CrossRef]
- G. Gouesbet, G. Gr’ehan, 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. Gr’ehan, and B. Maheu, "Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism," J. Opt. (Paris) 16, 83-93 (1985). [CrossRef]
- R. Kant, "Generalized Lorenz-Mie scattering theory for focused radiation and finite solids of revolution: case i: symmetrically polarized beams," J. Mod. Opt. 52, 2067-2092 (2005). [CrossRef]
- A. S. van de Nes and P. Torok, "Rigorous analysis of spheres in Gauss-Laguerre beams," Opt. Express 15, 360-13,374 (2007). [CrossRef]
- Y. A. Kravtsov, "Complex rays and complex caustics," Radiophys. and Quant. Elec. 10, 719-730 (1967). [CrossRef]
- F. A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. Royal Soc. Lond. A 366, 155-171 (1979). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999). [CrossRef]
- B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation," Opt. Lett. 24, 1543-1545 (1999). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 108-109.
- M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969), pp. 39-54.
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999), pp. 430-431.
- A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234-244 (1974). [CrossRef]
- M. A. Alonso, R. Borghi, and M. Santarsiero, "New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci," Opt. Express 14, 6894-6905 (2006). [CrossRef] [PubMed]

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