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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 2, Iss. 7 — Jul. 1, 2011
  • pp: 1893–1906
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Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces

Leonardo A. Ambrosio and Hugo E. Hernández-Figueroa  »View Author Affiliations


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

The generalized Lorenz-Mie theory (GLMT) is an extension of the Lorenz-Mie theory [1

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

] for describing the electromagnetic field components of an arbitrary laser beam in terms of spherical harmonic functions [2

2. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

,3

3. 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).

], the coefficients of which being called the beam shape coefficients (BSCs), responsible for correctly modeling the intensity profile of the beam [4

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) 19, 35–48 (1988).

]. Their numerical evaluation using quadratures [2

2. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

] or finite series [4

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) 19, 35–48 (1988).

], however, can be a pretty time-consuming, lengthy or awkward task, first because of numerical integrations over the spherical coordinates of the adopted coordinate system, and second because of the inexistence of a single expression, in the latter case. Thus, efficient techniques such as the localized interpretation and, as its natural consequence, the integral localized approximation (ILA), have been developed for calculating on and off-axis BSCs with insignificant difference and much faster than the above mentioned numerical integrations [5

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) 20, 31–43 (1989).

7

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]

]. Anyway, regardless of the scheme adopted for evaluating the BSCs, since the development of the GLMT plenty of applications have been benefited by this theoretical methodology (for a resume of the updated subject see, e.g., Refs. [8

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

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]

]. and references therein).

The localized approximation developed by Gouesbet et al. is based on the localization principle of van de Hulst [11

11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

] and was rigorously justified for on- and off-axis focused Gaussian beams and laser sheets [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

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]

]. Its improved version, viz., the integral localized approximation, has been used during the years for biomedical optics research such as in optical force and torque calculations exerted on both homogeneous and multilayered spherical dielectric particles in optical trapping systems [14

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

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]

]. Even optical forces exerted on hypothetical negative refractive index metamaterials by Gaussian beams have been analyzed with the ILA in the context of the GLMT [16

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]

].

In optical trapping experiments, however, there are actually other laser beams of relevant importance, due to their increasing interest because of their properties such as multiple trapping and angular momentum transfer [17

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

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

]. Among those we found the Bessel beams.

Single Bessel beams furnish an easy way for multiple trapping, although their extended focus makes a full three-dimensional trap unachievable. Several methods have been developed for theoretically predicting the behavior of dielectric spherical particles, under the influence of a Bessel beam, in the ray optics regime, where the wavelength 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]

]. Even recent works that use the GLMT for studying transverse dynamics of silicon particles under the influence of Bessel beams still formulate their theory based on slightly modified versions of quadratures [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]

], explicitly emphasizing the time-demanding character of such approach as a function of the particle radius. The inclusion of the ILA for multi-ringed beams in the GLMT may fulfill a bottleneck in this theory and, therefore, really seems to deserve some special attention.

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

Consider an ideal monochromatic zero-order Bessel beam (BB) propagating with speed 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
Fig. 1 Geometrical description of an ordinary Bessel beam propagating parallel to + z (out of the page). The optical axis makes an angle ϕ 0 relative to the x-axis and is displaced ρ 0 from the origin O of the coordinate system.
. 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 kz = kcosθa and kρ = ksinθa, respectively, θa being the associated axicon angle and 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]

]. If this beam is linearly polarized in x or in y, we can write its electric field as

E(ρ,ϕ,z)={x^y^}E0J0(kρρ2+ρ022ρρ0cos(ϕϕ0))eikzz
(1)

where 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]

]; i.e., the results presented here will represent good descriptions of real experiments if the axicon angle is small enough that a paraxial approximation can be assumed.

Er{xy}=E0J0[sinθa(kr)2sin2θ+ρ02k22(kr)ρ0sinθcos(ϕϕ0)]ei(kr)cosθacosθsinθ{cosϕsinϕ}
(2)
Hr{xy}H0cosθaJ0[sinθa(kr)2sin2θ+ρ02k22(kr)ρ0sinθcos(ϕϕ0)]ei(kr)cosθacosθsinθ{sinϕcosϕ}
(3)

where H0=E0/η=E0ε/μ and η is the intrinsic impedance of the propagating medium, 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 sinϕ or cosϕ factor.

From Eqs. (2) and (3), the beam shape coefficients gn,TEm and gn,TMm in the integral localized approximation are readily found from the following relations [7

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]

]:

gn,TEm=Znm2πH002πG^[Hr(r,θ,ϕ)]exp(imϕ)dϕ
(4)
gn,TMm=Znm2πE002πG^[Er(r,θ,ϕ)]exp(imϕ)dϕ
(5)

where n and m are integer numbers ranging from 1 ≤ n < ∞ and –n < m < n, respectively, corresponding to a particular spherical harmonic function Ynm(θ,ϕ)=Pnm(cosθ)exp(imϕ), Pnm(cosθ) being associated Legendre functions. In the integral localized approximation description of gn,TEm and gn,TMm, the localization operator G^changes the factor kr to (n + 1/2) and θ to π/2, while Znmare normalization factors that read as [7

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]

]

Zn0=2n(n+1)i2n+1,            m=0
(6)
Znm=(2i2n+1)|m|1,           m0.
(7)

By substituting Eq. (2) into Eq. (5) for m = 0, we find

gn,TM0{xy}=i2n(n+1)2π(2n+1)02πJ0[sinθa(n+1/2)2+ρ02k22(n+1/2)ρ0cos(ϕϕ0)]{cosϕsinϕ}dϕ
(8)

where we have used the normalization factor Zn0 from Eq. (6). To analytically solve the above integral, we perform the following expansion for a zero-order Bessel function [24

24. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, University Press, 1944), p. 358.

]

J0[ϖ2+ξ22ϖξcos(ϕϕ0)]=p=0εpJp(ϖ)Jp(ξ)cos[p(ϕϕ0)]
(9)

εp being 1 if p = 0 and 2 otherwise. Thus, letting ϖ=sinθa(n+1/2) and ξ=ρ0ksinθa, we have from Eq. (8)

gn,TM0{xy}=i2n(n+1)2π(2n+1)02πp=0εpJp(sinθa(n+1/2))Jp(ρ0ksinθa)cos[p(ϕϕ0)]{cosϕsinϕ}dϕ=i2n(n+1)2π(2n+1)p=0εpJp(sinθa(n+1/2))Jp(ρ0ksinθa)02πcos[p(ϕϕ0)]{cosϕsinϕ}dϕ=i2n(n+1)2π(2n+1)p=0εpJp(sinθa(n+1/2))Jp(ρ0ksinθa){cospϕ002π{cos(p1)ϕ+cos(p+1)ϕ20}dϕ+sinpϕ002π{0cos(p1)ϕcos(p+1)ϕ2}dϕ}=i2n(n+1)(2n+1)J1(sinθa(n+1/2))J1(ρ0ksinθa){cosϕ0sinϕ0},
(10)

which will always be zero whenever ρ 0 = 0 (on-axis case), because J1(ρ0ksinθa)=0.

gn,TMm0{xy}=12π(2i2n+1)|m|1p=0εpJp(sinθa(n+1/2))Jp(ρ0ksinθa)[cospϕ002π{cospϕcosmϕcosϕicospϕsinmϕsinϕ}dϕ+sinpϕ002π{isinpϕsinmϕcosϕsinpϕcosmϕsinϕ}dϕ]
           =12π(2i2n+1)|m|1p=0εpJp(sinθa(n+1/2))Jp(ρ0ksinθa)[cospϕ002π{cospϕcos|m|ϕcosϕicospϕsin|m|ϕsinϕ}dϕ+sinpϕ002π{isinpϕsin|m|ϕcosϕsinpϕcos|m|ϕsinϕ}dϕ]           =12(2i2n+1)|m|1[{1i}J|m|1(sinθa(n+1/2))J|m|1(ρ0ksinθa)[cos(|m|1)ϕ0isin(|m|1)ϕ0]+{1±i}J|m|+1(sinθa(n+1/2))J|m|+1(ρ0ksinθa)[cos(|m|+1)ϕ0isin(|m|+1)ϕ0]],
(11)

Notice that, in deriving Eqs. (10) and (11), we have made use of some basic orthogonality conditions for trigonometric functions. Also, because J | m | ± 1(ρ 0 ksinθa) = 0 whenever 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 gn,TEm and gn,TMm, both of which with a magnitude of (1/2)J0(sinθa(n+1/2)) that decreases with increasing n.

gn,TE0{xy}=i2n(n+1)(2n+1)J1(sinθa(n+1/2))J1(ρ0ksinθa){sinϕ0cosϕ0},
(12)
gn,TEm0{xy}=12(2i2n+1)|m|1[{i1}J|m|1(sinθa(n+1/2))J|m|1(ρ0ksinθa)[cos(|m|1)ϕ0isin(|m|1)ϕ0]+{±i1}J|m|+1(sinθa(n+1/2))J|m|+1(ρ0ksinθa)[cos(|m|+1)ϕ0isin(|m|+1)ϕ0]],
(13)

These are the beam-shape coefficients necessary to fully describe an 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

gn,TEm=14π(in1)Rjn(R)(n|m|)!(n+|m|)!0πsinθdθ02πdϕP(cosθ)n|m|exp(imϕ)Hr(R,θ,ϕ)H0
(14)
gn,TMm=14π(in1)Rjn(R)(n|m|)!(n+|m|)!0πsinθdθ02πdϕP(cosθ)n|m|exp(imϕ)Er(R,θ,ϕ)E0
(15)

Pnm(cosθ) being the associated Legendre polynomials, R = kr an auxiliary coordinate and jn(R) are spherical Bessel functions of integer order n. If Er and Hr represent an exact solution to Maxwell’s equations, then R is arbitrary because, in such a case, the integrals over the spherical angles θ and ϕ are proportional to jn(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

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]

,25

25. K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D thesis (Faculté des Sciences de L’Université de Rouen, 1995).

,26

26. C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

]. But some care must be exercised in choosing a specific 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

11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

]. 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

25. K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D thesis (Faculté des Sciences de L’Université de Rouen, 1995).

]. 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

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]

].

The number of steps for numerically evaluating the BSCs of Table 1, using methods F1 and F2, was set to 200 for integrating both in 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

25. K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D thesis (Faculté des Sciences de L’Université de Rouen, 1995).

].

From Table 1, one can see that the magnitudes of gn,TMm (or, equivalently, gn,TEm, as gn,TEm=ign,TMm for 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 gn,TM1/g1,TM1is small there is a dissimilitude between ILA, F1 and F2 methods (for example, g100,TM1|ILA = 0.280726, g100,TM1|F1 ≈0.279240 and g100,TM1|F2 ≈0.266963, representing a relative difference of 0.53% and 5.16%, respectively) which, overall, does not contribute significantly to the description of the field intensity profile of the Bessel beam with the parameters chosen. But we must point out that this does not represent a failure of the ILA method for high 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 g100,TM1|F1 ≈0.280659 and g100,TM1|F2 ≈0.268462, and increasing n necessarily imposes additional refinements in our numerical integration for F1 and F2.

As we have shown that the ILA can furnish the values of the most significant BSCs with great accuracy, we now turn our attention to the most outstanding advantage of using the integral localized approximation, viz., the computation time of these coefficients. Due to the absence of numerical integrations, calculating gn,TEm and gn,TMm using the ILA is much faster than using the quadrature methods F1 and F2. Table 2

Table 2. Elapsed time (in seconds) for computing the BSC’s gn,TM1 of an on-axis (ρ0 = ϕ0 = 0) zero-order Bessel beam with λ = 1064 nm and θa = 0.0141 rada

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shows the average time taken for numerically evaluating the BSCs presented in Table 1. These values were obtained by calculating each ten times and then taking the time average. Notice that, as 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.

Thus, the integral localized approximation proves to be a computationally efficient method for computing the beam-shape coefficients necessary to describe an ordinary BB in the framework of the generalized Lorenz-Mie theory. Although this approximation is known since de 90’s, it is noteworthy and remarkable that so few laser beams have ever been described with it [28

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

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]

]. It can be shown that good results are also achieved if we replace the x-polarized zero-order Bessel beam by a circularly polarized (xy plane) with the same parameters.

Now, suppose that we are given a set of beam-shape coefficients gn,TEm and gn,TMm, regardless of the numerical method used to evaluate them. The electric field components in the GLMT can be written as [13

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]

]:

Er(R,θ,ϕ)=iE0n=1(i)n(2n+1)jn(R)Rm=nngn,TMmπn|m|(θ)sinθexp(imϕ)
Eθ(R,θ,ϕ)=E0n=1(i)n(2n+1)n(n+1){jn(R)m=nngn,TEmimπn|m|(θ)exp(imϕ)+i[jn1(R)nRjn(R)]m=nngn,TMmτn|m|(θ)exp(imϕ)}Eϕ(R,θ,ϕ)=E0n=1(i)n(2n+1)n(n+1){jn(R)m=nngn,TEmτn|m|(θ)exp(imϕ)+i[jn1(R)nRjn(R)]m=nngn,TMmimπn|m|(θ)exp(imϕ)}
(16)

with similar expressions for the magnetic field components. In Eq. (16), πn|m|(θ)=sinθPnm(cosθ) and τn|m|(θ)=(1/sinθ)dPnm(cosθ)/dθ. Furthermore, consider an x-polarized zero-order Bessel beam propagating along + z. The dominant electric field component Ex is written, based on the above equation, as Ex(x,y,z)=Ersinθcosϕ+EθcosθcosϕEϕsinϕ but, but, if we further assume an observation point ρ 0 along the x-axis, we can express the electric field Ex in the same way as Ref. [13

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]

]:

Ex(x,y,z)={Er(r=|x|=|ρ0|,θ=π2,ϕ=0),        if x>0Er(r=|x|=|ρ0|,θ=π2,ϕ=π),     if x<0
(17)

Thus, for ordinary Bessel beams, the ILA is capable of reproducing the original electric field with great accuracy and reliability. We have also analyzed the intensity profile and relative difference for all the other electromagnetic field components using the ILA formulation. All of them can be reproduced by an adequate choice of nmax and mmax. Similar results were then obtained.

4. Application to optical force calculations

The success of the integral localized approximation for describing a BB can be used advantageously in biomedical applications, such as in the determination of the optical forces exerted over biological particles under the influence of such beams.

Notice, however, that according to Fig. 2, as long as the particle is assumed to be fixed at x = 0, nmax can be significantly reduced because at this point few n-iterations are necessary in calculating Cpr , x, 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 nmax = 200 and still get the same force profiles as shown.

As a second example, Fig. 6
Fig. 6 Radiation pressure cross-section Cpr,x (solid) for an x-polarized Bessel beam displaced along x using the ILA. The beam has λ = 802.7 nm and Δρ ≈2.35 μm in water (nm = 1.33). The beam intensity is shown as a dotted line. The silicon spheres have a refractive index np = 1.4496 and radii a = 1.15 μm (a), 2.15 μm (b), 2.50 μm (c) and 3.42 μm (d). Points of stable equilibrium are close to those predicted in Ref. [21], where a quadrature scheme was adopted for numerically implementing the GLMT.
shows the x component of the radiation pressure cross-section profile Cpr , x for silicon spheres (np = 1.4496) of four different radii immersed in water (nm = 1.33). The incident beam is a Bessel beam with Δρ ≈2.35 μm (θa ≈7.51°) and a wavelength 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]

], so that negative Cpr , x means an attractive force towards the optical axis of the beam.

Comparing our results with the above mentioned reference, we see approximately the same points of stable equilibrium. As for the simulation time, each Cpr , x curve took, in average, 512 s for nmax = 1300, mmax = 20 and 100 radial positions (we point out that, if we had implemented symmetry relations for calculating the BSCs for m < 0 in our code, or adopted a lower nmax, as pointed out above in deriving Fig. 5, this time would have been reduced further). These upper limits for 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

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]

], where it explicitly depended on the radius of the silicon spheres and computational calculations could last more than 3.5 hours for a silicon sphere of radius 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

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]

]. were performed on a higher performance PC). Obviously, here it is the number of BSCs which ultimately establishes the average time, as increasing nmax and mmax turn the evaluation of Eq. (16) and Cpr , x more time consuming and, furthermore, more and more BSCs have to be calculated.

In fact, the use of the ILA in the generalized Lorenz-Mie theory provides an efficient tool not only for force calculations, but also for other physical entities such as the electromagnetic field components themselves, as already shown in this paper, or angular momentum, scattering and absorbing cross-sections.

5. Conclusions

For the first time in the literature, the beam-shape coefficients for ordinary Bessel beams were numerically evaluated using the integral localized approximation in the context of the generalized Lorenz-Mie theory. Closed-form solutions were presented, extending the range of applicability of the ILA to beams other than Gaussian and laser sheets.

The method presented here furnishes almost the same results of those based on quadratures for the magnitude of the BSCs necessary to adequately describe the electromagnetic fields of zero-order Bessel beams. But the fundamental advantage lies on the incredibly reduced elapsed time to compute each of these coefficients. This ultimately justifies the adoption of this method for subsequent works.

Extending the range of applicability of the ILA certainly would represent a significant gain in the study of optical properties other than forces. We could use this method, for example, in optical torque calculations or simply to find absorbing, scattering and extinction cross-sections for a particle under the influence of a particular laser beam such as Laguerre-Gaussian, for example.

Acknowledgments

The authors wish to thank FAPESP—Fundação de Amparo à Pesquisa do Estado de São Paulo—under contracts 2009/54494-9 (L. A. Ambrosio’s post doctorate grant) and 2005/51689-2 (CePOF, Optics and Photonics Research Center), for supporting this work.

References and links

1.

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

2.

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

3.

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).

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) 19, 35–48 (1988).

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) 20, 31–43 (1989).

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]

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]

9.

G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1223–1238 (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]

11.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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]

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 419(6903), 145–147 (2002). [CrossRef] [PubMed]

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. 85(18), 4001–4003 (2004). [CrossRef]

20.

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

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]

23.

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

24.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, University Press, 1944), p. 358.

25.

K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D thesis (Faculté des Sciences de L’Université de Rouen, 1995).

26.

C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

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]

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]

29.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34(12), 2133–2143 (1995). [CrossRef] [PubMed]

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]

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

  1. G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–452 (1908). [CrossRef]
  2. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
  3. 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).
  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) 19, 35–48 (1988).
  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) 20, 31–43 (1989).
  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]
  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]
  9. G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1223–1238 (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]
  11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  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]
  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 419(6903), 145–147 (2002). [CrossRef] [PubMed]
  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. 85(18), 4001–4003 (2004). [CrossRef]
  20. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [CrossRef] [PubMed]
  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]
  23. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  24. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, University Press, 1944), p. 358.
  25. K. R. Fen, “Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques,” Ph.D thesis (Faculté des Sciences de L’Université de Rouen, 1995).
  26. C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  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]
  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]
  29. G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34(12), 2133–2143 (1995). [CrossRef] [PubMed]
  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]

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