Multipole expansion of circularly symmetric Bessel beams of arbitrary order for scattering calculations

Optics Communications 387 (2017) 102–109

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Multipole expansion of circularly symmetric Bessel beams of arbitrary order for scattering calculations

MARK

⁎

Jia Jie Wanga,b, , Thomas Wriedtb, Lutz Mädlerb, Yi Ping Hana, Peter Hartmannc a

School of Physics and Optoelectronic Engineering, Xidian University, 710071 Xi’an, China Foundation Institute of Material Science (IWT), Department of Production Engineering, University of Bremen, Badgasteiner Str. 3, 28359 Bremen, Germany c University of Applied Sciences, Applied Physics and Computer Science, Westsaechsische Hochschule Zwickau, 08056 Zwickau, Germany b

A R T I C L E I N F O

A BS T RAC T

Keywords: Bessel beam Light scattering Generalized Lorenz-Mie theory Beam shape coefficients

A rigorous, simple and efficient approach is derived in this paper for multipole expansion of a circularly symmetric Bessel beam. Different from the existing rigorous methods which are based on the plane wave spectrum of a Bessel beam, a straight-forward integral procedure is presented in a traditional way to obtain the analytical expressions of the expansion coefficients, also called beam shape coefficients (BSCs). The convergence and correctness of the BSCs are verified numerically in detail for both on-axis and off-axis cases. The results in this paper are useful in various analytical scattering theories, such as the generalized Lorenz-Mie theory and the Null-field method, when a Bessel beam is considered.

1. Introduction Analyses of interactions between shaped beams and small particles became more and more important in recent years due to their essential roles in optical characterization, optical trapping and manipulation, remote sensing and others [1,2]. Concerning shaped beams, there has been an increasing interest in Bessel beams [3,4], which is mainly due to its special properties, including propagation invariance, self-reconstruction, and the transfer of orbital angular momentum as well as spin angular momentum to matter. Prospective applications of Bessel beams can be found in wide range of fields, such as optical communication, biomedicine, optical manipulation, and imaging [5–7]. When describing a shaped beam for use in analytical scattering theories, such as the generalized Lorenz-Mie theories (GLMTs) [8] and the Null-field method [9], electric and magnetic fields are required to be expanded in terms of proper wave harmonics, e.g. vector spherical wave functions (VSWFs) for isotropic medium, or quasi-VSWFs for anisotropic medium [10]. The calculation of expansion coefficients, or the sub-coefficients which are called as beam shape coefficients (BSCs), is one of the key issues when dealing with any type of shaped beam. With decades of efforts devoted to the description of an arbitrary shaped beam, the BSCs of an arbitrary shaped beam can be evaluated by several methods [11], including quadratures, finite series, localized approximations (LA), and the integral localized approximation (ILA) [12]. In the case of Gaussian beams, whose field expressions are not

⁎

exact solutions to Maxwell's equations, the most efficient method for evaluating the BSCs has been the LA method [13]. The reconstructed fields based on LA BSCs are exact solutions to Maxwell's equations, which provide good approximations to the origin fields. The LA method has also been applied to the scattering of Bessel beams in several studies [14–16]. Application of LA for a Bessel beam is valid when the half-cone angle of the Bessel beam is relatively small [15]. However, significant errors occur when the half-cone angle is sufficiently large [17,18]. Therefore, a rigorous and efficient way for the calculation of BSCs of a Bessel beam is needed. Accurate BSCs of a Bessel beam can be obtained by a double quadrature over spherical coordinates, which is the original method used in the GLMTs [19,20]. Numerical evaluation of double quadrature for a zero-order Bessel beam was used by Preston et al. [21], and was also applied to a high-order Bessel beam by Mitri [22]. Although it is very time-consuming and complex in the numerical evaluation, this method provides accurate BSCs which can be used for validation of BSCs obtained using other approximate methods [14]. Double quadrature can be reduced to single quadrature for a zero-order Bessel beam, as shown by Cizmar et al. [23]. This reduction of quadrature was achieved based on an angular spectrum representation (ASR) of a Bessel beam, where the Bessel beam is regarded as a superposition of partial plane waves with delta distribution in polar angle δ (α − α0 ). Based on the ASR description of a zero-order Bessel beam, the results were further improved by Taylor and Love [24] who derived an

Corresponding author at: School of Physics and Optoelectronic Engineering, Xidian University, 710071 Xi’an, China. E-mail address: [email protected] (J.J. Wang).

http://dx.doi.org/10.1016/j.optcom.2016.11.038 Received 12 September 2016; Received in revised form 15 November 2016; Accepted 17 November 2016 Available online 23 November 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.

Optics Communications 387 (2017) 102–109

J.J. Wang et al.

partial plane waves, and (b) the Davis procedure which obtains the fields from a polarized vector potential. Although the two different procedures give two seemingly different answers for the fields, it turns out that the functional dependence of the two answers is identical for circularly symmetric Bessel beams. A general description for circularly symmetric Bessel beams was derived recently [29,30]. This generalization of the description makes the Davis type Bessel beam and the ASR type Bessel beam merely the two simplest cases of an infinite number of possible circularly symmetric Bessel beams, corresponding to different values of the arbitrary function g (α0 ). The electric field components of a general circularly symmetric Bessel beam with its beam center locating at an arbitrary point (x 0 , y0 , z 0 ) are

analytical expression of BSCs without integral. The BSCs were obtained by a superposition of the expansion coefficients of partial plane waves which consist of the plane wave spectrum of a Bessel beam. The calculation of BSCs of Bessel beams based on the angular spectrum representation was also analyzed by Lock [25], where a general zeroorder Bessel beam was considered. The same procedure was also extended to the case of polarized Bessel beams of arbitrary order by Chen et al. [26], and was applied by Ma and Li [27] to a study of an unpolarized Bessel beam. BSCs in analytical form allow scattering calculations to be carried out considerably more efficiently and accurately. This is of advantage for problems where a large number of scattering calculations must be performed, e.g. forces and torque prediction in an optical tweezers [28]. The existing approaches for calculating BSCs analytically are based on the ASR description of a Bessel beam. In these approaches, a Bessel beam is required to be represented as a plane wave spectrum using the ASR, and then analytical expressions of BSCs are obtained by a superposition of the expansion coefficients of each partial plane wave. In this paper, we show that the analytical expressions of BSCs of a circularly symmetric Bessel beam (whose energy density and Poynting vector component along its propagation direction are circularly symmetric in the transverse plane) can be obtained in a straightforward way by performing the integrals directly, it is a simpler approach than the existing methods. The other parts of this paper is organized as follows. Derivations of analytical expressions of BSCs is presented in Section 2 for a circularly symmetric Bessel beam. The correctness and convergence of the BSCs are verified numerically in detail in Section 3. Conclusions are given in Section 4.

⎧ 1 Ex(1,0) = E0 g (α0 ) e−ikz (z − z0) ⎨ (1 + cosα0 )(−i )l eilϕG Jl (σG ) − (1 − cosα0 ) × ⎩ 2 [(−i )l −2 ei (l −2) ϕG Jl −2 (σG )+(−i )l +2ei (l +2) ϕG Jl +2 (σG )]} Ey(1,0) = E0 g (α0 ) e−ikz (z − z0)

1 (1 − cos α0 ) 2i

[(−i )l −2 ei (l −2) ϕG Jl −2 (σG ) − (−i )l +2ei (l +2) ϕG Jl +2 (σG )] Ez(1,0) = −E0 g (α0 ) e−ikz (z − z0) sin α0 [(−i )l −1ei (l −1) ϕG Jl −1 (σG ) + (−i )l +1ei (l +1) ϕG Jl +1 (σG )],

(1)

where superscript (1, 0) which is reminiscent of x-polarization is used, and σG = kt ρG , ρG = [(x − x 0 )2 + (y − y0 )2]1/2 , ϕG = tan−1[(y − y0 )/(x − x 0 )]. The transverse and longitudinal wave numbers are kt = k sin α0 and kz = k cos α0 , respectively. The l-order Bessel function of the first kind is denoted as Jl (⋅). The wavenumber is k , and α0 is the half-cone angle of the Bessel beam which is defined with respect to the axis of wave propagation. When g (α0 ) = (1 + cosα0 )/4 , the expressions in Eq. (1) reduce to those of a Davis circularly symmetric Bessel beam used in [25,31]. When g (α0 ) = 1/2 , they reduce to those of an ASR Bessel beam used in [23,24,26]. The expressions for magnetic fields are not presented for the sake of brevity, since they can be obtained from electric fields in Eq. (1) by the relation B (r) = (i / ω)∇ × E (r). The time dependence exp(iωt ) is assumed in this paper. Following the theoretical treatments in the GLMT, the radial electric and magnetic field components derived using the Bromwich scalar potentials are (Sec. III.3 in [8])

2. Derivations of beam shape coefficients A geometry of a spherical particle illuminated by an off-axis Bessel beam is shown in Fig. 1. Two Cartesian coordinate systems, Oxyz and Ob uvw , are used. The Oxyz is attached to the particle and the Ob uvw is attached to the Bessel beam. The axes Ob u , Ob v and Ob w are parallel to the axes Ox , Oy and Oz , respectively. The coordinates of Ob in Oxyz are denoted as (x 0 , y0 , z 0 ). In the description of an ideal Bessel beam, two different procedures are commonly applied to obtain the fields of an l-order Bessel beam: (a) the ASR procedure which obtains the fields by a superposition of

Fig. 1. Geometry of a spherical particle illuminated by an off-axis Bessel beam. A Cartesian coordinate system OXYZ is attached to the particle, and a Cartesian coordinate system Ob uvw is attached to the Bessel beam.

103

Optics Communications 387 (2017) 102–109

J.J. Wang et al.

⎡ Er ⎤ ⎢H ⎥ = ⎣ r⎦

⎡ E0 g m ⎤ n (n + 1) n, TM ⎥ jn (kr ) Pn|m| (cos θ ) eimϕ , cnpw ⎢ r ⎢⎣ H0 gnm, TE ⎥⎦ n =1 m =−n ∞

+n

gnm, TM =

∑ ∑

d mPn (x ) . dx m

(3)

⎡ (−i )l +1Jm − l −1 (kρ ρ ) Jm (kρ ρ) eimϕe−i (m − l −1) ϕ0 ⎤ ⎫ ⎪ 0 ⎥⎬ − cos θ sin α0 ⎢ ⎢⎣+(−i )l −1Jm − l +1 (kρ ρ0 ) Jm (kρ ρ) eimϕe−i (m − l +1) ϕ0 ⎥⎦ ⎪ ⎭

Considering the orthogonality relations for exponentials and those for associated Legendre functions, the BSCs gnm, TM and gnm, TE can be calculated as a two-dimensional integral (Sec. III.3 in [8])

⎡g m ⎤ n +1 (n − m )! kr ⎢ nm, TM ⎥ = i 4π (n + m )! jn (kr ) ⎢⎣ gn, TE ⎥⎦

(10) Reformulating the Eq. (10), we have

⎡ Er (kr )/ E0 ⎤ m ⎢ ⎥ Pn (cosθ ) e−imϕ φ =0 ⎣ Hr (kr )/ H0 ⎦

π

∫θ=0 ∫

2π

gnm, TM =

d

2n + 1 ( n − m ) !

e−ik (z − z0)cos α0 2n (n + 1) (n + m ) ! j

π

r

n (kr )

∫θ =0 Pnm (cosθ )sin θdθ

(kρ ρ)

To obtain the BSCs, only the radial components Er and Hr of the electromagnetic fields are required, which can be obtained from Eq. (1) by

⎡ Ey ⎤ ⎡ Ez ⎤ ⎡ Er ⎤ ⎡ Ex ⎤ ⎢ H ⎥ = ⎢ H ⎥ sin θ cos ϕ + ⎢ H ⎥ sin θ sin ϕ + ⎢ H ⎥ cos θ. ⎣ r⎦ ⎣ x⎦ ⎣ z⎦ ⎣ y⎦

g (α 0 ) cnpw

⎧ m (−i )l ⎨sin θ [Jm − l −1 (kρ ρ0 ) e−i (m − l −1) ϕ0 + Jm − l +1 (kρ ρ0 ) e−i (m − l +1) ϕ0] k ρ Jm ρ ⎩

(4)

sin θdθdϕ.

π

∫θ =0 Pnm (cosθ ) sin θdθ

⎧ ⎡ Jm − l −1 (kρ ρ ) Jm −1 (kρ ρ) ei (m −1) ϕe−i (m − l −1) ϕ0 ⎤ ⎪ 0 ⎥ ⎨ 1 sin θ (−i )l (1 + cosα0 ) ⎢ ⎪2 ⎢⎣+Jm − l +1 (kρ ρ0 ) Jm +1 (kρ ρ) ei (m +1) ϕe−i (m − l +1) ϕ0 ⎥⎦ ⎩ l +2 J i (m +1) ϕe−i (m − l −1) ϕ0 ⎤ ⎡ m − l −1 (kρ ρ0 ) Jm +1 (kρ ρ ) e (1 − cosα0 ) (− i ) ⎢ ⎥ − sin θ l −2 2 ⎢⎣+(−i ) Jm − l +1 (kρ ρ0 ) Jm −1 (kρ ρ) ei (m −1) ϕe−i (m − l +1) ϕ0 ⎥⎦

(2)

where cnpw = (−i )n+12n + 1/ kn (n + 1). The first kind spherical Bessel function is jn (kr ). The associated Legendre function Pnm (x ) is defined as

Pnm (x ) = (−1)m (1 − x 2 )m /2

g (α0 ) −ik (z − z0 )cos α0 2n + 1 (n − m ) ! r e 2n (n + 1) (n + m ) ! jn (kr ) cnpw

+ [Jm − l −1 (kρ ρ0 ) e−i (m − l −1) ϕ0 − Jm − l +1 (kρ ρ0 ) e−i (m − l +1) ϕ0] ×[sin θ cosα0 J ′m (kρ ρ) + i cos θ sin α0 Jm (kρ ρ)]}. (11) Considering the equation

(5)

d [Jm (kt ρ) e−ikz cos α0] = e−ikz cos α0 [Jm′ (kt ρ) kr sin θ cos α0 dα0

To perform the two-dimensional integral in Eq. (4) analytically, the field components of a Bessel beam should be reformulated using the Neumann-Graf's translational addition theorem for the Bessel functions [32]

+ ikr cos θ sin α0 Jm (kt ρ)],

(12)

we obtain

∞

Jl (σG

) eilϕG

=

∑

Jp (σ0 ) Jp + l

(σ ) ei (l + p) ϕe−ipϕ0 ,

gnm, TM =

(6)

p =−∞

1/2

= E 0 g (α 0

⎧

) e−ik (z − z0)cos α0 ⎨ (−i )l (1 ⎩

+

+ [Jm − l −1 (σ0 ) e−i (m − l −1) ϕ0 − Jm − l +1 (σ0 ) e−i (m − l +1) ϕ0] ⎫ 1 d [Jm (kt ρ) e−ikz cos α0] ⎬. ⎭ kr dα0

∞ cosα0 ) ∑ p =−∞ Jp (kρ ρ0 ) Jp + l (kρ ρ)

ei (l + p) ϕe−ipϕ0 −

∞ (1 − cosα0 ) ⎡ ⎢ (−i )l +2 ∑ p =−∞ Jp (kρ ρ0 ) Jp + l +2 (kρ ρ) ei (l +2+ p) ϕe−ipϕ0 2 ⎣

(13)

Then the integral of θ in Eq. (13) can be performed analytically by applying the equation [33]

∞

π

+(−i )l −2 ∑ p =−∞ Jp (kρ ρ0 ) Jp + l −2 (kρ ρ)

∫θ=0 sin θdθPnm (cos θ ) e−ikrcosα

⎤⎫ ei (l −2+ p) ϕe−ipϕ0 ⎥ ⎬, ⎦⎭

0 cos θ J

m (kr

sin α0 sin θ )

= 2(−i )n − m Pnm (cos α0 ) jn (kr ).

(14)

In this way, the spherical Bessel function jn (kr ) in Eq. (13) is canceled analytically, the expression of BSCs becomes

(7)

Ey(1,0) = E0 g (α0 ) e−ik (z − z0)cos α0 (−

π

∫θ=0 Pnm (cosθ ) sin θdθ

⋅(−i )l × {[Jm − l −1 (σ0 ) e−i (m − l −1) ϕ0 + Jm − l +1 (σ0 ) e−i (m − l +1) ϕ0] m Jm (σ ) e−ikz cos α0 kr sin α0

where σ0 = kt ρ0 , ρ0 = (x 02 + y02 ) , ϕ0 = tan−1(y0 / x 0 ), ρ = (x 2 + y 2 )1/2 , σ = kt ρ , and ϕ = tan−1 (y / x ). Inserting Eq. (6) into Eq. (1), we can obtain

Ex(1,0)

i n +1 (n − m )! kr g (α0 ) eikz0 cos α0 2 (n + m )! jn (kr )

∞ 1 − cos α0 ⎡ ) ⎢ (−i )l +2 ∑ p =−∞ Jp (kρ ρ0 ) 2i ⎣

(n − m ) !

gnm, TM = −g (α0 )(−1)(m − m )/2 (n + m ) ! eikz z0 {i l − m +1ei (l − m +1) ϕ0 Jl − m +1 (σ0 )[τnm (cos α0 ) + mπnm (cos α0 )]

Jp + l +2 (kρ ρ) ei (l +2+ p) ϕe−ipϕ0 ⎤ ∞ − (−i )l −2 ∑ p =−∞ Jp (kρ ρ0 ) Jp + l −2 (kρ ρ) ei (l −2+ p) ϕe−ipϕ0 ⎥ , ⎦

+ i l − m −1ei (l − m −1) ϕ0 Jl − m −1 (σ0 )[τnm (cos α0 ) − mπnm (cos α0 )]}. (8)

(15)

where the general Legendre functions are introduced

∞

Ez(1,0) = −E0 g (α0 ) e−ik (z − z0)cos α0 sin α0 [(−i )l +1 ∑ p =−∞ Jp (kρ ρ0 ) Jp + l +1 (kρ ρ)

πnm (cosα0 ) =

ei (l + p +1) ϕe−ipϕ0

Pnm (cosα0 ) , sin α0

τnm (cosα0 ) =

dPnm (cosα0 ) . dα0

(16)

Following the same procedure, the gnm, TE can be obtained as

⎤ ∞ + (−i )l −1 ∑ p =−∞ Jp (kρ ρ0 ) Jp + l −1 (kρ ρ) ei (l + p −1) ϕe−ipϕ0 ⎥ . ⎦

(n − m ) !

gnm, TE = ig (α0 )(−1)(m − m )/2 (n + m ) ! eikz z0

(9)

{i l − m +1ei (l − m +1) ϕ0 Jl − m +1 (σ0 )[τnm (cos α0 ) + mπnm (cosα0 )]

Inserting Eqs. (7)–(9) into Eq. (5), we can obtain the radial components Er . Then inserting Er into Eq. (4), we can obtain an integral expression for the BSCs gnm, TM . The integral of ϕ in the twodimensional integral expression of gnm, TM can then be performed analytically to obtain

− i l − m −1ei (l − m −1) ϕ0 Jl − m −1 (σ0 )[τnm (cos α0 ) − mπnm (cosα0 )]}.

(17)

Furthermore, in some light scattering methods, e.g. the Null-field method [9], shaped beams are commonly expanded in terms of the VSWFs as 104

Optics Communications 387 (2017) 102–109

J.J. Wang et al. ∞

E = E0 ∑

n

∑

(1) amn M(1) mn(k r) + bmn N mn(k r).

Without giving a special description, the position of the Bessel beam center is assumed to be moved along the x -axis in the simulations. Values of gnm, TM versus terms n with different m are displayed in Fig. 3 for an off-axis Bessel beam of order l = 4 and an off-axis Bessel beam of order l = 5. As shown in Fig. 3, the larger the value of m is, the faster the BSCs converge. The values of BSCs for m = 4 are much smaller than those for m = 1. Additionally, the values of BSCs for m larger than 5, which are not plotted in the figure, are so small that they might be negligible in some practical applications, e.g. light scattering by small particles. Values of gnm, TM and gnm, TE versus terms n for an off-axis Bessel beam with different positions of beam center are displayed in Fig. 4. As shown in Fig. 4(a) and (b), gnm, TE converges faster than gnm, TM in general. As the Bessel beam center moves farther away from the origin of the coordinates along x -axis, it seems that gnm, TE converges a little faster, while the shift of beam center position has little influence on the convergence of gnm, TM for the case under study. Furthermore, it is interesting to find that the oscillations in the convergence curves of gnm, TM for different beam center positions are synchronized, as shown in Fig. 4(a), while the amplitude of gnm, TM becomes smaller as x 0 increases. The convergence behavior of gnm, TM and gnm, TE seems to switch with each other when the the Bessel beam center moves farther away from the origin of the coordinates along y -axis, which can be seen from Fig. 4(c) and (d). Values of gnm, TM and gnm, TE versus terms n are displayed in Fig. 5 for an off-axis Bessel beam with different half-cone angles α0 . It shows that the change of half-cone angle has no obvious influence on the convergence of BSCs as n increases, while stronger oscillations occur in the convergence curves of BSCs for Bessel beams with larger halfcone angle. It should be noticed that the BSCs of small values of m converge rather slowly, e.g. m = 1 for an on-axis Bessel beam of order l = 2 as shown in Fig. 2(a), and m = 1 for an off-axis Bessel beam of order l = 4 as shown in Fig. 3. Thus, a sufficiently large number of n should be taken into consideration in order to reconstruct a field to obtain a good approximation of the original field. To verify the convergence and correctness of the BSCs, comparisons between the original field of Eq. (1) and reconstructed field from BSCs using expressions in Eq. (2) or those in Eq. (18) are performed. The explicit expressions for incident field components Eθ and Eφ are

(18)

n =1 m =−n

Definitions of VSWFs are a little different in the literature due to different individual taste. The expressions of VSWFs used here are the same as those used in Ref. [34] (Eqs. (1) and (2) there) m m m imϕ M(1) mn(kr , θ , ϕ ) = (−1) [imπn (cos θ ) i θ − τn (cos θ ) i ϕ] jn (kr ) e m N(1) mn(kr , θ , ϕ ) = (−1) {

iθ +

n (n + 1) jn (kr ) Pnm (cos kr

imπnm (cos

θ ) ir +

1 d [rjn (kr )] }[τnm (cos kr dr

θ)

θ ) i ϕ] eimϕ . (19)

The expansion coefficients amn , bmn in Eq. (18) can be obtained by comparing the radial components in Eq. (2) and those in Eq. (18) m ⎡ bmn ⎤ (n − m )! ⎡⎢ gn, TM ⎤⎥ pw . ⎢ a ⎥ = kcn (−1)(m − m )/2 ⎣ mn ⎦ (n − m )! ⎢⎣− ignm, TE ⎥⎦

(20)

When g (α0 ) = 1/2 , the expansion coefficients in Eq. (20) reduce to those of an ASR Bessel beam, which are the same as those given by Chen et al. [26]. For a zero-order Bessel beam l = 0 , they are the same as those given by Taylor [24]. When g (α0 ) = (1 + cosα0 )/4 , they reduce to those of the Davis Bessel beam, which are the same as those given by Lock [25], except that a different time-dependent constant was used. 3. Numerical verifications of BSCs In this section, the convergence and correctness of the BSCs are numerically discussed in detail. Simulations are performed using Matlab based on the expressions of BSCs in Eqs. (15) and (17). The wavelength of the beam is assumed as λ = 1.064μm , the refractive index of the medium is n = 1.33. For an on-axis Bessel beam of l-order, all the BSCs are zero except m = l ± 1. The BSCs gnm, TM and gnm, TE satisfy the relationship l −1 gnl,−1 TM = − ign, TE ,

l +1 gnl,+1 TM = ign, TE .

(21)

gnm, TM

for an on-axis Bessel Thus, only convergence behavior of BSCs beam are plotted in Fig. 2. As shown in Fig. 2, for a Bessel beam of order l, the BSCs of m = l + 1 converge much faster than those of m = l − 1, e.g. curves in Fig. 2(b) converges much faster than those in Fig. 2(a). For Bessel beams with different beam order, it seems that the BSCs of higher order Bessel beam converge faster than that of lower order Bessel beam, e.g. curves in Fig. 2(c) converges faster than those in Fig. 2(a). But it should also be noticed that BSCs of higher order Bessel beam have larger m . Thus, we claim that the larger m is, the faster the BSCs converge. Furthermore, for a Bessel beam with a larger half-cone angle, stronger oscillations can be found in the values of BSCs than that for smaller half-cone angle. Additionally, as shown in Fig. 2, the scales on the y-axis are vastly different for BSCs of Bessel beams with different beam orders. This is due to the fact that a factorial (n − m) ! is generated when calculating the

Eθ =

E0 r

+n

∞

∑ ∑

cnpw [gnm, TM ψn′(kr ) τn|m| (cos θ ) + gnm, TE ψn (kr ) mπn|m| (cos θ )]

n =1 m =−n

eimϕ ,

Eϕ =

E0 r

(22) +n

∞

∑ ∑

cnpw [ignm, TM ψn′(kr ) mπn|m| (cos θ ) + ignm, TE ψn (kr ) τn|m| (cos θ )]

n =1 m =−n

(23)

eimϕ .

For better programming, Er in Eq. (2), Eθ in Eq. (22) and Eφ in Eq. (23) are reformulated as

(n + m ) !

BSCs in the multipole expansion procedure, as shown in Eqs. (15) and (17). This is a common orthogonality factor occurs in the expansion description of a shaped beam. In some literature, where the Bessel fields are expanded using the angular spectrum decomposition [26,35], BSCs for different modes m are usually normalized, e.g. isolating a ⎡ 2n + 1 (n − m) ! ⎤1/2 normalization factor Emn = ⎢ n (n + 1) (n + m ) ! ⎥ from Eqs. (15) and (17). ⎣ ⎦ By doing this it is then convenient to ensure that the intensity of the incoming beams is kept constant with respect to the convergence angle and winding number when one integrate over a cone of plane waves. As opposed to the on-axis case, the values of BSCs of a l-order Bessel beam in an off-axis case are not zero for other m besides m = l ± 1. The relationships between gnm, TM and gnm, TE in Eq. (21) are no longer valid for an off-axis case. Convergence of BSCs gnm, TM and gnm, TE for an off-axis Bessel beam are plotted in Figs. 3–5 for various cases.

∞

Er = E0 ∑ (−i )n +1 (2n + 1) gn0, TM n =1 ∞

+ E0

jn (kr ) 0 Pn (cos θ ) kr

Nstop

∑ ∑

(−i )n +1 (2n + 1)

m =1 n = m m −imϕ e [gnm, TM eimϕ + gn−, TM ],

jn (kr ) |m| Pn (cos θ ) kr (24)

∞ E0 2n + 1 ∑n =1 (−i )n +1 n (n + 1) gn0, TM ψn′(kr ) τn0 (cos θ ) kr ∞ Nstop 2n + 1 E0 m −imϕ ∑m =1 ∑n = m (−i )n +1 n (n + 1) ψn′(kr ) τn|m| (cos θ )[gnm, TM eimϕ + gn−, TM e ] kr ∞ Nstop 2n + 1 E0 |m| m −m −imϕ n +1 imϕ ∑m =1 ∑n = m (−i ) n (n + 1) ψn (kr ) mπn (cos θ )[gn, TE e − gn, TE e ], kr

Eθ = + +

(25) 105

Optics Communications 387 (2017) 102–109

J.J. Wang et al.

Fig. 2. Values of BSCs gnm, TM versus terms n for an on-axis Bessel beam with different half-cone angles. (a) l =2, m = 1; (b) l =2, m = 3; (c) l =3, m = 2 ; (d) l =3, m = 4 ; (e) l =4, m = 3; (f)

l =4, m = 5.

Eϕ =

E0 kr

∞

∑ (−i )n n =1

E + 0 kr

∞

deviation is less than the order of 1.0 × 10−6 , which occur at the corner of the square area under consideration. The original field is almost the same as the reconstructed field which is not displayed. Additional comparisons between the original field components and reconstructed field components, i.e. Ex and Ez , have also been performed, and very good agreements were achieved. These additional comparisons are not displayed here for the sake of brevity.

2n + 1 0 g ψ (kr ) τn0 (cos θ ) n (n + 1) n, TE n

Nstop

∑ ∑ m =1 n = m

(−i )n

2n + 1 ψ ′(kr ) mπn|m| (cos θ ) n (n + 1) n

m −imϕ [gnm, TM eimϕ − gn−, TM ]+ e

E0 kr

∞

Nstop

∑ ∑ m =1 n = m

m −imϕ τn|m| (cos θ )[gnm, TE eimϕ + gn−, TM e ],

(−i )n

2n + 1 ψ (kr ) n (n + 1) n (26)

4. Conclusions

The case of an off-axis Bessel beam of order l = 2 is displayed for demonstration in Fig. 6. Since the field distribution in an arbitrary transverse plane is the same for an ideal non-diffracting Bessel beam, z and z 0 are assumed to be 0.0 without loss of generality. In the computations, the series have been truncated at a number determined by Nmax = kr + 4.05(kr )1/3 + 2 , where r is the radius of the area taken into consideration. In Fig. 6, we have r = 2 a /2 , a is the square's side length. Relative deviation Exoriginal − Exreconstucted / Exoriginal between the amplitude of original field and that of reconstructed field is also displayed in Fig. 6(b). As shown in Fig. 6(b), the maximum relative

In this paper, a rigorous and simple approach is derived for the multipole expansion of a circularly symmetric Bessel beam. Different from the existing rigorous methods which are based on the plane wave spectrum of a Bessel beam, the approach reported in this paper is derived using a straightforward integral procedure. The convergence and correctness of the BSCs are carefully verified for both on-axis and off-axis cases. Generally speaking, the larger the value of m is, the faster the BSCs converge. For a Bessel beam with a larger half-cone angle, stronger oscillations in the convergence curves of BSCs can be found. Furthermore, comparisons 106

Optics Communications 387 (2017) 102–109

J.J. Wang et al.

Fig. 3. Values of gnm, TM versus terms n with different m (m =1, 2, 3, 4) for an off-axis Bessel beams of beam order l = 4 and an off-axis Bessel beams of beam order l = 5.

Fig. 4. Values of gnm, TM and gnm, TE versus terms n for an off-axis Bessel beam with different positions of beam center. (a)-(b): beam center is moved along x -axis; (c)-(d): beam center is moved along y -axis.

between the original field and the reconstructed field from BSCs are performed. Very good agreement is achieved that indicate the correctness of the BSCs. The results in this paper are useful in various analytical scattering

theories, such as the generalized Lorenz-Mie theories and the Null-field method, where a Bessel beam is considered. The code for the calculation of BSCs is available upon request.

107

Optics Communications 387 (2017) 102–109

J.J. Wang et al.

Fig. 5. Values of gnm, TM and gnm, TE versus terms n for an off-axis Bessel beam with different half-cone angles α0 .

Fig. 6. Spatial distributions of Ex in xoy plane for an off-axis Bessel beam of order l=2. Beam center position is x 0 = y0 = 2.0μm , z0 = 0.0μm , half-cone angle α0 = 20° . (a) Reconstructed

Ex ; (b) Relative deviation between the amplitude of original field and that of reconstructed field Exoriginal − Exreconstucted / Exoriginal .

Acknowledgements [10]

This work was supported by the National Natural Science Foundation of China (Grant no. 61501350), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2015JQ6264) and a grant from the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. This work was also partially supported by the German Research Foundation (DFG) within Priority Programme SPP 1934 (DiSPBiotech) (Project WR 22/53-1, MA 3333/12-1), and a Scholarship from China Scholarship Council.

[11]

[12] [13]

[14]

References

[15] [16]

[1] G. Gouesbet, F.R.A. Onofri, Preface: laser-light and interactions with particles (lip) 2014, J. Quant. Spectrosc. Radiat. Transf. 162 (2015) 1–7. [2] M.I. Mishchenko, N.T. Zakharova, N.G. Khlebtsov, G. Videen, T. Wriedt, Comprehensive thematic T-matrix reference database: a 2014–2015 update, J. Quant. Spectrosc. Radiat. Transf. 178 (2016) 276–283. [3] D. McGloin, K. Dholakia, Bessel beams: diffraction in a new light, Contemp. Phys. 46 (1) (2005) 15–28. [4] D.L. Andrews, Structured Light And Its Applications: An Introduction To Phasestructured Beams And Nanoscale Optical Forces xiii, Academic, Amsterdam; Boston, 2008, p. 341. [5] V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam, Nature 419 (6903) (2002) 145–147. [6] F.O. Fahrbach, P. Simon, A. Rohrbach, Microscopy with self-reconstructing beams, Nat. Photonics 4 (11) (2010) 780–785. [7] Q.W. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photonics 1 (1) (2009) 1–57. [8] G. Gouesbet, G. Gréhan, Generalized Lorenz-Mie Theories, Springer, Berlin, 2011. [9] A. Doicu, T. Wriedt, Y.A. Eremin, Light Scattering by Systems of Particles Null-

[17]

[18]

[19]

[20]

[21]

[22]

108

Field Method with Discrete Sources: Theory and Programs, Springer-Verlag, Berlin, 2006. J.J. Wang, Y.P. Han, Z.F. Wu, L. Han, T-matrix method for electromagnetic scattering by a general anisotropic particle, J. Quant. Spectrosc. Radiat. Transf. 162 (2015) 66–76. G. Gouesbet, G. Grehan, B. Maheu, Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods, Appl. Opt. 27 (1988) 4874–4883. K.F. Ren, G. Gouesbet, G. Grehan, Integral localized approximation in generalized Lorenz-Mie theory, Appl. Opt. 37 (19) (1998) 4218–4225. G. Gouesbet, J.A. Lock, G. Grehan, Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: localized approximations and localized beam models, J. Quant. Spectrosc. Radiat. Transf. 12 (2011) 1–27. L.A. Ambrosio, H.E. Hernandez-Figueroa, Integral localized approximation description of ordinary Bessel beams and application to optical trapping forces, Biomed. Opt. Express 2 (7) (2011) 1893–1906. R.X. Li, L.X. Guo, C.Y. Ding, Z.S. Wu, Scattering of an axicon-generated Bessel beam by a sphere, Opt. Commun. 307 (2013) 25–31. F.G. Mitri, R.X. Li, L.X. Guo, C.Y. Ding, Resonance scattering of a dielectric sphere illuminated by electromagnetic Bessel non-diffracting (vortex) beams with arbitrary incidence and selective polarizations (vol 361, pg 120, 2015)Ann. Phys. 363 (2015) 562–563. G. Gouesbet, J.A. Lock, Comments on localized and integral localized approximations in spherical coordinates, J. Quant. Spectrosc. Radiat. Transf. 179 (2016) 132–136. L.A. Ambrosio, J.J. Wang, On the validity and reliability of the integral localized approximation for zeroth-order Bessel beams and associated radiation pressure forces, Appl. Opt. (2016) Submited for publication. G. Gouesbet, B. Maheu, G. Grehan, Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation, J. Opt. Soc. Am. A 5 (1988) 1427–1443. J.P. Barton, D.R. Alexander, S.A. Schaub, Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam, J. Appl. Phys. 64 (1988) 1632–1639. P.C. Preston, B.J. Mason, J.P. Reid, D. Luckhaus, R. Signorell, Size-dependent position of a single aerosol droplet in a Bessel beam trap, J. Opt. 16 (2) (2014) 025702. F.G. Mitri, Electromagnetic wave scattering of a high-order Bessel vortex beam by a

Optics Communications 387 (2017) 102–109

J.J. Wang et al. dielectri sphere, IEEE Trans. Antennas Propag. 59 (11) (2011) 4375–4379. [23] T. Cizmar, E. Kollarova, Z. Bouchal, P. Zemanek, Sub-micron particle organization by self-imaging of non-diffracting beams, New J. Phys. 8 (2006). [24] J.M. Taylor, G.D. Love, Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations, J. Opt. Soc. Am. a-Opt. Image Sci. Vis. 26 (2) (2009) 278–282. [25] J.A. Lock, Angular spectrum and localized model of Davis-type beam, J. Opt. Soc. Am. A 30 (3) (2013) 489–500. [26] J. Chen, J. Ng, P. Wang, Z.F. Lin, Analytical partial wave expansion of vector Bessel beam and its application to optical binding, Opt. Lett. 35 (10) (2010) 1674–1676 (erratum: 36(7)(: 1243-1243). [27] X.B. Ma, E.B. Li, Scattering of an unpolarized Bessel beam by spheres, Chin. Opt. Lett. 8 (12) (2010) 1195–1198. [28] S.K. Song, N. Wang, W.L. Lu, Z.F. Lin, Optical force on a large sphere illuminated by Bessel beams: comparisons between ray optics method and generalized LorenzMie theory, J. Opt. Soc. Am. A 31 (10) (2014) 2192–2197. [29] J.J. Wang, T. Wriedt, J.A. Lock, L. Madler, General description of circularly

[30]

[31] [32] [33]

[34] [35]

109

symmetric Bessel beams of arbitrary order, J. Quant. Spectrosc. Radiat. Transf. 184 (2016) 218–232. J.J. Wang, T. Wriedt, J.A. Lock, Y.C. Jiao, General description of transverse mode Bessel beams and construction of basis Bessel fields, J. Quant. Spectrosc. Radiat. Transf. (2016). http://dx.doi.org/10.1016/j.jqsrt.2016.10.010i. S.R. Mishra, A vector wave analysis of a bessel beam, Opt. Commun. 85 (2–3) (1991) 159–161. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1922. A.A.R. Neves, A. Fontes, L.A. Padilha, E. Rodriguez, C.H.D. Cruz, L.C. Barbosa, C.L. Cesar, Exact partial wave expansion of optical beams with respect to an arbitrary origin, Opt. Lett. 31 (16) (2006) 2477–2479. G. Gouesbet, T-matrix formulation and generalized Lorenz-Mie theories in spherical coordinates, Opt. Commun. 283 (4) (2010) 517–521. X. Zambrana-Puyalto, Control and Characterization of Nano-Structures with the Symmetries of Light, in Department of Physics & Astronomy, Macquarie University, New South Wales, Australia, 2014.