Resonance scattering of a dielectric sphere illuminated by electromagnetic Bessel non-diffracting (vortex) beams with arbitrary incidence and selective polarizations

Annals of Physics 361 (2015) 120–147

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Resonance scattering of a dielectric sphere illuminated by electromagnetic Bessel non-diffracting (vortex) beams with arbitrary incidence and selective polarizations F.G. Mitri a,∗ , R.X. Li b,c,∗∗ , L.X. Guo b,c , C.Y. Ding b a

Chevron, Area 52 Technology – ETC, 5 Bisbee Ct., Santa Fe, NM 87508, USA

b

School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

c

Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China

article

info

Article history: Received 29 April 2015 Accepted 8 June 2015 Available online 17 June 2015 Keywords: Generalized Lorenz–Mie scattering theory Vector Bessel vortex beams Angular spectrum decomposition Dielectric sphere

∗ ∗∗

abstract A complete description of vector Bessel (vortex) beams in the context of the generalized Lorenz–Mie theory (GLMT) for the electromagnetic (EM) resonance scattering by a dielectric sphere is presented, using the method of separation of variables and the subtraction of a non-resonant background (corresponding to a perfectly conducting sphere of the same size) from the standard Mie scattering coefficients. Unlike the conventional results of standard optical radiation, the resonance scattering of a dielectric sphere in air in the field of EM Bessel beams is examined and demonstrated with particular emphasis on the EM field’s polarization and beam order (or topological charge). Linear, circular, radial, azimuthal polarizations as well as unpolarized Bessel vortex beams are considered. The conditions required for the resonance scattering are analyzed, stemming from the vectorial description of the EM field using the angular spectrum decomposition, the derivation of the beam-shape coefficients (BSCs) using the integral localized approximation (ILA) and Neumann–Graf’s addition theorem, and the determination of the scattering coefficients of the sphere using Debye series. In contrast with the standard scattering theory, the resonance method presented here allows the quantitative description

Corresponding author. Corresponding author at: School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China E-mail addresses: [email protected] (F.G. Mitri), [email protected] (R.X. Li).

http://dx.doi.org/10.1016/j.aop.2015.06.004 0003-4916/© 2015 Elsevier Inc. All rights reserved.

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of the scattering using Debye series by separating diffraction effects from the external and internal reflections from the sphere. Furthermore, the analysis is extended to include rainbow formation in Bessel beams and the derivation of a generalized formula for the deviation angle of high-order rainbows. Potential applications for this analysis include Bessel beam-based laser imaging spectroscopy, atom cooling and quantum optics, electromagnetic instrumentation and profilometry, optical tweezers and tractor beams, to name a few emerging areas of research. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Breakthroughs and advances in physics have been mostly accomplished by using wave scattering phenomena, as one of the most efficient strategies to probe and characterize a particle of a collection of particles. Moreover, diffraction in systems involving any kind of wave propagation (i.e., electromagnetic/optical [1], mechanical/acoustical, gravitational) is recognized as one of the limiting factors for various industrial and technological advances. For example, in electromagnetic (EM)/optical laser beams and pulses, diffraction, manifested by a gradual spatial broadening of the beam, decreases image resolution and collimation quality in imaging, holography, microscopy, tweezers and lithography to name a few applications. Thus, it is of utmost importance to develop novel adequate techniques, operational devices and apparatuses capable of alleviating the beam distortion and the resulting degradation effects. Those challenges have fueled physicists and engineers to investigate unconventional beam solutions and improved instrumentation tools that resist diffraction over an extended region in space. At present, such ‘‘non-diffracting’’ waves [2] are well established both theoretically and experimentally, and innovative applications in fundamental and applied physics are increasingly burgeoning, demonstrating the ability to resist not only diffraction but also the simultaneous effects of attenuation and dispersion in liquid, elastic and viscoelastic dispersive media. A particular example that received significant attention is known as the Bessel beam (BB), which originates in the scalar wave diffraction theory as an exact solution of the wave equation [3–6], 2 Ψi = 0, where Ψi is a scalar wave function which describes the propagating field, the d’Alembertian operator is denoted by 2 = ∇ 2 − c −2 ∂/∂ t, and c is the wave speed in the medium of wave propagation. For a BB propagating in a Cartesian coordinate system along the axial z direction, the generalized mathematical   expression for the scalar field of vortex (spiraling) type is given by Ψi = ΨBB = Ψ0 Jm kρ ρ ei(kz z +mφ−ωt ) , where Ψ0 is the field amplitude, Jm (·) is the cylindrical Bessel function of order m, which determines the order (known also as the topological charge) of the beam. The fundamental solution (m = 0) has a maximum in amplitude (or intensity) at the center of the beam,  whereas the higher-order solutions (|m| ̸= 0) possess a central null [6,7]. The x2 + y2 is the distance to a point in the transverse plane (x, y), the azimuthal parameter ρ = −1 angle is φ = tan (y/x), the exponential e−iωt denotes the time-dependence where ω is the angular frequency, kρ = k sin α0 and kz = k cos α0 are the radial and axial wavenumbers, respectively, k is the wavenumber, and α0 is the half-cone angle defined with respect to the axis of wave propagation z, such that α0 = 0 corresponds to plane waves propagating along z. Note that ΨBB is the result of a superposition of plane waves over a cone with half-angle α0 [4], so as the resulting interference on the axis of wave propagation z produces an amplitude (or intensity) maximum for the zeroth-order beam (denoted in the following by J0 ) resulting from a constructive interference when the plane waves are all in phase, or a null in axial amplitude (or intensity) for the higher-order beams (denoted in the following by Jm ) resulting from a destructive interference when the plane waves are phase-shifted with respect to one another, in such a way that the phase shift around the cone is equal to 2π m.

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Moreover, ΨBB is described by the product of two separable functions (excluding the timedependence). The ‘‘axial’’ function eikz z describes the field’s variations along the direction of wave  propagation z, and the ‘‘transverse’’ function Jm kρ ρ eimφ describes the field’s properties in the transverse plane, perpendicular to the axis of wave propagation. The form of the axial function (which is equivalent to the propagator for unbounded plane waves) anticipates that the beam propagates while its phase only (and not the amplitude) varies with z without spreading (i.e., nondiffracting) while its transverse shape, described by the transverse function, is unchanged [8,9]. Other properties, such as the ability to reform after encountering an obstruction [10–12], the immobilization in the ‘‘hollow’’ axial region (for higher-order beams) [13] and the angular momentum-induced rotation of particles [14] have made BBs the focus of noteworthy investigations over the past decade and has poised them to make a significant impact in the advancement and development of new technologies. In most (if not all) applications, the total (incident + scattered) field interacting with a particle (or a collection of particles) located arbitrarily along the beam’s path, lead to the resonance scattering phenomenon, described originally in the framework of the nuclear resonance scattering theory [15–19] and heavy ion scattering [20–22]. It has also been thoroughly described using semi-analytical (i.e., involving numerical integration procedures to evaluate the beam-shape coefficients (BSCs) in the arbitrary/off-axial configuration) formalisms in acoustics [23,24]. In the electromagnetic (EM) context, it can be studied by means of the Lorenz–Mie [25] theory for plane waves [26–31], or the rigorous generalized Lorenz–Mie theories (GLMTs) for arbitrary-shaped beams [32–35]. Despite some investigations dedicated to EM/optical BBs from the standpoint of EM scattering (dealing with unpolarized [36] and linearly polarized [37] J0 and unpolarized Jm [38] BBs), it is of some importance to consider a complete and thorough analysis considering the scattering phenomenon with particular emphasis on the state of polarization of the EM field, spanning the operational frequency ranges of Rayleigh (particle size much smaller than the wavelength), Mie (particle size comparable and/or exceeding the wavelength) or ray optics (particle size much larger than the wavelength) regimes. The aim here is to tackle this subject and provide a comprehensive study, treating the interaction of an illuminating (vector) BB of any order m with a dielectric sphere, in the framework of the GLMT and resonance scattering for widespread applications mentioned previously. 2. Polarized BBs in the framework of the GLMT and resonance scattering Adequate description of polarized EM radiation requires a vectorial treatment [39,40], based on suitable mathematical solutions of Maxwell’s equations. In the case of BBs, several analytical (vectorial) approaches have been considered [36,38,41–46], nevertheless, the theoretical treatments may not be easily applicable to polarized BBs. To further simplify the analytical description of BBs, the angular spectrum decomposition (ASD) used previously for an axicon-generated linearly-polarized BB [47,48] is extended in the following to include the cases of linear, circular, radial and azimuthal polarizations as well as an unpolarized beam and investigate the polarization effects on the scattering by a dielectric sphere (Fig. 1). 2.1. Angular spectrum decomposition (ASD) and vector EM beams The ASD method [49] describes mathematically the EM fields of an arbitrary beam by means of weighted sums of individual (monochromatic) plane waves (including evanescent waves, which are disregarded in this analysis because they do not contribute to the scattering in the far-field) propagating in different directions in a homogeneous nonelectrically conducting, nonmagnetic medium. Assuming a time-variation in the form e−iωt , the vector electric E and magnetic B fields are expressed, respectively, as [47,48] E(r, θ , φ) = B(r, θ , φ) =



αmax

α=0

1 iω



2π

β=0

Epw eik·r | sin α cos α|dα dβ,

∇ × E(r, θ , φ),

(1)

(2)

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Fig. 1. Scattering geometry. A Bessel beam incident upon a lossless homogeneous dielectric sphere in air with arbitrary incidence.

where the wave vector k = (k sin α cos β, k sin α sin β, k cos α), r is the vector position, and Epw is known as the angular spectrum function. Its expression is given by, Epw = QP (α, β),

(3)

where k (= ω/c ) is the scalar wavenumber, ω is the angular frequency, c is the speed of light, α and β denote, respectively, the polar and azimuthal angles of the wave vector, αmax < π /2, and P (α, β) describes the incident beam’s profile [48] and the vector complex function Q is expressed as,

   px (cos α cos2 β + sin2 β) − py (1 − cos α) sin β cos β  Q = −px (1 − cos α) sin β cos β + py (cos α sin2 β + cos2 β) ,   −px sin α cos β − py sin α sin β

(4)

with (px , py ) being the complex polarization parameters, which determine the polarization character of the resulting beam. For a BB of order l, the angular spectrum function is expressed as, Epw = QP (α, β)

= Epw0 (α0 , β) Q

δ(α − α0 ) ilβ e , |sin α0 | cos α0

(5)

where Epw0 (α0 , β) is the amplitude of the electric field component of the plane waves propagating over a cone with half-angle α0 and azimuthal angle β . Substituting Eq. (5) into Eq. (1), the electric field for the BB can be therefore expressed as, E(r, θ , φ) =

2π



β=0 2π

 =

β=0

Epw eik·r dβ Epw0 Q|α=α0 eilβ eik·r dβ.

(6)

Consequently, if follows that the magnetic field is written as, B(r, θ , φ) =



2π

β=0



2π

= β=0

with Bpw0 = Epw0 ωk .

Bpw eik·r dβ Bpw0 k × Q|α=α0 eilβ eik·r dβ,

(7)

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Now that the general expressions for the electric and magnetic fields are obtained in Eqs. (6) and (7), respectively, the following polarizations for BBs are considered. 2.1.1. Linearly x-polarized BB For a BB polarized along the x-direction, the polarization parameters (px , py ) = (1, 0). Therefore, the vector complex function Q becomes,



Q|α=α0

 (cos α cos2 β + sin2 β) = Q = −(1 − cos α) sin β cos β  , − sin α cos β α=α x

(8)

0

where the superscript x denotes the state of polarization.  2π Using the properties of the integral 0 eilβ eiσ cos(β−φ) dβ = 2π il Jl (σ )eilφ and the following trigonometric identities: cos2 β =

 −i β

1 2

[1 + cos 2β ], sin2 β =

1 2

[1 − cos 2β ], cos β =

1 2

eiβ + e−iβ , and





1 eiβ − e , the substitution of Eq. (8) into Eqs. (6) and (7), respectively, leads after some sin β = 2i algebraic manipulation to the expressions for the electric and magnetic field components as,



   1  il Jl (σ )eilφ − P⊥ il+2 Jl+2 (σ )ei(l+2)φ + il−2 Jl−2 (σ )ei(l−2)φ      2     x ikz z   1 Ey = EB0 e , l+2 i(l+2)φ l −2 i(l−2)φ −P ⊥ i Jl+2 (σ )e − i Jl−2 (σ )e     x    2i   Ez       −P|| il+1 Jl+1 (σ )ei(l+1)φ + il−1 Jl−1 (σ )ei(l−1)φ   1      x P⊥ il+2 Jl+2 (σ )ei(l+2)φ − il−2 Jl−2 (σ )ei(l−2)φ −       2i Bx      x ikz z   1 By = BB0 e , l ilφ l +2 i(l+2)φ l −2 i(l−2)φ i Jl (σ )e + P⊥ i Jl+2 (σ )e + i Jl−2 (σ )e      x   2   Bz      l +1  iP|| i Jl+1 (σ )ei(l+1)φ − il−1 Jl−1 (σ )ei(l−1)φ  x  Ex  

(9)

(10)

where EB0 = π Epw0 (1 + cos α0 ), BB0 = ωk EB0 , σ = kρ ρ = kρ sin α0 , and the BB coefficients P⊥ and sin α 1−cos α P|| are respectively given by P⊥ = 1+cos α0 and P|| = 1+cos0α . Note that Eqs. (9), (10) describe the EM 0 0 field components in the Cartesian coordinate system (x, y, z ). When the center of the BB is translated and relocated at (x0 , y0 , z0 ) rather than at the origin, the EM field components become,

   1  il Jl (σG )eilφG − P⊥ il+2 Jl+2 (σG )ei(l+2)φG + il−2 Jl−2 (σG )ei(l−2)φG       2   x  1  l +2 Ey = EB0 eikz (z −z0 ) , (11) i(l+2)φG l −2 i(l−2)φG −P⊥ i Jl+2 (σG )e − i Jl−2 (σG )e      x   2i   Ez       −P|| il+1 Jl+1 (σG )ei(l+1)φG + il−1 Jl−1 (σG )ei(l−1)φG     1   x  − P⊥ il+2 Jl+2 (σG )ei(l+2)φG − il−2 Jl−2 (σG )ei(l−2)φG     B    2i   xx    ik ( z − z )   0 By = BB0 e z 1 , (12) l +2 i(l+2)φG l ilφG l −2 i(l−2)φG + i Jl−2 (σG )e     Bx   i Jl (σG )e + 2 P⊥ i Jl+2 (σG )e      z     iP|| il+1 Jl+1 (σG )ei(l+1)φG − il−1 Jl−1 (σG )ei(l−1)φG   where σG = kρ ρG = kρG sin α0 , ρG = (x − x0 )2 + (y − y0 )2 = ρ 2 + ρ02 − 2ρρ0 cos(φ − φ0 ),  ρ0 = x20 + y20 , φG = φ +ψ , φ0 = tan−1 (y0 /x0 ), φ = tan−1 (y/x) and φG = tan−1 [(y − y0 )/(x − x0 )]  x  Ex  

(see Fig. 1 for the definition of these parameters).

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The expressions given by Eqs. (11), (12) can be further simplified by applying Neumann–Graf’s translational addition theorem for the cylindrical Bessel functions [50], such that, Jl (σG )eilψ =

∞ 

Jp (kρ ρ0 )Jp+l (kρ ρ)eip(φ−φ0 ) ,

(13)

Jp (kρ ρ0 )Jp+l (kρ ρ)ei(l+p)φ e−ipφ0 ,

(14)

p=−∞

and Jl (σG )eilφG =

∞  p=−∞

to finally express the translated EM field components in the new coordinate system as

    ∞ l i(l+p)φ −ipφ0     i J ( k ρ ) J ( k ρ) e e p ρ 0 p+l ρ        p=−∞            ∞        1     l +2 i(l+2+p)φ −ipφ0   P i J ( k ρ ) J ( k ρ) e e − ,     ⊥ p ρ 0 p + l + 2 ρ   2       p =−∞            ∞      1 l −2   i(l−2+p)φ −ipφ0   − P⊥ i Jp (kρ ρ0 )Jp+l−2 (kρ ρ)e e      x   2   p =−∞   E       ∞  x    l +2 i(l+2+p)φ −ipφ0 x ikz (z −z0 ) i Jp (kρ ρ0 )Jp+l+2 (kρ ρ)e e Ey = EB0 e ,        x    p=−∞ 1      Ez  −P ⊥     , ∞    2i     l −2 i(l−2+p)φ −ipφ0     − i J ( k ρ ) J ( k ρ) e e   p ρ 0 p + l − 2 ρ       p =−∞       ∞        l +1 i(l+1+p)φ −ipφ0   i Jp (kρ ρ0 )Jp+l+1 (kρ ρ)e e           p=−∞         − P||  ,    ∞          l −1 i(l−1+p)φ −ipφ0   +i Jp (kρ ρ0 )Jp+l−1 (kρ ρ)e e  

(15)

    ∞  l +2 i(l+2+p)φ −ipφ0    i J ( k ρ ) J ( k ρ) e e p ρ 0 p + l + 2 ρ            p=−∞ 1         P , − ⊥    ∞    2i       l − 2 i ( l − 2 + p )φ − ip φ 0   − i J ( k ρ ) J ( k ρ) e e   p ρ 0 p+l−2 ρ       p=−∞       ∞      l i(l+p)φ −ipφ0     i J ( k ρ ) J ( k ρ) e e p ρ 0 p + l ρ         p=−∞     x        B     x ∞      1   l +2 i(l+2+p)φ −ipφ0  x ikz (z −z0 ) + P i J ( k ρ ) J ( k ρ) e e By = BB0 e , .   ⊥ p ρ 0 p+l+2 ρ  2       x   p=−∞     Bz       ∞       1 l −2    i ( l − 2 + p )φ − ip φ 0   + P⊥ i J ( k ρ ) J ( k ρ) e e   p ρ 0 p+l−2 ρ     2   p=−∞       ∞        l + 1 i ( l + 1 + p )φ − ip φ 0   i J ( k ρ ) J ( k ρ) e e p ρ 0 p+l+1 ρ           p=−∞         , iP ||     ∞          l − 1 i ( l − 1 + p )φ − ip φ 0   − i J ( k ρ ) J ( k ρ) e e   p ρ 0 p+l−1 ρ

(16)

p=−∞

p=−∞

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2.1.2. Linearly y-polarized BB For a BB polarized along the y-direction, the polarization parameters become (px , py ) = (0, 1), so that the vector complex function Q is expressed as,

 −(1 − cos α) sin β cos β = Qy =  cos α sin2 β + cos2 β  . − sin α sin β α=α 

Q|α=α0

(17)

0

Performing the required algebraic manipulations as explicitly described in Section 2.1.1, the translated electric and magnetic field components for a BB polarized along the y-direction are given, respectively, by,

 y  Ex   Ey

y   y 

=

By

y   y 

cBxy

 

Ez

 y  Bx  

 x   Ey  

=−

cBxz

(18)

 −1 x   c Ex   Bxx

 

Bz

,

 

c

.

(19)



−1 x 

Ez

2.1.3. Circularly polarized BB For a circularly-polarized BB, the polarization parameters become (px , py ) = (1, ±i) corresponding to left (+) and right (−) polarizations. Consequently, the vector complex function Q is expressed as, Q|α=α0 = Qcirc

= Qx ± i Qy ,

(20)

so that the translated electric and magnetic field components for a circularly polarized BB become, respectively,

 circ   Ex  

 x  y  Ex ± iEx   x y Eycirc = Ey ± iEy ,    circ     x  Ez ± iEzy Ez  circ   x  y  Bx    Bx ± iBx   x y B ± iB Bcirc = . y y y    circ     x y B ± iB Bz z z

(21)

(22)

2.1.4. Radially polarized BB For a radially-polarized BB, the polarization parameters become (px , py ) = (cos β, sin β), and consequently, the vector complex function Q is expressed as, cos α cos β





Q|α=α0 = Qrp =  cos α sin β 





− sin α

.

(23)

α=α0

Performing the adequate algebraic manipulations following the methodology described in Section 2.1.1, the translated electric and magnetic field components for a radially-polarized BB are

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given, respectively, by,

 −Ezx cot α0     − cBxz cot α0 rp Ey = , ∞        rp  ikz (z −z0 ) l i(l+p)φ −ipφ0    i Jp (kρ ρ0 )Jp+l (kρ ρ)e e Ez −2P|| EB0 e 

(24)

  x   Bz / sin α0  Brp = Byz / sin α0 . y      rp    0

(25)

    

 rp    Ex 

p=−∞

 rp   Bx   Bz

2.1.5. Azimuthally polarized BB The polarization parameters for an azimuthally-polarized BB are defined as (px , py ) = (− sin β, cos β), so that the vector complex function Q becomes,

− sin β = Q = cos β 

Q|α=α0

 .

ap

0

(26)

α=α0

Following the procedure described previously, the electric and magnetic field components for an azimuthally-polarized BB are found respectively, to be,

 x    cBz / sin α0   Eyap = −Ezx / sin α0 ,    ap      0  ap   Ex  

(27)

Ez

 ap   Bx   Bap y

 

Bap z

=

 

    

c −1 Ezx cot α0 Bxz cot α0

    

. ∞     ikz (z −z0 ) l i(l+p)φ −ipφ0    i Jp (kρ ρ0 )Jp+l (kρ ρ)e e 2P|| BB0 e 

(28)

p=−∞

2.1.6. Unpolarized BB An unpolarized BB, expressed using the ASD method, may be considered as the synthesis of both x- and y-polarized beams [51]. Therefore, the polarization parameters become (px , py ) = (1, 1), and the vector complex function Q is expressed as, Q|α=α0 = Qunp

= Qx + Qy , x

(29)

y

where Q and Q are given explicitly by Eqs. (8) and (17), respectively. It follows that the EM field components are expressed as,

 unp   Ex  

 x  y  Ex + Ex   Eyunp = Eyx + Eyy ,     unp     x Ez + Ezy Ez  unp   x  y  Bx    Bx + Bx   x y Bunp B + B = . y y y    unp     x y Bz Bz + Bz

(30)

(31)

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2.2. Beam-shape coefficients (BSCs) and the integral localized approximation (ILA) As discussed in the GLMT for the scattering by a sphere [32], partial-wave coefficients, termed as BSCs, are used to describe the incident beam regardless of the presence of the sphere. There exist several methods to evaluate the BSCs based on quadratures [33,36,38], finite series and localized approximations [32]. An improved method, known as the integral localized approximation (ILA) [52] combines the features of quadratures and prescriptions from the localized approximation [32], which will be used here. The analytical derivation for the BSCs stems from the determination of the radial components Er pol pol and Br of the EM field forming the BB. According to each state of polarization, Er = Er and Br = Br can be determined from the Cartesian components in the translated system of coordinates as,



Erpol



 =

Bpol r

Expol



 sin θ cos φ +

Bpol x

Eypol Bpol y



 sin θ sin φ +

Ezpol Bpol z

 cos θ ,

(32)

where the superscript ‘‘pol’’ determines the polarization state (i.e., ‘‘pol = x, y, circ, rp, ap, and unp’’, respectively, as defined previously). Subsequently, the ILA method [52,53] is utilized, which consists in applying a localized operator pol pol to the components Er and Br . Namely, the localized operator transforms the parameters kρ and θ (which appear explicitly in the expressions of the electric and magnetic field components in Cartesian pol pol coordinates) to (n + 1/2) and π /2, respectively, in the expressions of Er and Br , leading to the pol

expressions of the localized radial fields (denoted by an overbar) as Er pol Br

pol Bx

=

cos φ +

pol By

= Expol cos φ + Eypol sin φ , and

sin φ .

pol

m,pol

pol

m,pol

After obtaining Er and Br for the desired type of BB polarization, the BSCs gn,TM and gn,TE can be directly obtained by single integration over the azimuthal angle [52],



m,pol

gn,TM



m,pol gn,TE

=

Znm 2π

2π

 0

  E pol (r , θ , φ)/E  B0

r



pol Br

(r , θ , φ)/BB0 

e−imφ dφ,

(33)

where

 2n(n + 1)i    − 2n + 1 |m|−1 Znm =   2i   2n + 1

m=0 (34) m ̸= 0.

2.2.1. BSCs for a linearly x-polarized BB For a BB polarized along the x-direction, the BSCs are determined by inserting Eqs. (15) and (16) into Eq. (32), applying the localization operator to obtain Er x and Br x , then substituting the result in Eq. (33) and performing the integration. This procedure leads to the mathematical expressions of the BSCs as,



m,x

gn,TM m,x gn,TE

 =

Znm −ikz z0 l e i 2

   −i(1−l+m)φ0 −i(−1−l+m)φ0 J (ε) J (ϖ ) e + J (ε) J (ϖ ) e   1 − l + m 1 + m − 1 − l + m − 1 + m      ,     +P⊥ J−1−l+m (ε)J1+m (ϖ )e−i(−1−l+m)φ0 + J1−l+m (ε)J−1+m (ϖ )e−i(1−l+m)φ0    , (35) ×    P⊥ J1−l+m (ε)J−1+m (ϖ )e−i(1−l+m)φ0 − J−1−l+m (ε)J1+m (ϖ )e−i(−1−l+m)φ0      ,  i −J−1−l+m (ε)J−1+m (ϖ )e−i(−1−l+m)φ0 + J1−l+m (ε)J1+m (ϖ )e−i(1−l+m)φ0 where ε = ρ0 k sin α0 and ϖ = (n + 1/2) sin α0 .

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2.2.2. BSCs for a linearly y-polarized BB As for the BB polarized along the y-direction, the BSCs are determined by inserting Eqs. (18), (19) into Eq. (32), applying the localization operator to obtain Er y and Br y , then substituting the result in Eq. (33) and performing the integration. It follows that the BSCs are expressed as, m,y



gn,TM

 =

m,y

m,x





gn,TE

.

,x −gnm,TM

gn,TE

(36)

2.2.3. BSCs for a circularly polarized BB The BSCs for a circularly-polarized BB follow immediately by direct superposition as in the case of the EM field components given by Eqs. (21), (22), and their substitution into Eq. (32). After applying the localization operator to obtain Er circ and Br circ , then substituting the result into Eq. (33) and performing the integration, the BSCs are expressed as, m,circ



gn,TM



gn,TM

=

m,circ

m,x





m,x

gn,TE

gn,TE

m,y

 ±i

gn,TM m ,y

gn,TE

 .

(37)

2.2.4. BSCs for a radially polarized BB For the case of a radially-polarized BB, the BSCs are obtained by substituting Eqs. (24), (25) into Eq. (32), applying the localization operator to obtain Er rp and Br rp , then substituting the result in Eq. (33) and performing the integration. This procedure leads to the following expressions, m,rp



gn,TM

 =

m,rp

gn,TE

Znm P|| e−ikz z0 Jm−l

(ε)i

l+1 −i(m−l)φ0

e



cot α0 [Jm+1 (ϖ ) − Jm−1 (ϖ )] ,



(i/sinα0 ) [Jm+1 (ϖ ) + Jm−1 (ϖ )] ,

.

(38)

2.2.5. BSCs for an azimuthally polarized BB The BSCs for an azimuthally-polarized BB are derived by inserting Eqs. (27), (28) into Eq. (32), using the localization operator to obtain Er ap and Br ap , then substituting the result in Eq. (33) and performing the integration, leading to the expressions,



m,ap

gn,TM

 =

m,ap

m,rp



gn,TE

gn,TE

,rp −gnm,TM

 .

(39)

2.2.6. BSCs for an unpolarized BB Finally, the BSCs for an unpolarized BB are derived by inserting Eqs. (30), (31) into Eq. (32), using the localization operator to obtain Er unp and Br unp , then substituting the result in Eq. (33) and performing the integration, leading to the following expressions,



m,unp

gn,TM

m,ap



 =

m,x

m,y

gn,TM + gn,TM m,x

m,y

gn,TE + gn,TE

gn,TE

 .

(40)

2.2.7. BSCs for BBs having α0 = 0 For the particular case when α0 = 0 (corresponding to plane waves for the zeroth-order BB), the parameters ε and ϖ vanish. For the zeroth-order beam (l = 0), this leads  to the vanishing of the m,{rp,ap}

BSCs for the radially- and azimuthally polarized beams, such that gn,{TM ,TE }  α =0 = 0. Moreover, for 0



l=0

the higher-order  BBs (l ̸= 0), the BSCs for the linear, circular and unpolarized BBs vanish such that m,{x,y,circ ,unp}

gn,{TM ,TE }

  α0 =0 = 0. l̸=0

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2.3. Electromagnetic resonance scattering in the framework of the GLMT Once the BSCs for the BB with its chosen polarization state are obtained, the far-field scattered intensity can be evaluated as [32], lim

kρ→∞

pol Isc

(ρ, θ , φ) =

1 2ck2 ρ 2

 2  2   pol   pol  S1 (θ , φ) + S2 (θ , φ) ,

(41)

where pol

S1 = pol S2

∞  +n   2n + 1  m,pol m,pol man gn,TM πn|m| (cos θ ) + ibn gn,TE τn|m| (cos θ ) exp(imφ), n(n + 1) n=1 m=−n

∞  +n   2n + 1  m,pol m,pol an gn,TM τn|m| (cos θ ) + imbn gn,TE πn|m| (cos θ ) exp(imφ), = n(n + 1) n=1 m=−n

(42)

|m|

πn|m| (cos θ ) =

Pn (cos θ ) sin θ

, (43)

|m|

τn (cos θ ) = |m|

dPn (cos θ ) dθ

,

and the parameters an and bn are the standard Mie scattering coefficients for a dielectric sphere, given by [25], an =

bn =

m1 ψn (m1 x) ψn′ (x) − ψn (x) ψn′ (m1 x) (1)′

m1 ψn (m1 x) ξn

(x) − ξn(1) (x) ψn′ (m1 x)

m1 ψn (x) ψn′ (m1 x) − ψn (m1 x) ψn′ (x) (1)′

(1)

m1 ξn (x) ψn′ (m1 x) − ψn (m1 x) ξn

( x)

,

(44)

,

(45)

where, x = ka is the non-dimensional size parameter of the sphere of radius a with relative refractive index m1 , ψn (x) = xjn (x) is the spherical Riccati–Bessel function of the first kind with jn (.) being the (1) (1) spherical Bessel function of the first kind, ξn (x) = xhn (x) is the spherical Riccati–Hankel function (1) of the first kind and hn (.) being the spherical Hankel function of the first kind. It is convenient to define a normalized scattering intensity in the far-field such that, pol 2 2 I∞, sc (θ , φ) = 2ck ρ





pol lim Isc (ρ, θ , φ)

kρ→∞

 2  2      = S1pol (θ , φ) + S2pol (θ , φ) .

(46)

A dielectric sphere is an example of a ‘‘dielectric resonator’’ [54], for which the complex eigenfrequencies of the EM eigen-vibrations may be evaluated from a characteristic equation [54]. A detailed connection between the resonance scattering and eigen-vibrations using plane EM waves [55] showed that distinct resonance features arise through a mechanism in which surface waves [56] circumnavigate repeatedly the sphere’s surface, leading to a resonance reinforcement [57] and an EM radiation enhancement during the scattering process. A thorough analysis for a dielectric sphere [57] showed that for each partial-wave n, ‘‘pure’’ resonances are superimposed on a nonresonance background attributed to a perfectly conducting sphere, so that their distinct contributions can be explicitly expressed by the Mie scattering coefficients (see Eq. (22) in [55]). Therefore, appropriate isolation of the resonances requires subtracting this non-resonance background from the Mie coefficients, which is extended in the framework of the GLMT, and applied in the following for a BB with any state of polarization.

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131

Consequently, the resonance Mie scattering coefficients are defined as, cond ares , n = an − an

(47)

cond bres , n = bn − bn

where acond and bcond are the Mie scattering coefficients for a perfectly conducting sphere, given by, n n

ψn′ (x)

acond = n

ξn(1) (x) ψn (x) ′

bcond = n

ξn(1) (x)

, (48)

.

It follows that the normalized far-field resonance scattering intensity can be evaluated as,

 

res,pol

res,pol I∞, sc (θ , φ) = S1

2  2     (θ , φ) + S2res,pol (θ , φ) ,

(49)

where res,pol

∞  +n   2n + 1  res m,pol |m| m,pol |m| man gn,TM πn (cos θ ) + ibres n gn,TE τn (cos θ ) exp(imφ), n(n + 1) n=1 m=−n

S1

=

res,pol S2

∞  +n   2n + 1  res m,pol |m| m,pol |m| an gn,TM τn (cos θ ) + imbres = n gn,TE πn (cos θ ) exp(imφ). n(n + 1) n=1 m=−n

(50)

Other observables of interest are the efficiency factors Qsca , Qext and Qabs , which represent the relative amounts of the incident energy that is scattered, extinguished and absorbed by the sphere, respectively. Their expressions are given explicitly in Ch. III of [32]. Their resonance counterparts are res deduced as follows by replacing an and bn , by ares n and bn , respectively, such that,

2  ∞  n  2n + 1 (n + |m|)!  res 2 m,pol 2 2 m,pol 2 |an | |gn,TM | + |bres , n | |gn,TE | ka n(n + 1) (n − |m|)! n=1 m=−n  2  ∞  n   m,pol 2  2n + 1 (n + |m|)!   res  m,pol 2 2 ℜ an |gn,TM | + ℜ bres |gn,TE | , = n ka n(n + 1) (n − |m|)! n=1 m=−n

res,pol Qsca =

res,pol

Qext

res,pol

Qabs



2

res,pol res,pol = Qext − Qsca .

(51)

(52) (53)

pol

Moreover, a particular attention is given to the scattering efficiency factor Qsca by substituting Eq. (47) into its expression given in [32]. After algebraic operation, it is expressed as, pol res,pol cond,pol interf ,pol Qsca = Qsca + Qsca + Qsca ,

(54)

where, cond,pol Qsca =



2  ∞  n  2n + 1 (n + |m|)!  cond 2 m,pol 2 ,pol 2 |an | |gn,TM | + |bcond |2 |gnm,TE | , n ka n(n + 1) (n − |m|)! n=1 m=−n 2

(55)

and, interf ,pol Qsca =

∞  n  2n + 1 (n + |m|)! 2 n (ka) n=1 m=−n (n + 1) (n − |m|)!    m,pol 2  res cond∗  m,pol 2  cond∗ × ℜ ares a | g | + ℜ bn bn |gn,TE | . n n n,TM

8

(56)

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Eq. (54) reveals an important property in the scattering process; that is, the scattering efficiency factor (or alternatively the scattering cross section) is not the mere sum of the ‘‘resonance’’ and ‘‘conducting’’ efficiency factors. A similar phenomenon exists also in acoustics [58]. There is an interf ,pol interference efficiency factor Qsca describing the interaction of the diffracted or specularly reflected waves with the surface resonance waves. Resonance occurs when a surface wave has an exact number of wavelengths around the sphere. When the resonance waves are in phase with the diffracted or specularly reflected waves, resonances appear as maxima in the scattering efficiency. Moreover, the width of the resonance is related to a ‘‘dwell time’’, interpreted as a measure of the duration the surface wave reradiates as it progressively decays during the resonance scattering process [59]. 3. Numerical results and discussions 3.1. Numerical validation The analysis is started by performing a numerical validation of the ILA method used to derive the BSCs, since they are used to evaluate the physical observables, such as the scattering efficiency and its resonance counterpart, as well as the scattered far-field intensity from the sphere. The numerical verification is focused on comparing the computations for the magnitude of the ‘‘exact’’ incident translated electric field [given by Eq. (11)], with its corresponding reconstructed one from the BSCs according to the following transformation [32],

 pol   Ex   Eypol

 

 pol 

Ez

sin θ cos φ

cos θ cos φ

 =  sin θ sin φ cos θ

cos θ sin φ



− sin θ

  pol  E − sin φ   r    pol cos φ  Eθ ,   pol   0

(57)

Eφ

with

 pol  E   r   pol

Eθ

 

 pol 

Eφ

=

EB0 kρ

  ∞ +n 1     m,pol |m| n+1   i k ρ) P ( cos θ ) exp ( im φ) 2n + 1 g ψ (− ) ( ) n,TM n (   n     k ρ   n=1 m=−n       ∞  +n     2n + 1 m,pol ′ m,pol |m| |m| gn,TM ψn (kρ) τn (cos θ ) + mgn,TE ψn (kρ) πn (cos θ ) exp(imφ) . ×     n=1 m=−n n(n + 1)       ∞ + n      2n + 1      m , pol m , pol ′ | m | | m |  mgn,TM ψn (kρ) πn (cos θ ) + gn,TE ψn (kρ) τn (cos θ ) exp(imφ)  i n ( n + 1 ) n=1 m=−n (58) The same can be done for the magnetic field, but it has been omitted for brevity. The case of an incident first-order Bessel beam polarized along the x-direction is considered, for which the wavelength of the incident radiation is 0.785 µm, with a half-cone angle α0 = 5°, and the offset of the beam is x0 = y0 = 1 µm with z0 = 0. Computations for the x-component of the electric field are evaluated, and Fig. 2 displays the results of the original field [panel (A)] and the reconstructed one [panel (B)]. The maximal numerical error is in the order of ∼10–5 which is considered negligible. Additional tests and computations (not shown here for the sake of brevity) and comparisons between the original and reconstructed additional components (i.e., y and z) of the incident electric field, as well as for all the considered polarizations and half-cone angle α0 of the first-order Bessel beam have been performed and the maximum numerical error did not exceed 10−5 . In the computations, the

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133

Fig. 2. The spatial distribution in the transverse (x, y) plane for the electric field of a first-order Bessel beam (l = 1) computed from the exact solution [panel (A)] and the reconstructed one [panel (B)]. The maximal numerical error is in the order of ∼10−5 .

series have been truncated at a maximal nmax = N = ka + 4(ka)1/3 + 2, a condition that ensures adequate convergence of the series [60]. 3.2. Resonance extinction (or scattering) efficiency The analysis is started by evaluating the extinction efficiency and its resonance counterpart [i.e., Eq. (52)] of a dielectric lossless sphere (with a refractive index m1 = 1.4), which is a measure of the ability of the sphere to capture power from the incident Bessel beam and redirect it as an extinct power. Note that for a non-absorptive (lossless) sphere, the (resonance) extinction efficiency equals the (resonance) scattering efficiency. The panels in Fig. 3A display the extinction efficiency and its resonance counterpart versus the non-dimensional frequency ka in the range 0 < ka ≤ 25, for linear (x, y), circular (circ), radiallypolarized (rp), azimuthally-polarized (ap) and unpolarized (unp) zeroth-order Bessel beams centered on a dielectric sphere (i.e., on-axis configuration), respectively, with a half cone-angle α0 = 0°. Note that a zeroth-order Bessel beam with α0 = 0° corresponds to plane waves. The upper panels in Fig. 3A for the x- and y-polarizations display similar results. Moreover, the results for the circularly polarized and unpolarized plane wave are equal, whereas the azimuthally- and radially-polarized plane wave results are zero. In addition, the plots reveal the importance of the background subtraction method that shows the capability of isolating resonances of the dielectric sphere, especially in the res,pol after range 0 < ka ≤ 5, such that individual resonance peaks tend to appear in the plots of Qext res,pol as ka increases from the background subtraction. There is also an initial increase in the plots of Qext res,{x,y} Rayleigh limit (ka ≪ 1). For the linearly polarized case (i.e. the upper panels), Qext approach the geometrical optics limit of 2 as ka → ∞, whereas this value is doubled for the circularly polarized and unpolarized cases. Furthermore, the frequency spectrum consists of rapid oscillations superimposed on a slowly varying background, which is removed for the resonance extinction efficiency plots. Similarly to Fig. 3A, the panels in Fig. 3B display the extinction efficiency and its resonance counterpart versus ka but for zeroth-order Bessel beams centered on the dielectric sphere having a half cone-angle α0 = 10°, with emphasis on polarization. For the linearly- and circularly-polarized res,pol as well as the unpolarized beams, the amplitudes of the resonance peaks in the plots for Qext are significantly diminished for ka > 10, which is not the case for the radially- and azimuthally-polarized beams. The effect of increasing the order (or topological charge) of the Bessel beams to l = 1 is shown in res,{x,y,circ ,unp} the panels of Fig. 3C for α0 = 0°, and Fig. 3D for α0 = 10°, respectively. In Fig. 3C, Qext = 0, res,{rp,ap} while the plots for Qext show resonance peaks which tend to be narrower at higher ka. When res,pol α0 > 0°, all the plots for Qext display resonances in Fig. 3D, which can be better displayed without

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Fig. 3A. The extinction efficiency and its resonance counterpart versus the non-dimensional frequency ka in the range 0 < ka ≤ 25, for linear (x, y), circular (circ), radially-polarized (rp), azimuthally-polarized (ap) and unpolarized (unp) zerothorder Bessel beams centered on a dielectric sphere (i.e., on-axis configuration), respectively, with a half cone-angle α0 = 0°.

Fig. 3B. The same as in Fig. 3A, but α0 = 10°.

the slowly varying background. These results particularly demonstrate the Bessel beam polarizations effects on the extinction efficiency and its resonance counterpart. res,pol

Additional computations for the resonance extinction efficiency Qext of the dielectric sphere are performed in the ranges 0° ≤ α0 ≤ 40° and 0 < ka ≤ 25, for a zeroth- and a first-order Bessel beam centered on the dielectric sphere, and the results are displayed in the panels of Fig. 4A for the zeroth-order Bessel beam and Fig. 4B for the first-order Bessel beam. The panels of Figs. 4C, 4D show res,pol the Qext plots for the dielectric sphere shifted off the beam’s axis by (kx0 , ky0 ) = (12.2; 12.2). As observed from these panels, significant differences are noticed between the axial and off-axial cases. In res,pol particular, for the off-axial cases, the maxima in the Qext plots are noticed at α0 values not exceeding 10°.

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135

Fig. 3C. The same as in Fig. 3A, but for a first-order Bessel vortex beam.

Fig. 3D. The same as in Fig. 3A, but for a first-order Bessel vortex beam with α0 = 10°.

3.3. Far-field directivity patterns for the normalized scattered intensity from a dielectric sphere and rainbow formation in Bessel beams Next, the effect of polarization is investigated on the normalized scattering intensity in the farfield [i.e., Eq. (46)] for the homogeneous lossless dielectric sphere (with a refractive index m1 = 1.4) in Bessel beams with α0 = 30° chosen as an example. The 3D computational plots are displayed for a zeroth-order Bessel beam in Fig. 5A, and for a first-order Bessel beam in Fig. 5B for ka = 15. As pol shown in the plots, the effect of beam polarization on the directivity of I∞,sc (θ , φ) is generally very significant. In all the figures, the arrows on the left-hand side of each panel indicate the direction of the incident beam. In Fig. 5A, the linear, circular, and unpolarized beams show comparable spatial distribution of the scattered intensity with a maximum central lobe (in the forward scattering direction θ = 0), whereas

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res,pol

Fig. 4A. The plots for the resonance extinction efficiency Qext of the dielectric sphere in the ranges 0° ≤ α0 ≤ 40° and 0 < ka ≤ 25, for a zeroth-order Bessel beam centered on the dielectric sphere.

Fig. 4B. The same as in Fig. 4A but for a first-order Bessel beam centered on the dielectric sphere.

the radially- and azimuthally-polarized beams’ results show an intensity null along the axis of wave propagation. Note also that the second intensity lobe occurs about θ ∼ 30°, which equals the halfcone angle value α0 = 30°. Increasing the beam’s order to l = 1 also affects the directivity patterns for the scattered intensity, as shown in Fig. 5B. The intensity maximum along the axis of wave propagation for the linear, circular, and unpolarized beams observed in Fig. 5A is transformed into an intensity null for the first-order Bessel beam centered on the sphere. However, concerning the radially- and azimuthally-polarized beams’ results, the main central lobe possesses a nonzero intensity in the forward scattering direction

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Fig. 4C. The same as in Fig. 4A but for a zeroth-order Bessel beam shifted off-axially with respect to the center of the sphere.

Fig. 4D. The same as in Fig. 4A but for a first-order Bessel beam shifted off-axially with respect to the center of the sphere.

(θ = 0) along the axis of wave propagation, with the secondary lobe occurring about θ ∼ 30° still emphasized in the plots. The effect of increasing the size parameter to ka = 30 is shown in Figs 5C and 5D for the zerothand first-order Bessel beam, respectively. In Fig. 5C, the main central lobe in the forward direction observed for the linear, circular, and unpolarized zeroth-order Bessel beams increases, and becomes narrower, compared to the results of Fig. 5A. This is not the case for the radially- and azimuthally polarized beams’ results as shown in the panels of Fig. 5C which still display an intensity null along the axis of wave propagation. Nevertheless, the secondary lobe occurring about θ ∼ 30° remains emphasized for all polarizations.

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Fig. 5A. The plots for the normalized scattering intensity in the far-field from the homogeneous lossless dielectric sphere (with a refractive index m1 = 1.4) centered on the axis of a zeroth-order Bessel beam with α0 = 30° and ka = 15.

Fig. 5B. The same as in Fig. 5A but for a first-order Bessel beam.

As for Fig. 5D, the intensity null along the axis of wave propagation, observed for the linear, circular, and unpolarized beams as shown in Fig. 5B remains, and is less enhanced in the 3D directivity pattern plots than the secondary scattering lobe. Furthermore, concerning the radially- and azimuthallypolarized beams’ results, the forward scattering lobe increases in magnitude, which can be expected since the non-dimensional frequency ka has increased. Moreover, as a general statement, the 3D scattering intensity directivity patterns show spatial azimuthal symmetry with respect to the center of the sphere. This is not the case when the sphere is shifted off the axis of the Bessel beams. To illustrate this case, the beam is shifted off-axially in both the x and y directions, such that the offset in non-dimensional units is (kx0 , ky0 ) = (12.2; 12.2). The results are plotted in Figs. 6A, 6C for the zeroth-order Bessel beam, and Figs. 6B, 6D for the first-order Bessel beam, with α0 = 30° and ka = 15 and 30, respectively. It is obvious that the 3D scattering directivity patterns for the off-axial scattering of the normalized intensity in the far-field show significant differences from the on-axis case displayed in Figs. 5A–5D. As observed in the panels of Figs. 6A–6D, the azimuthal symmetry is broken, with the generation of more scattering lobes when ka increases [i.e., the panels of Figs. 6C, 6D].

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Fig. 5C. The same as in Fig. 5A but for a zeroth-order Bessel beam with α0 = 30° and ka = 30.

Fig. 5D. The same as in Fig. 5A but for a first-order Bessel beam with α0 = 30° and ka = 30.

The effect of increasing ka is investigated by computing the normalized scattering intensity in the pol far-field I∞,sc (θ , π /2) for a homogeneous dielectric sphere having a refractive index m1 = 1.4, for ka = 500, φ = π /2 (corresponding to the direction in the horizontal plane) and values of the halfcone and polar angles, varying in the ranges 0° ≤ α0 ≤ 40° and 0° ≤ θ ≤ 180°, respectively. The plots are displayed in the logarithmic scale, and the color bar minimum and maximum bounds have been set to vary between –5 and 15, respectively, for enhanced visualization. The panels in Figs. 7A, 7B, display the computational results for a zeroth- and a first order Bessel beam, respectively, with different polarizations and centered on the liquid sphere. There appear distinctive features especially for the y- and radially-polarized (rp) beam results for which a reduction in the normalized scattering intensity is observed, starting about θ ∼ 146.75° for α0 = 0°, and having the form of a V-shape as α0 increases. The same observation is noticed in the panels of Figs. 7C, 7D, for a zeroth- and a first order Bessel beam, respectively, shifted off-axially by (kx0 , ky0 ) = (12.2; 12.2). Moreover, a large linear enhancement in the normalized scattering intensity starting at (θ , α0 ) = (0°, 0°) and ending at (θ , α0 ) = (40°, 40°) is present in all the plots. To further investigate the physical conditions for this enhancement, the resonance scattering method is used, in which the standard Mie scattering coefficients for a dielectric sphere an and bn given previously by Eqs. (44), (45) are first

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Fig. 6A. The plots for the normalized scattering intensity in the far-field from the homogeneous lossless dielectric sphere (with a refractive index m1 = 1.4) shifted off-axially from the center of a zeroth-order Bessel beam with α0 = 30° and ka = 15.

Fig. 6B. The same as in Fig. 6A but for a first-order Bessel beam.

rewritten in terms of Debye series [37,61–63] as, an bn

 =

1 2

 1−

R212 n

−

∞  p=1

 Tn21



p−1 R121 n



Tn12

,

(59)

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Fig. 6C. The same as in Fig. 6A but ka = 30.

Fig. 6D. The same as in Fig. 6A but for a first-order Bessel beam with ka = 30.

with Tn21 =

m1 2i m2 Dn

,

(1)′

R212 = n Tn12 =

C1 ξn

(60)

(κ2 )ξn(1) (κ1 ) − C2 ξn(1) (κ2 )ξn(1) (κ1 ) ′

Dn 2i Dn

,

,

(61) (62)

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Fig. 7A. The plots for the normalized scattering intensity in the far-field from the homogeneous lossless dielectric sphere (with a refractive index m1 = 1.4) centered on a zeroth-order Bessel beam with ka = 500, and φ = π/2, in the ranges 0° ≤ α0 ≤ 40° and 0° ≤ θ ≤ 180°.

Fig. 7B. The same as in Fig. 7A but for a first-order Bessel beam.

(2)′

C1 ξ n

(κ2 )ξn(2) (κ1 ) − C2 ξn(2) (κ2 )ξn(2) (κ1 ) ′

,

(63)

Dn = −C1 ξn(2) (κ2 )ξn(1) (κ1 ) + C2 ξn(2) (κ2 )ξn(1) (κ1 ),

(64)

R121 n

=

Dn ′

′

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Fig. 7C. The same as in Fig. 7A but for a zeroth-order Bessel beam shifted off axially with respect to the center of the sphere.

Fig. 7D. The same as in Fig. 7A but for a first-order Bessel beam shifted off-axially with respect to the center of the sphere.

κj = mj (ka) , j = 1, 2  for TE 1, C1 = m1  , for TM , m2

(65) C2 =

  m1 , m2  1,

for TE for TM

(66)

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Fig. 8. The plots for the normalized scattering intensity in the far-field and its resonance counterpart for a zeroth-order Bessel beam polarized along the x-direction with a half-cone angle α0 = 10° and φ = π/2. The enhancement in the plot x for I∞, sc (θ, π/2) is clearly emphasized at θ ∼ α0 = 10°.

where the prime indicates the derivative of the function with respect to its argument and m2 is (2) the index of refraction of the medium surrounding the sphere. The function ξn (·) is the spherical Riccati–Hankel function of the second kind. Next, the Mie scattering coefficients for a perfectly conducting sphere are also written in terms of Debye series as, acn



bcn

=

1 2

1 − Rcn,212 ,



(67)

where,

Rnc ,212

 (1)′  − ξn ′ (κ2 )   (2) ξn (κ2 ) = (1)   (κ2 ) ξ  − n (2) ξn (κ2 )

for TE (68) for TM.

A detailed explanation of each of the coefficients contributing to Eq. (59) can be found in [64], which are summarized here. The first term 1/2 [1] describes the diffraction  of the incident Bessel [or alternatively 1/2 −Rcn,212 for the perfectly beam around the sphere. The second term 1/2 −R212 n conducting sphere given in Eq. (67)] describes outgoing spherical waves reflected from surface of the sphere. The third term in Eq. (59) is a series which describes the contribution of all other modes of refraction. Each term in the series represents the contribution of that mode which has undergone p − 1 internal reflections, and then emerged from the sphere. Subsequently, the resonance Mie scattering coefficients, defined previously in Eq. (47), can be expressed as, ares n bres n

 =

1 2

 Rcn,212

−

R212 n

−

∞ 

 Tn21



p−1 R121 n



Tn12

.

(69)

p=1

Eq. (69) shows a very important result, demonstrating the quantitative description of the resonance scattering using Debye series and its success in separating the diffraction term from the reflection and internal reflections terms. This result was not achievable using the standard scattering theory [64]. Finally, to further illustrate the analysis, a numerical example is considered by plotting the normalized scattering intensity in the far-field with and without background subtraction, for a zeroth-order Bessel beam polarized along the x-direction with a half-cone angle α0 = 10° and φ = π /2. Eqs. (46), x res,x (49), (59) and (69) are used to plot I∞, sc (θ , π /2) and I∞,sc (θ , π /2) in the bandwidth 5° ≤ θ ≤ 15°. The result is displayed in Fig. 8 which clearly shows the enhancement in the normalized scattering

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intensity for θ ∼ α0 = 10°. When evaluating the resonance (normalized) intensity [using the coefficients given by Eq. (69)], this enhancement is removed. Since Eq. (69) clearly shows the lack of the contribution of the diffraction term (described by 1/2 [1]), but the existence of reflected waves from both the surface and the internal core of the sphere, it is concluded that the scattering enhancement, starting at (θ , α0 ) = (0°, 0°) and ending at (θ , α0 ) = (40°, 40°), which has been observed previously in Figs. 7A–7D, is due to diffracted surface waves circumnavigating the surface of the sphere and scattered in the forward direction of the individual plane-wave components forming the Bessel beam. Similar results (not shown here) for other beam polarizations and beam order showed analogous effects. Another observation in the plots of Figs. 7A–7D reveals the presence of V-shape enhancements, starting around θ ∼ 146.75°. These enhancements are attributed to rainbow formation in Bessel beams [37]. The rainbow can be formed when an incident beam of light is displaced by reflection and dispersed by refraction in a liquid dielectric sphere [65] in the direction of the backward hemisphere [66]. The rainbow features can be quantitatively described using the ray (geometrical) theory and its association with partial waves using Debye series [61–63]. For example, the contribution of high-order rainbows (corresponding to multiple internal reflections taking place inside a liquid sphere) to the scattering has been developed for a Gaussian beam [34]. In the present analysis, one notices the appearance of two primary rainbows [37] for all polarizations and beam order, symmetrical with respect to the rainbow starting at α0 = 0°, although less enhanced for the y- and radially-polarized beams. Note that the deviation angle for the primary rainbow of a plane wave (i.e., α0 = 0° for the zeroth-order Bessel beam) can be predicted from the equation [65,67], D1 = π + 2αinc − 4 sin−1 (sin αinc /m1 ) ,

(70)

where αinc is the angle of incidence. Using Descartes–Snell law, the  angle of incidence can be calculated 

by taking (dD1 /dαinc ) = 0, so that αinc = cos−1



m21 − 1 /3 . For the homogeneous dielectric



sphere having an index of refraction m1 = 1.4 considered here, the deviation angle is found to be D1 = 146.75°, which correlates exactly with the value shown in the figures. Eq. (70) can be further extended to predict the deviation angles for the two primary rainbows resulting from the plane-wave components forming the Bessel beams with a half-cone angle α0 . It is expressed as, −1 D∓ (sin αinc /m1 ) ∓ α0 , 1 = π + 2αinc − 4 sin

(71)

where D1 and D1 denote the deviation angles of the two primary rainbows for which θ decreases or increases, respectively, as α0 > 0. Eq. (71) may be also generalized to predict the deviation angles of two high-order (kth-order) rainbows in Bessel beams such that, −

+

−1 D∓ (sin αinc /m1 ) ∓ α0 , k = kπ + 2αinc − 2 (k + 1) sin

(72)

which reduces to Eq. (1.6) of [65] for the case of plane waves (i.e., α0 = 0). In the present analysis, the half-cone angle chosen in the computations of Figs. 7A–7D has been limited to α0 < 40°, which basically corresponds to the paraxial limit of the Bessel beam solutions derived in Section 2.1. Beyond that limit, the ILA may not be used to evaluate the BSCs since the z-components of the electric and magnetic fields become significant. Note that the contribution of the z-components of the EM field has not been taken into account here in the ILA description of the BSCs [see the expressions of the localized fields in the paragraph before Eq. (33)]. Above this limit (i.e., α0 > 40°), the linearity of the scattering enhancement (resulting from the diffracted surface waves) breaks down (see Fig. 6 in [37]), which is not entirely physical. A recent investigation on the multiple scattering of light by a pair of uniaxial anisotropic spheres (using a Gaussian numerical integration to calculate the BSCs) [68] observed a linear behavior for half-cone angle values 0° ≤ α0 < 90°, nevertheless, erroneously attributed the linearity of the scattering enhancement to ‘‘the effect of two spheres’’. It should be emphasized here that the discrepancy between the results presented in [37] and those of [68] for α0 > 40° results from the approximation used in the calculation of the BSCs (i.e., the ILA), but not the presence of two spheres. This has also been recently and independently confirmed

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in [69]. As explained previously, the linear enhancement in the scattering is induced by the diffracted surface waves circumnavigating the surface of the sphere, and scattered in the forward direction of the individual plane-wave components forming the Bessel beam having a half-cone angle α0 . 4. Conclusion In summary, light scattering from a homogeneous dielectric sphere in Bessel beams has been investigated using a modified version of the GLMT which considers resonance effects. Adequate subtraction of a non-resonant background (corresponding to a perfectly conducting sphere of the same size) from the standard Mie scattering coefficients and computations of the resonance extinction efficiencies for linear, circular, radial, azimuthal and unpolarized Bessel (vortex) beams, leads to the isolation of ‘‘pure’’ resonance peaks. Furthermore, the analysis based on the resonance method allows the quantitative description of the scattering using Debye series by separating the diffraction from the reflection and internal reflections terms of the sphere. In addition, the analysis is extended to include rainbow formation in Bessel beams, and a generalized formulation for the deviation angle of the two primary rainbows is obtained, with further extension to any high-order rainbows. Acknowledgment R.X. Li, L.X. Guo, and C.Y. Ding acknowledge the support from the National Science Foundation for Distinguished Young Scholars of China (Grant No. 61225002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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