## Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces |

Biomedical Optics Express, Vol. 2, Issue 7, pp. 1893-1906 (2011)

http://dx.doi.org/10.1364/BOE.2.001893

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### Abstract

Ordinary Bessel beams are described in terms of the generalized Lorenz-Mie theory (GLMT) by adopting, for what is to our knowledge the first time in the literature, the integral localized approximation for computing their beam shape coefficients (BSCs) in the expansion of the electromagnetic fields. Numerical results reveal that the beam shape coefficients calculated in this way can adequately describe a zero-order Bessel beam with insignificant difference when compared to other relative time-consuming methods involving numerical integration over the spherical coordinates of the GLMT coordinate system, or quadratures. We show that this fast and efficient new numerical description of zero-order Bessel beams can be used with advantage, for example, in the analysis of optical forces in optical trapping systems for arbitrary optical regimes.

© 2011 OSA

## 1. Introduction

1. G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. **330**(3), 377–452 (1908). [CrossRef]

7. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. **37**(19), 4218–4225 (1998). [CrossRef] [PubMed]

8. J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transf. **110**(11), 800–807 (2009). [CrossRef]

10. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models,” J. Quant. Spectrosc. Radiat. Transf. **112**(1), 1–27 (2011). [CrossRef]

12. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A **11**(9), 2503–2515 (1994). [CrossRef]

13. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A **11**(9), 2516–2525 (1994). [CrossRef]

14. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. **108**(4-6), 343–354 (1994). [CrossRef]

15. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. **155**(1-3), 169–179 (1998). [CrossRef]

16. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express **1**(5), 1284–1301 (2010). [CrossRef] [PubMed]

17. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**(4-6), 239–245 (2001). [CrossRef]

20. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**(9), 2787–2809 (2004). [CrossRef] [PubMed]

*λ*is much smaller than the diameter d of the particle (

*λ*<< d), and in the case where

*λ*>> d, for which a dipole model can be assumed. Also, in the Rayleigh-Gans regime, simple formulas can be used to predict the optical trapping forces with relative success [21

21. A. N. Rubinov, A. A. Afanas’ev, I. E. Ermolaev, Y. A. Kurochkin, and S. Y. Mikhnevich, “Localization of spherical particles under the action of gradient forces in the field of a zero-order Bessel beam. Rayleigh-Gans approximation,” J. Appl. Spectrosc. **70**(4), 565–572 (2003). [CrossRef]

22. G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a Bessel beam,” Opt. Express **15**(21), 13972–13987 (2007). [CrossRef] [PubMed]

## 2. The GLMT and the ILA applied to ordinary Bessel beams

*c*and angular frequency

*ω*

_{0}parallel to +

*z*, with its optical axis displaced

*ρ*

_{0}= (

*x*

_{0},

*y*

_{0}) from the

*z*-axis and making an angle

*ϕ*

_{0}relative to the

*x*-axis, according to Fig. 1 . Although we could have shifted the beam in a three-dimensional fashion, we must take into account that, as long as the ideal definition is considered and that the corresponding shift in

*z*be constricted to the focal length (thus keeping the radial Bessel profile still valid), this implies in a more complete but unnecessary mathematical formulation. It is well known that the longitudinal and transverse wave numbers for a BB are given by

*k*=

_{z}*k*cos

*θ*and

_{a}*k*=

_{ρ}*k*sin

*θ*, respectively,

_{a}*θ*being the associated axicon angle and

_{a}*k*=

*ω*

_{0}/

*c*the wavenumber [23

23. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

*x*or in

*y*, we can write its electric field as

*J*

_{0}(.) is the ordinary Bessel function and

*E*

_{0}is the electric field strength. A factor exp(iω0t) has been omitted for convenience. The field profile in Eq. (1) may be considered to be valid as long as β = sinθa/(1 + cosθa) << 1 [22

22. G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a Bessel beam,” Opt. Express **15**(21), 13972–13987 (2007). [CrossRef] [PubMed]

*E*and

_{r}*H*of the beam as functions of spherical coordinates (

_{r}*r*,

*θ*,

*ϕ*) relative to the origin

*O*in Fig. 1 (in the case of optical trapping, for instance,

*O*would coincide with the center of a spherical scatterer, as will be the case in Section 4). The radial component

*E*of the electric field for this BB is readily found by putting a multiplicative factor sin

_{r}*θ*cos

*ϕ*(sin

*θ*sin

*ϕ*) to the right of Eq. (1) in the case of an

*x*-polarized (

*y*-polarized) beam, and replacing

*ρ*by

*r*sin

*θ*and

*z*by

*r*cos

*θ*. Then, for the radial component

*H*of the magnetic field, we recall Faraday’s law in the frequency domain and in its differential form, so that we can write

_{r}*E*and

_{r}*H*, after some simple algebra, as

_{r}*ε*and

*μ*being, respectively, the permittivity and permeability of this medium. The slashes represent either an

*x*- or a

*y*-polarized beam, with its corresponding sin

*ϕ*or cos

*ϕ*factor.

7. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. **37**(19), 4218–4225 (1998). [CrossRef] [PubMed]

*n*and

*m*are integer numbers ranging from 1 ≤

*n*< ∞ and –

*n*<

*m*<

*n*, respectively, corresponding to a particular spherical harmonic function

*kr*to (

*n*+ 1/2) and

*θ*to π/2, while

7. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. **37**(19), 4218–4225 (1998). [CrossRef] [PubMed]

*ε*being 1 if

_{p}*p*= 0 and 2 otherwise. Thus, letting

*ρ*

_{0}= 0 (on-axis case), because

*J*

_{|}

_{m}_{| ± 1}(

*ρ*

_{0}

*k*sin

*θ*) = 0 whenever

_{a}*ρ*

_{0}= 0 and

*m*≠ 1, as expected for an on-axis (

*z*-axis) beam, the only non-zero beam-shape coefficients for

*ρ*

_{0}= 0 are

*n*.

*x*- or

*y*- polarized (on- or off-axis) ordinary Bessel beam in the GLMT using the integral localized approximation. They may be greatly simplified if, for example,

*ϕ*

_{0}= 0 for

*x*-polarization, or

*ϕ*

_{0}= π/2 together with an

*y*-polarized BB, or other analogous particular cases. Notice that we can readily extend the set of equations (10)–(13) to circularly polarized beams by means of simple symmetry relations and writing each of the new TM and TE BSC’s as functions of both TM and TE BSC’s in Eqs. (10)–(13) [15

15. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. **155**(1-3), 169–179 (1998). [CrossRef]

## 3. Numerical results for the BSC’s of an ordinary Bessel beam

*R*=

*kr*an auxiliary coordinate and

*j*(

_{n}*R*) are spherical Bessel functions of integer order

*n*. If

*E*and

_{r}*H*represent an exact solution to Maxwell’s equations, then

_{r}*R*is arbitrary because, in such a case, the integrals over the spherical angles

*θ*and

*ϕ*are proportional to

*j*(

_{n}*R*)/

*R*[12

12. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A **11**(9), 2503–2515 (1994). [CrossRef]

13. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A **11**(9), 2516–2525 (1994). [CrossRef]

*R*when numerically evaluating Eqs. (14) and (15). It is recommended, because of the van de Hulst principle, and we have adopted it here, to impose

*R*=

*n*+ 1/2 [11]. This holds because the most significant contribution of the integrands in Eqs. (14) and (15) in the case of a zero-order Bessel beam also happens around

*R*=

*n*+ 1/2, as one can verify by plotting several curves of them and performing the same analysis as those for a focused Gaussian beam [25]. By recalling orthogonality relations of the spherical Bessel functions, the above equations may be written as triple integrals, thus getting rid of the

*R*dependence, with a corresponding undesirable computational cost [12

12. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A **11**(9), 2503–2515 (1994). [CrossRef]

13. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A **11**(9), 2516–2525 (1994). [CrossRef]

*ϕ*and

*θ*. Furthermore, for method F2, the integral over

*R*was performed from 0 to 200 for

*n*= 1 and up to 10, from 0 to 500 for

*n*= 15 and 20, and from 0 to 1500 for

*n*= 50 and 100, all of them with 500 integration points. This ensures a reasonable convergence of both quadrature methods [25].

*m*≠ 1 for

*ρ*

_{0}= 0.This happens because, as already mentioned in the previous section,

*J*

_{|}

_{m}_{| ± 1}(

*ρ*

_{0}

*k*sin

*θ*) = 0. It is easily verified that, for off-axis BBs, significant values for the BSC’s appear for

_{a}*m*≠ 0, as also expected [12

**11**(9), 2503–2515 (1994). [CrossRef]

27. K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A **11**(6), 1812–1817 (1994). [CrossRef]

*ρ*

_{0}= 0) from the ILA method are very close to those obtained by means of quadratures, when we take into account only the most relevant BSCs in magnitude (low

*n*). In fact, for all the BSCs presented, whenever the relative magnitude

*n*, being solely a matter of raising the number of numerical points in

*θ*,

*ϕ*(for F1 and F2) and

*R*(for F2) to ensure an adequate numerical convergence. For example, using 500 points in

*θ*and

*ϕ*leads to

*n*necessarily imposes additional refinements in our numerical integration for F1 and F2.

*n*increases, so does the average time for F1 and F2. This happens in our simulations because our subroutine uses recursive relations for the spherical Bessel functions, so that, for example, for

*n*= 200, 200 iterations are performed to compute

*j*

_{200}(

*R*) in these methods.

28. F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Sys. Characterization. **8**(1-4), 222–228 (1991). [CrossRef]

30. G. Gouesbet, “Validity of the localized approximations for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A **16**(7), 1641–1650 (1999). [CrossRef]

*x*-polarized zero-order Bessel beam by a circularly polarized (

*xy*plane) with the same parameters.

**11**(9), 2516–2525 (1994). [CrossRef]

*x*-polarized zero-order Bessel beam propagating along +

*z*. The dominant electric field component

*E*is written, based on the above equation, as

_{x}*ρ*

_{0}along the

*x*-axis, we can express the electric field

*E*in the same way as Ref. [13

_{x}**11**(9), 2516–2525 (1994). [CrossRef]

*ϕ*= 0 and

*ρ*

_{0}= 0, 30, 60 and 90 μm for different ranges of

*n*and with

*m*= 15. There is a clear compromise between

_{max}*x*and the maximum

*n*entering superposition Eq. (16) for

*E*from which a good prediction of the Bessel profile is achieved. For the parameters chosen, a maximum

_{r}*n*= 700 is sufficient, as long as the physical analysis is limited to 0 <

*x*< 120 μm. If

*x*is to be kept below approximately 60 μm (i.e., in cases where the physical analysis does not demand the knowledge of the field for distances above

*x*= 60 μm), then 1 ≤

*n*< 500 could be used to make field calculations faster. Note that this is in contradiction with some previous analysis for focused Gaussian beams, where off-axis descriptions using the ILA may possess serious limitations due to the approximation of the beam itself, i.e., the truncation of the Davis’ series description of this type of beam [13

**11**(9), 2516–2525 (1994). [CrossRef]

6. 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**(6), 998–1007 (1990). [CrossRef]

*ρ*

_{0}’s,

*m*may also have to assume higher values.

_{max}*n*and

_{max}*m*. Similar results were then obtained.

_{max}## 4. Application to optical force calculations

*x*for

*n*= 1.1. The force intensity profile for

_{rel}*k*= 0.1, i.e.,

_{ρ}a*a*≈1.20 μm, is also shown for comparison. In this normalized plot,

*F*

_{x}_{,max|}

_{kρa}_{= 3.5}/

*F*

_{x}_{,max|}

_{kρa}_{= 0.1}= 2858.9. To reproduce these slopes, we have set

*n*= 700 and

_{max}*m*= 15, thus ensuring the adequate description of the Bessel beam profile, as already pointed out in previous sections.

_{max}21. A. N. Rubinov, A. A. Afanas’ev, I. E. Ermolaev, Y. A. Kurochkin, and S. Y. Mikhnevich, “Localization of spherical particles under the action of gradient forces in the field of a zero-order Bessel beam. Rayleigh-Gans approximation,” J. Appl. Spectrosc. **70**(4), 565–572 (2003). [CrossRef]

*x*= 0,

*n*can be significantly reduced because at this point few

_{max}*n*-iterations are necessary in calculating

*C*

_{pr}_{,}

*, to achieve an adequate description of the fields and, consequently, of the optical forces. In this way, for plotting Fig. 5, for instance, we could have safely chosen*

_{x}*n*= 200 and still get the same force profiles as shown.

_{max}*x*component of the radiation pressure cross-section profile

*C*

_{pr}_{,}

*for silicon spheres (*

_{x}*n*= 1.4496) of four different radii immersed in water (

_{p}*n*= 1.33). The incident beam is a Bessel beam with Δ

_{m}*ρ*≈2.35 μm (

*θ*≈7.51°) and a wavelength

_{a}*λ*= 802.7 nm, both in water. These parameters were chosen to best fit the theoretical data of Ref. [22

22. G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a Bessel beam,” Opt. Express **15**(21), 13972–13987 (2007). [CrossRef] [PubMed]

*C*

_{pr}_{,}

*means an attractive force towards the optical axis of the beam.*

_{x}*C*

_{pr}_{,}

*curve took, in average, 512 s for*

_{x}*n*= 1300,

_{max}*m*= 20 and 100 radial positions (we point out that, if we had implemented symmetry relations for calculating the BSCs for

_{max}*m*< 0 in our code, or adopted a lower

*n*, as pointed out above in deriving Fig. 5, this time would have been reduced further). These upper limits for

_{max}*n*and

*m*are again necessary to ensure that the set of BSCs reproduce the electromagnetic fields for this particular BB within the range 0 ≤

*x*≤ 20 μm with negligible difference, thus providing the correct picture for the radiation pressure cross-section. This simulation time is radically in contrast to the elapsed time observed in Ref. [22

**15**(21), 13972–13987 (2007). [CrossRef] [PubMed]

*a*= 10 μm (for the same PC configuration, this would represent a multiplicative factor of 25.2. In practice, however, this factor is even higher, as numerical simulations of Ref. [22

**15**(21), 13972–13987 (2007). [CrossRef] [PubMed]

*n*and

_{max}*m*turn the evaluation of Eq. (16) and

_{max}*C*

_{pr}_{,}

*more time consuming and, furthermore, more and more BSCs have to be calculated.*

_{x}## 5. Conclusions

## Acknowledgments

## References and links

1. | G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. |

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

3. | B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) |

4. | G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) |

5. | G. Gouesbet, G. Gréhan, and B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) |

6. | 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. | K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. |

8. | J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transf. |

9. | G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. |

10. | G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models,” J. Quant. Spectrosc. Radiat. Transf. |

11. | H. C. van de Hulst, |

12. | G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A |

13. | G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A |

14. | K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. |

15. | H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. |

16. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express |

17. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

18. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

19. | V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. |

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

21. | A. N. Rubinov, A. A. Afanas’ev, I. E. Ermolaev, Y. A. Kurochkin, and S. Y. Mikhnevich, “Localization of spherical particles under the action of gradient forces in the field of a zero-order Bessel beam. Rayleigh-Gans approximation,” J. Appl. Spectrosc. |

22. | G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a Bessel beam,” Opt. Express |

23. | J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

24. | G. N. Watson, |

25. | K. R. Fen, “ |

26. | C. F. Bohren and D. R. Huffmann, |

27. | K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A |

28. | F. Corbin, G. Gréhan, and G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Sys. Characterization. |

29. | G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. |

30. | G. Gouesbet, “Validity of the localized approximations for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(290.4020) Scattering : Mie theory

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

(290.5825) Scattering : Scattering theory

**ToC Category:**

Optical Traps, Manipulation, and Tracking

**History**

Original Manuscript: April 4, 2011

Revised Manuscript: June 7, 2011

Manuscript Accepted: June 7, 2011

Published: June 9, 2011

**Citation**

Leonardo A. Ambrosio and Hugo E. Hernández-Figueroa, "Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces," Biomed. Opt. Express **2**, 1893-1906 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-7-1893

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### References

- G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–452 (1908). [CrossRef]
- G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
- B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
- G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
- G. Gouesbet, G. Gréhan, and B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
- 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(6), 998–1007 (1990). [CrossRef]
- K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998). [CrossRef] [PubMed]
- J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 800–807 (2009). [CrossRef]
- G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1223–1238 (2009). [CrossRef]
- G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models,” J. Quant. Spectrosc. Radiat. Transf. 112(1), 1–27 (2011). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
- G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994). [CrossRef]
- G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994). [CrossRef]
- K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994). [CrossRef]
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