General description of circularly symmetric Bessel beams of arbitrary order

Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Journal of Quantitative Spectroscopy & Radiative Transfer

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General description of circularly symmetric Bessel beams of arbitrary order

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Jia Jie Wang a,b,n, Thomas Wriedt b, James A. Lock c, Lutz Mädler b

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a

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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 Department of Physics, Cleveland State University, Cleveland, OH 44115, USA b

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a r t i c l e i n f o

abstract

Article history: Received 24 May 2016 Received in revised form 13 July 2016 Accepted 13 July 2016

A general description of circularly symmetric Bessel beams of arbitrary order is derived in this paper. This is achieved by analyzing the relationship between different descriptions of polarized Bessel beams obtained using different approaches. It is shown that a class of circularly symmetric Davis Bessel beams derived using the Hertz vector potentials possesses the same general functional dependence as the aplanatic Bessel beams generated using the angular spectrum representation (ASR). This result bridges the gap between different descriptions of Bessel beams and leads to a general description of circularly symmetric Bessel beams, such that the Davis Bessel beams and the aplanatic Bessel beams are merely the two simplest cases of an infinite number of possible circularly symmetric Bessel beams. Additionally, magnitude profiles of the electric and magnetic fields, the energy density and the Poynting vector are displayed for Bessel beams in both paraxial and nonparaxial cases. The results presented in this paper provide a fresh perspective on the description of Bessel beams and cast some insights into the light scattering and lightmatter interactions problems in practice. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Bessel beam Hertz vector potential Angular spectrum representation Light scattering

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1. Introduction

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Along with a wide application of various laser-based optical instruments, such as Phase Doppler Anemometry (PDA), Laser Doppler Velocimetry (LDV), Optical Tweezers and many others, the investigation of interactions between shaped laser beams and small particles becomes a very hot topic in recent years, which attracts attention of researchers from lots of areas [1,2]. In the analysis of various shaped beams, there has been an increasing interest in Bessel beams which were introduced by Durnin and co-workers [3,4]

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

almost three decades ago. Although ideal Bessel beams cannot be generated in reality, high quality quasi-Bessel beams can be generated using an axicon lens [5,6], spatial light modulator (SLM) [7,8], or a combination of an axicon and a spatial light modulator [9]. The geometry of a quasi-Bessel beam generated by an axicon is shown in Fig. 1. Due to the special properties of Bessel beams, including propagation invariance [10], self-reconstruction, long focal depth of field [11,12] as well as the transfer of orbital angular momentum and spin angular momentum to matter [13], prospective applications of Bessel beams can be found in various fields, such as optical communication, accurate optical measurement, optical manipulation of small particles, and imaging [14,15]. The description of shaped beams is a fundamental issue. It plays a key role in the analysis of beam properties,

http://dx.doi.org/10.1016/j.jqsrt.2016.07.011 0022-4073/& 2016 Elsevier Ltd. All rights reserved.

61 Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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Fig. 1. Geometry of a quasi-Bessel beam generated using an axicon lens. Half-cone angle of the Bessel beam is α0 . A Cartesian coordinate system ðX; Y; ZÞ and a corresponding cylindrical coordinate system ðρ; ϕ; zÞ are used.

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beam propagation as well as light-matter interactions. As an exact solution of the scalar wave equation, a basic description of Bessel beams in scalar version [3,4] was applied when the Bessel beam was introduced. For an onaxis Bessel beam propagating along the z axis, the general expression for the scalar field is described by ψðρ; ϕ; z; tÞ ¼ ψ 0 J n ðkt ρÞeinϕ e  iðkz z  ωtÞ , where ψ 0 is the amplitude of the field, and J n ð U Þ is the n-order Bessel pffiffiffiffiffiffiffiffiffiffiffiffiffiffi function of the first kind. The parameters ρ ¼ x2 þ y2 and ϕ ¼ tan  1 ðy=xÞ are the radial distance and the azimuthal angle in the transverse plane ðx; yÞ, respectively. The transverse and longitudinal wave numbers are kt ¼ k sin α0 and kz ¼ k cos α0 , respectively. 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. If α0 ¼ 0, the scalar Bessel beam reduces to a scalar plane wave. The time-dependent part of the wave expðiωtÞ is used and omitted throughout in this paper, with ω being the angular frequency. So far, there are a number of studies based on the scalar field description [16–19], which gives satisfactory results under the paraxial conditions, e.g the spot size of the beam is much larger than the wavelength. A vectorial treatment is required for an adequate description of polarized electromagnetic wave radiation and scattering, especially in nonparaxial cases where tightly focused Bessel beams are used, e.g. in optical tweezers where small particles are manipulated by a tightly focused laser beam [11,13,20]. The intensity profile of a scalar Bessel beam is circularly symmetric, while the intensity profile of a polarized Bessel beam can be circularly symmetric or asymmetric [21,22]. The Bessel beams with a circularly symmetric distribution of energy density have been called circularly symmetric Bessel beams, whose Poynting vector component along the propagation direction is also circular symmetric. Although ideal Bessel beams can hardly be generated in reality, it is common practice to start with the simplest theoretical assumption of idealized fields, which can cast insights into practical analysis where quasi-Bessel beams are applied. Several vectorial approaches have been proposed to describe ideal Bessel beams, with exact vectorial solutions to the Maxwell's equations. Bouchal and Olivík

[23] derived expressions for polarized Bessel beams of arbitrary order as the solution to the vector Helmholtz equation, in which radial, azimuthal, circular and linear polarizations were analyzed. Recently, facilitated by the application of the Hertz vector potential [24], the Bessel beams of transverse magnetic (TM) and transverse electric (TE) mode [25,26] and the linearly and circularly polarized Bessel beams [27] were derived in a rather simple way. This is due to the fact that the potentials are more fundamental quantities than the electric and magnetic fields. Once the potentials are known, the fields can be obtained by differentiation. In this procedure the derivation of the fields is implemented in the Lorenz condition when linearly polarized vector potentials are used, which is similar to the procedure used by Davis [28] for the development of a Gaussian beam model. Thus the Bessel beam derived using the vector potential has been called a Davis Bessel beam [29] to distinguish it from the Bessel beam obtained using the angular spectrum representation (ASR), which is commonly called an aplanatic Bessel beam since it was originally proposed for an aplanatic optical system [30]. The ASR method was introduced by Cizmar et al. [31] to describe a focused zero-order aplanatic Bessel beam generated by an axicon lens. Later it was extended to the description of Bessel beams of higher-order by Chen et al. [32,33]. More detailed expressions of higher-order aplanatic Bessel beams were presented by Mitri et al. [34] in a study of resonance scattering of a dielectric sphere and used recently by Yang and Li [35] to calculate the optical force exerted on a Rayleigh particle. Although various descriptions of polarized Bessel beams derived using different approaches are available in the literature, different approaches give seemingly different answers for the fields. This situation casts confusion and sometimes leads to a misuse of Bessel beam expressions. Thus a clear picture of the connection between different descriptions of Bessel beams is necessary for easier applications in practice as well as providing some insights into the nature of an ideal Bessel beam. This was recently done for a zero-order Bessel beam by Lock [29]. The aplanatic Bessel beam of zero-order generated with the ASR was found to have a same functional dependence

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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as the Davis zero-order Bessel beam, which led to a general mathematical description of zero-order Bessel beams. In this paper, to further simplify the description of Bessel beams, a vector analysis of polarized Bessel beams of arbitrary order is implemented using the Hertz vector potentials and the ASR. In order to achieve a fundamental generalization of different descriptions of polarized Bessel beams, particular attention is paid to the relationship between different descriptions of Bessel beams derived using different methods. This paper is organized as follows. Using the transverse Hertz vector potentials, a class of Davis type circularly symmetric Bessel beams of arbitrary order is derived in Section 2. The fields of an aplanatic circularly symmetric Bessel beam generated using the ASR are presented in Section 3. A general description of circularly symmetric Bessel beams is proposed in Section 4, which makes the two versions of the Bessel beams presented in Sections 2 and 3 merely the two simplest cases of infinite number of possible n-order circularly symmetric Bessel beams. Conclusions are given in Section 5.

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2. Davis Bessel beams using Hertz vector potentials The derivation of polarized Bessel beams using Hertz vector potentials is very efficient and simple. Actually, the Hertz vector potentials have some advantages in the solution of radiation and propagation problems [24] since they are more fundamental than the electric and magnetic fields. The generation of beams using the Hertz vector potentials is applicable in the optical range as well as in the electromagnetic range, which covers the circularly symmetric Bessel beams generated using axicon [5,6] or using the spatial light modulator (SLM) [7,8], the Bessel beam of TE and TM modes generated using wave guides, and those asymmetric Bessel beams [21,22] which can be generated using diffractive optical elements (DOE) or SLM. In this section, based on the Hertz vector potentials, a vector analysis of polarized Bessel beams of arbitrary order is revisited firstly for the linearly and circularly polarized Bessel beams in Cartesian coordinates. Then this procedure is extended to the derivation of a class of Davis type circularly symmetric Bessel beams of arbitrary order. Considering an electromagnetic wave propagating in an isotropic, dielectric, non-magnetic and linear medium, Maxwell's equations for the electric field E and the magnetic field B can be written as ∇  E ¼ iωB;

∇  B ¼ iωεμE

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∇ U εE ¼ 0;

∇ UB ¼ 0

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where ε and μ are the permittivity and permeability of the medium, respectively. Using the Lorenz gauge condition, and introducing the Hertz vector potential [36], the vector Helmholtz equation is expressed as

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2

∇2 Π þ k Π ¼ 0

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E-type waves 2

Ee ¼ ðk þ ∇2 ÞΠe ;

Be ¼ iωμε∇  Πe

ð3Þ

ð1Þ

ð2Þ

and its two types of independent solutions, the electric potential Πe and the magnetic potential Πm , resulting in two independent sets of waves

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H-type waves 2

Bm ¼ ðk þ ∇2 ÞΠm

Em ¼  iω∇  Πm ;

ð4Þ

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Considering the problem in a right-hand Cartesian coordinate system, the vector Helmholtz wave equation simplifies to

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∂2 Π ∂2 Π ∂2 Π 2 þ þ þk Π ¼ 0 ∂x2 ∂y2 ∂z2

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ð5Þ

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2.1. Linearly x-polarized Bessel beam The linearly x-polarized Bessel beam of arbitrary order can be generated using the vector potential polarized along the positive y axis if Π takes the form Πm ¼ Π m ey ¼ J n ðkt ρÞð  iÞn einϕ e  ikz z ey

ð6Þ

Substituting Eq. (6) into Eq. (5) and after some straightforward algebra, we obtain the expressions for the electric and magnetic fields as

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n inϕ  ikz z EðxÞ e x ¼ E 0 kkz J n ðkt ρÞð iÞ e

EðxÞ y ¼0 EðxÞ z ¼ kE0 BðxÞ x ¼ B0

  ix ix þy kt J n þ 1 ðkt ρÞ  2 nJ n ðkt ρÞ ð  iÞn einφ e  ikz z ρ ρ

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ð7Þ

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 2 ix iy2 þ 2xy 2 ðn nÞJ n ðkt ρÞ ρ4

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 iny2 inx2 þ2xy xy 2 k J ðk ρÞ  k J ðk ρÞ þ t t t nþ1 ρ3 ρ2 t n

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n inϕ  ikz z

ð  iÞ e e # (" 2 2 y2  x2 þ 2ixy 2 k x2 þkz y2 ðxÞ J n ðkt ρÞ By ¼ B0 ðn  nÞ þ ρ4 ρ2  y2  x2  2inxy kt J n þ 1 ðkt ρÞ ð  iÞn einφ e  ikz z þ ρ3   x  iy y k BðxÞ nJ ðk ρÞ þ i J ðk ρÞ ð iÞn einφ e  ikz z t t t n n þ 1 z ¼ kz B0 ρ ρ2

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where the superscript “ðxÞ” indicates x-polarized, the amplitude of the electric field is E0 , and the amplitude of the magnetic field is B0 ¼ ωk E0 .  The time-averaged energy density hwi ¼ εE UE þ  B U B =μ0 =16π can be obtained (" 2 2 4 4 1 ðk þ k Þn2 þ k x2 þ kz y2 2 2 2 ϵE0 k kz þ z ⟨w⟩ ¼ 2 16π ρ # 2 2 2 2 ðn  nÞ þ ðkz y  k x2 Þ 2 þ 2ðn2  nÞ J n ðkt ρÞ ρ4

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ðn2 þ 1Þ þ k x2 þ kz y2 2 2 kt J n þ 1 ðkt ρÞ þ ρ2 2

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ðn  1Þkz y2 þ ðn þ 1Þk x2 þnðn  1Þ2 2 kt J n ðkt ρÞJ n þ 1 ðkt ρÞ ρ3



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ð9Þ

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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The time-averaged Poynting vector power density hSi is given by 0 1 0 1 Ey U H z  Ez UH y hSx i   B B S C 1 B E U H  E U H  C x ð10Þ @ y A ¼ Re@ z x z C A 2 Ex U Hy  Ey UH x hSz i with explicit expressions ( 2 2 1 ðn2 nÞ þ k x2 þ kz y2 yknJ 2n ðkt ρÞ ⟨Sx ⟩ ¼ 2 ρ4 y2 þ x2  4nx2 nykkt J n ðkt ρÞJ n þ 1 ðkt ρÞ ρ5 ) 2 2nx2 ykt 2 þ kJ ðk ρÞ nþ1 t ρ4 þ

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⟨Sy ⟩ ¼

( 2 1 y2 kt n2 þ n 2 x xknJ 2n ðkt ρÞ  kkz 2 nJ 2n ðkt ρÞ 2 ρ4 ρ þ

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y2  x2 2 nxkkt J 2n þ 1 ðkt ρÞ ρ4

2nx2  2ny2  x2  y2 xnkkt J n ðkt ρÞJ n þ 1 ðkt ρÞg þ ρ5 " 1 y2  x2 2 ðn  nÞJ n ðkt ρÞ ⟨Sz ⟩ ¼ kkz J n ðkt ρÞ 2 ρ4 2

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ð11Þ

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k x2 þkz y2 y2  x2 J n ðkt ρÞ þ kt J n þ 1 ðkt ρÞ þ 2 ρ ρ3

) ð12Þ

# ð13Þ

From Eqs. (9)–(13), we can see that both the timeaveraged energy density and the Poynting vector power density along the propagation direction are functions of coordinates ðx; yÞ that they are not circularly symmetric. For the purpose of demonstration, magnitude profiles of electric and magnetic fields of a linearly x-polarized

Bessel beam are displayed in Figs. 2 and 3, for a paraxial case corresponding to a half-cone angle of 10°, and a nonparaxial case corresponding to a half-cone angle of 80°, respectively. The magnitude distributions of fields are displayed in the range of ½  15λ; 15λ for the paraxial case, while a much smaller range of ½ 2:5λ; 2:5λ is used for the nonparaxial case where the beam is tightly focused. In the simulations, the distributions of H instead of B are plotted. This is because the values of H are much closer to those of E, thus it is easier to make a comparison between electric field and magnetic field. In the paraxial case shown in Fig. 2, both the components of Ex and H y have a circular symmetry, while the other field components are axial symmetric about the x and also the y axis. The magnitudes of Etotal and H total seem to be circularly symmetric, which is because the circularly symmetric components Ex and H y dominate in this case. However, in the nonparaxial case displayed in Fig. 3, only Ex possesses a circular symmetry. The magnitude distribution pattern of H y in the nonparaxial case is quite different from that in the paraxial case. The magnitudes of Etotal and H total are asymmetric in this case since the dominant components Ez and H y are now asymmetric. The magnitude profiles of energy density o w4 and Poynting vector power density o S 4 are plotted in Fig. 4 for both paraxial case (same parameters as Fig. 2) and nonparaxial case (same parameters as Fig. 3). As we can see, in the paraxial case, all the Poynting vector components are circularly symmetric. Nevertheless, this symmetry no longer exists in the nonparaxial case.

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2.2. Linearly y-polarized Bessel beam

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The linearly y-polarized Bessel beam of arbitrary order can be generated using the vector potential polarized

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Fig. 2. Magnitude profiles of the electric and magnetic fields for a linearly x-polarized Bessel beam of n ¼ 2 in a paraxial case with half-cone angle 10°. Length scales in unit of wavelength λ are used. The magnitude is plotted in arbitrary unit.

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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Fig. 3. Magnitude profiles of the electric and magnetic fields for a linearly x-polarized Bessel beam of n ¼ 2 in a nonparaxial case with half-cone angle 80°. Length scales in unit of wavelength λ are used. The magnitude is plotted in arbitrary unit.

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Fig. 4. Magnitude profiles of the transverse, longitudinal, total components of the time-averaged Poynting vector hSi, and the time-averaged energy density hwi for a linearly x-polarized Bessel beam of n ¼ 2 in a paraxial case (a)–(d) with the same parameters as Fig. 2, and a nonparaxial case (e)–(h) with the same parameters as Fig. 3, respectively. The magnitude is plotted in arbitrary unit.

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along the positive x axis if Π takes the form 51 53

Πm ¼ Π m ex ¼ J n ðkt ρÞð  iÞn einϕ e  ikz z ex

ð14Þ

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Substituting Eq. (14) into Eq. (5) and after some straightforward algebra, we obtain the expressions for the electric and magnetic fields as

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EðyÞ x ¼0

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n inϕ  ikz z EðyÞ e y ¼  E0 kkz J n ðkt ρÞð  iÞ e   iy x iy EðyÞ nJ n ðkt ρÞ  kt J n þ 1 ðkt ρÞ ð  iÞn einφ e  ikz z z ¼ kE0 2 ρ ρ

ð15Þ

BðyÞ x ¼ B0

þ

# 2 2 x2  y2  2ixy 2 k y 2 þ kz x2 ðn  nÞ þ J n ðkt ρÞ ρ2 ρ4

x2  y2 þ 2inxy kt J n þ 1 ðkt ρÞ ρ3



ð  iÞn einϕ e  ikz z

 2 ix  iy2 þ2xy 2 xy 2 BðyÞ ¼ B ðn nÞJ n ðkt ρÞ  2 kt J n ðkt ρÞ 0 y ρ4 ρ þ

 iny2 inx2 þ2xy kt J n þ 1 ðkt ρÞ ð  iÞn einφ e  ikz z 3 ρ

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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Πm ¼ Π m ey ¼ J n ðkt ρÞð  iÞn einφ e  ikz z ey

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where the superscript “ðyÞ” indicates y-polarized.

Following the derivation given in Sections 2.1and 2.2, the explicit expressions for the electric and magnetic fields of a circularly symmetric Bessel beam are obtained

2.3. Circularly polarized Bessel beam

Exð1; 0Þ

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  x  iy x nJ n ðkt ρÞ  kt J n þ 1 ðkt ρÞ ð iÞn einφ e  ikz z 2 ρ ρ ð16Þ

ð19Þ

BðyÞ z ¼  ikz B0

The circular polarization states can be constructed by a superposition of two orthogonal linear polarizations with equal magnitude and additional phase. That is to say, a right-circularly or a left-circularly polarized Bessel beam can be obtained if Πm takes the form Πm ¼ Π m ex 7 iΠ m ey ¼ J n ðkt ρÞð iÞn einϕ e  ikz z ðex 7 iey Þ

ð17Þ

where “þ” corresponds to a right-circular polarization, and “  ” corresponds to a left-circular polarization. The expressions for the electric and magnetic fields can then be obtained by substituting Eq. (17) into Eq. (5) followed by some straightforward algebra. They can also be obtained by a superposition of the expressions of the two orthogonal linearly polarized cases as follows 2 ðcircÞ 3 2 ðxÞ 3 2 ðcircÞ 3 2 ðxÞ 3 Ex Bx Ex 7 iEðyÞ Bx 7 iBðyÞ x x 6 ðcircÞ 7 6 ðxÞ 6 ðcircÞ 7 6 ðxÞ ðyÞ 7 ðyÞ 7 6 Ey 7 ¼ 6 Ey 7 iEy 7; 6 By 7 ¼ 6 By 7 iBy 7 ð18Þ 4 5 4 5 4 5 4 5 EðcircÞ z

ðyÞ EðxÞ z 7 iEz

BðcircÞ z

ðyÞ BðxÞ z 7 iBz

where the superscript “ðcircÞ” denotes the circular polarization, the “þ” corresponds to a right-circular polarization, and the “  ” corresponds to a left-circular polarization. The detailed expressions are not presented here for the sake of conciseness. Additionally, it is worth mentioning that the expressions for the electric and magnetic fields in the case of Πe ¼ Π e ex , Πe ¼ Π e ey , Πe ¼ Π e ðex þiey Þ, and Πe ¼ Π e ðe x iey Þ can be obtained from the results of cases Πm ¼ Π m ex , Πm ¼ Π m ey , Πm ¼ Π m ðex þ iey Þ, and Πm ¼ Π m ðex iey Þ, respectively, using the duality transformations E-B, B-  E. The choice of Πm yields a linearly or circularly polarized electric field E, while the choice of Πe leads to a linearly or circularly polarized magnetic field B [27]. As we can see from Eqs. (7) and (8) as well as from Fig. 3, the distributions of ow 4 and hSz i for an x-polarized Bessel beam are not circularly symmetric, since its field components E and B are not symmetric. The results are the same for a y-polarized Bessel beam (Eqs. (10) and (11)). Nevertheless, to describe a circularly symmetric Bessel beam one would expect E and B to be symmetric, which can be constructed as follows.

" 2 2 1 x2  y2  2ixy 2 k y2 þ kz x2 ¼ E0 ðn  nÞJ n ðkt ρÞ þ J n ðkt ρÞ 4 2 2 ρ ρ

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 x  y þ 2inxy kt J n þ 1 ðkt ρÞ ð iÞn einϕ e  ikz z þ k U kz J n ðkt ρÞ þ 3 2

2

ρ

ð20Þ  2 1 ix iy2 þ 2xy 2 xy 2 ðn  nÞJ n ðkt ρÞ  2 kt J n ðkt ρÞ Eyð1; 0Þ ¼ E0 2 ρ4 ρ  iny2  inx2 þ 2xy n inϕ  ikz z kt J n þ 1 ðkt ρÞ ð  iÞ e e þ ρ3

ð21Þ

ð22Þ

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87 ð23Þ

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 ð  iÞn einϕ e  ikz z

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ð25Þ

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where the superscript ð1; 0Þ which is reminiscent of x-polarization is used since the polarization of the circularly symmetric Bessel beam described in Eqs. (20)–(25) is predominantly along the x-axis. By setting n ¼ 0, the expressions in Eqs. (20)–(25) reduce to those of a zeroorder circularly symmetric Davis Bessel beam derived by Mishra [38], which was widely applied in a number of light scattering problems [39–41].  The time-averaged energy density hwi ¼ εE UE þ  B UB =μ0 =16π can be obtained

2 1 k þkz n ϵ0 E20 ⟨w⟩ ¼ 2ðk þ kz Þ2 J 2n ðkt ρÞ þðk  kz Þ2 16π 4 h i h io 2  J 2n þ 2 ðkt ρÞ þJ 2n  2 ðkt ρÞ þ 2kt J 2n þ 1 ðkt ρÞ þ J 2n  1 ðkt ρÞ

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  1 iy iy  x 0Þ Bð1; ¼ ðk þ kz ÞB0 kt J n þ 1 ðkt ρÞ  2 nJ n ðkt ρÞ ð  iÞn einϕ e  ikz z z 2 ρ ρ

2.4. Circularly symmetric Bessel beams of polarization (1,0)

ð26Þ

A circularly symmetric (as opposed to circularly polarized) Davis Bessel beam can be constructed by averaging the linearly x-polarized Davis fields and the dual fields [37,38], which are generated using the following potentials, respectively.

The time-averaged Poynting vector power density hSi is given by 0 1 0 1 Ey U H z  Ez UH y hSx i C B S  C 1 B Ez U H x Ex U H z C ð27Þ @ y A ¼ ReB A 2 @   Ex UH y  Ey UH x hSz i

Πe ¼ Π e ex ¼  J n ðkt ρÞð  iÞn einϕ e  ikz z ex

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" 2 2 1 y2  x2 þ 2ixy 2 k x2 þ kz y2 0Þ Bð1; ¼ B0 ðn  nÞJ n ðkt ρÞ þ J n ðkt ρÞ y 2 ρ2 ρ4 y2  x2  2inxy kt J n þ 1 ðkt ρÞ ρ3

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 2 1 ix iy2 þ 2xy 2 Bxð1; 0Þ ¼ B0 ðn nÞJ n ðkt ρÞ 2 ρ4

þ kkz J n ðkt ρÞ þ

71

75

  1 ix ix þ y Ezð1; 0Þ ¼ ðk þ kz ÞE0 kt J n þ 1 ðkt ρÞ  2 nJ n ðkt ρÞ ð  iÞn einϕ e  ikz z 2 ρ ρ

iny2 inx2 þ2xy kt J n þ 1 ðkt ρÞ þ ρ3  xy 2  2 kt J n ðkt ρÞ ð iÞn einϕ e  ikz z ρ

67

101 103 105 107 109 111 113 115 117 119 121 123

61 Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

J.J. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 3 5

With explicit expressions  1 2 ⟨Sx ⟩ ¼ k kt sin ϕ ðk þ kz ÞJ n ðkt ρÞ J n þ 1 ðkt ρÞ þJ n  1 ðkt ρÞ 2  þ ðk  kz Þ J n þ 1 ðkt ρÞJ n þ 2 ðkt ρÞ þ J n  1 ðkt ρÞJ n  2 ðkt ρÞ ð28Þ

7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

 1 2 ⟨Sy ⟩ ¼  k kt cos ϕ ðk þ kz ÞJ n ðkt ρÞ J n þ 1 ðkt ρÞ þ J n  1 ðkt ρÞ 2  þ ðk  kz Þ J n þ 1 ðkt ρÞJ n þ 2 ðkt ρÞ þ J n  1 ðkt ρÞJ n  2 ðkt ρÞ

7

The magnitude profiles of energy density ow 4 and Poynting vector power density o S 4 of a circularly symmetric Bessel beam of polarization ð1; 0Þ are plotted in Fig. 7 for both paraxial case (same parameters as Fig. 5) and nonparaxial case (same parameters as Fig. 6). As we can see, different from the results in Fig. 4 for a linearly polarized Bessel beam, all the distributions of Poynting vector components and energy density for a circularly symmetric Bessel beam are circularly symmetric in both paraxial case and nonparaxial case, as we expected.

ð30Þ To give a clear view of the properties of the circularly symmetric Bessel beams as well as to illustrate their different behaviors compared to that of linearly polarized Bessel beams, the magnitude distributions of circularly symmetric Bessel beams in Eqs. (20)–(25) are displayed in Figs. 5 and 6, for a paraxial case corresponding to a halfcone angle of 10°, and a nonparaxial case corresponding to a half-cone angle of 80°, respectively. Beam order n ¼ 2 is considered for demonstration. As shown in Fig. 5, for the paraxial case, the magnitude distributions of field components Ex and H y are circularly symmetric. Ey and H x have the same distribution pattern, which can be verified from the mathematical expressions. The magnitude distributions of Ez and H z are similar but shifted by Δϕ ¼ π=2. For the nonparaxial case displayed in Fig. 6, the circular symmetry of Ex and H y no longer exists. The magnitude distributions of Ex and H y are similar but shifted by Δϕ ¼ π=2, which is also the same case for the magnitude distributions of Ez and H z . While Ey and H x have the same distribution pattern.

65 67 69 71 73

ð29Þ ( ) i 1 2 ðk  kz Þ2 h 2 2 2 2 J n þ 2 ðkt ρÞ þ J n  2 ðkt ρÞ ⟨Sz ⟩ ¼ k ðk þ kz Þ J n ðkt ρÞ  2 2 4

63

2.5. Circularly symmetric Bessel beams of polarization (0,1) 75 A circularly symmetric (as opposed to circularly polarized) Bessel beam can also be constructed by averaging the linearly y-polarized Davis fields and the dual fields, which are generated using the following potentials

79

Πm ¼ Π m ex ¼ J n ðkt ρÞð iÞn einϕ e  ikz z ex

81

Πe ¼ Π e ey ¼ J n ðkt ρÞð  iÞn einφ e  ikz z ey

ð31Þ

The explicit field expressions are  2 1 ix  iy2 þ 2xy 2 ðn  nÞJ n ðkt ρÞ Exð0; 1Þ ¼ E0 2 ρ4 þ

iny2  inx2 þ 2xy xy 2 kt J n þ 1 ðkt ρÞ  2 kt J n ðkt ρÞ ρ3 ρ

83 85 

ð iÞn einϕ e  ikz z

87 ð32Þ

y2  x2  2inxy kt J n þ 1 ðkt ρÞ ρ3

89 91

" 2 2 1 y2  x2 þ 2ixy 2 k x2 þ kz y2 1Þ Eð0; ¼ E0 ðn  nÞJ n ðkt ρÞþ J n ðkt ρÞ y 2 ρ2 ρ4 þ k U kz J n ðkt ρÞ þ

77

93

 ð  iÞn einϕ e  ikz z

95 ð33Þ

  1 x  iy iy 1Þ Eð0; ¼ E0 ðk þ kz Þ nJ n ðkt ρÞ þ kt J n þ 1 ðkt ρÞ ð  iÞn einϕ e  ikz z z 2 ρ ρ2

97 ð34Þ 99

39

101

41

103

43

105

45

107

47

109

49

111

51

113

53

115

55

117

57

119 121

59 61

Fig. 5. Magnitude profiles of the electric and magnetic fields for a circularly symmetric Bessel beam of polarization ð1; 0Þ with beam order n ¼ 2 in a paraxial case with half-cone angle 10°. Length scales in unit of wavelength are used. The magnitude is plotted in arbitrary unit.

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

123

J.J. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

1

63

3

65

5

67

7

69

9

71

11

73

13

75

15

77

17

79

19

81

21

Fig. 6. Magnitude profiles of the electric and magnetic fields for a circularly symmetric Bessel beam of polarization ð1; 0Þ with beam order n ¼ 2, in a nonparaxial case with half-cone angle 80°. Length scales in unit of wavelength are used. The magnitude is plotted in arbitrary unit.

83

23

85

25

87

27

89

29

91

31

93

33

95

35

97

37

99

39

101

41

103

43

105

45 47 49

Fig. 7. Magnitude profiles of the transverse, longitudinal, total components of the time-averaged Poynting vector hSi, and the time-averaged energy density hwi for a circularly symmetric Bessel beam of polarization ð1; 0Þ for order n ¼ 2 in a paraxial cases (a)–(d) with the same parameters as Fig. 5, and a nonparaxial case (e)–(h) with the same parameters as Fig. 6, respectively. The magnitude is plotted in arbitrary unit.

" 2 2 1 x2  y2 2ixy 2 k y2 þ kz x2 1Þ Bð0; ¼  B0 ðn  nÞJ n ðkt ρÞþ J n ðkt ρÞ x 2 ρ2 ρ4

51

ρ

ð35Þ

53 55 57 59 61

where the superscript ð0; 1Þ which is reminiscent of y-polarization is used since the polarization of the circularly symmetric Bessel beams of Eqs. (32)–(37) is predominantly along the y-axis.

 x2  y2 þ 2inxy n inϕ  ikz z þ k U kz J n ðkt ρÞ þ kt J n þ 1 ðkt ρÞ ð  iÞ e e 3

 1 ix2  iy2 þ 2xy 2 xy 2 Byð0; 1Þ ¼ B0  ðn  nÞJ n ðkt ρÞ þ 2 kt J n ðkt ρÞ 2 ρ4 ρ  iny2  inx2 þ 2xy  kt J n þ 1 ðkt ρÞ ð iÞn einϕ e  ikz z ð36Þ ρ3 1Þ Bð0; z

  1 ix þ y ix ¼ B0 ðk þ kz Þ nJ n ðkt ρÞ kt J n þ 1 ðkt ρÞ ð iÞn einϕ e  ikz z 2 2 ρ ρ

ð37Þ

107 109 111 113 115

2.6. Circularly symmetric Bessel beams of polarization ð1; iÞ and ð1; iÞ A circularly symmetric Bessel beam of polarization ð1; iÞ can be obtained by a superposition of the fields derived using the vector potentials Πe ¼ Π e ð ex þ iey Þ and Πm ¼ Π m ð  iex þ ey Þ. It can also be obtained by a superposition of the circularly symmetric Bessel beams of

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

117 119 121 123

J.J. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 3 5

polarization ð1; 0Þ and that of ð0; 1Þ 2 ð1; iÞ 3 2 ð1; 0Þ 3 2 ð1; iÞ 3 2 ð1; 0Þ 3 Bx Ex Ex Bx þ iExð0; 1Þ þ iBxð0; 1Þ 6 ð1; iÞ 7 6 ð1; 0Þ 7 6 7 6 7 6 Ey 7 ¼ 6 Ey 6 ð1; iÞ 7 6 ð1; 0Þ þ iByð0; 1Þ 7 þ iEyð0; 1Þ 7 4 5 4 5; 4 B y 5 ¼ 4 B y 5 Ezð1; 0Þ þ iEzð0; 1Þ Bzð1; 0Þ þ iBzð0; 1Þ Ezð1; iÞ Bzð1; iÞ ð38Þ

7 The explicit expressions are 9 11

3 ðk þ kz Þ2 iyÞ2 2 J n ðkt ρÞ þ ðx þ kt J n þ 2 ðkt ρÞ 2 2ρ2 7 6 7 6 ð1; iÞ 7 1 6 2 2 2 6 Ey 7 ¼ E0 ð  iÞn einϕ e  ikz z 6 iðk þ kz Þ J n ðkt ρÞ  iðx þ iyÞ kt J n þ 2 ðkt ρÞ 7 2 5 2 7 4 6 2ρ2 5 4 ð1; iÞ ix  y Ez ρ kt ðk þ kz ÞJ n þ 1 ðkt ρÞ 2

iÞ Eð1; x

2

3

13 2

15 17 19 21 23 25 27 29 31 33

39 41 43 45 47

2

3

6 6 ð1; iÞ 7 1 6 6 By 7 ¼ B0 ð  iÞn einϕ e  ikz z 6 5 2 4 6 4 ð1; iÞ Bz

2

2

iyÞ  iðk þ2kz Þ J n ðσÞ  iðx þ kt J n þ 2 ðkt ρÞ 2ρ2 2

ðk þ kz Þ2 iyÞ2 2 J n ðσÞ  ðx þ kt J n þ 2 ðkt ρÞ 2 2ρ2 x þ iy ρ kt ðk þ kz ÞJ n þ 1 ðkt ρÞ

3 7 7 7 7 5

ð40Þ

where the superscript ð1; iÞ which is reminiscent of leftcircular polarization is used. Similarly, a general description of a circularly symmetric Bessel beam of polarization ð1; iÞ can be obtained by a superposition of the fields derived using the vector potentials Πe ¼  Π e ðex þiey Þ and Πm ¼ Π m ðiex þ ey Þ. It can also be obtained by a subtraction of the circularly symmetric Bessel beams of polarization ð1; 0Þ and that of ð0; 1Þ 2

Eð1; x 6 ð1; 6 Ey 4 Eð1; z

 iÞ 3

2

ð0; 1Þ 3

2

Bð1; x 6 ð1; 6 By 4 Bð1; z

0Þ Eð1;  iEx x 7 6 ð1; 0Þ 1Þ 7 7 ¼ 6 Ey 7;  iEð0; y 5 4 5 ð1; 0Þ ð0;  iÞ Ez  iEz 1Þ  iÞ

 iÞ 3

ð0; 1Þ 3

2

0Þ Bð1;  iBx x 7 6 ð1; 0Þ 1Þ 7 7 ¼ 6 By 7  iBð0; y 5 4 5 ð1; 0Þ ð0;  iÞ Bz  iBz 1Þ  iÞ

ð41Þ

The explicit expressions are 2

3 ðk þ kz Þ2 iyÞ2 2 J n ðσÞ þ ðx  kt J n  2 ðσÞ 2 2ρ2 7 6 7 1 7 6 2 2 2 7 ¼ E0 ð  iÞn einϕ e  ikz z 6  iðk þ kz Þ J n ðσÞþ iðx  iyÞ kt J n  2 ðσÞ 7 2 5 2 7 6 2ρ2 5 4  iÞ  ix ρ iykt ðk þ kz ÞJ n  1 ðkt ρÞ

 iÞ Eð1; x 6 ð1;  iÞ 6 Ey 4

Eð1; z

35 37

iÞ Bð1; x

ð39Þ

2

Bð1; x 6 ð1; 6 By 4 Bð1; z

 iÞ

3

3

2

2

6 7 1 6 7 ¼ B0 ð  iÞn einϕ e  ikz z 6 5 2 6 4  iÞ  iÞ

2

2

iyÞ iðk þ2kz Þ J n ðσÞþ iðx  kt J n  2 ðσÞ 2ρ2 2

ðk þ kz Þ2 iyÞ2 2 J n ðσÞ ðx  kt J n  2 ðσÞ 2 2ρ2 x  iy ρ kt ðk þ kz ÞJ n  1 ðσÞ

ð42Þ

3 7 7 7 7 5

ð43Þ

where the superscript ð1; iÞ which is reminiscent of right-circular polarization is used. Furthermore, if we considered the vector Helmholtz equation in Eq. (2) in a cylindrical coordinate system, a class of circularly symmetric Bessel beams of transverse magnetic (TM) and transverse electric (TE) mode can also be generated using the vector potential polarized along z axis, which will be discussed in a separate paper.

49 51

3. Aplanatic Bessel beams described by the angular spectrum representation

53 55 57 59 61

The angular spectrum representation (ASR) [30,42,43] is a very useful approach to the description of laser beam propagation and focusing, in which the electromagnetic field of an arbitrary beam is described as a superposition of plane and evanescent waves propagating in different directions with different weights (amplitudes). This method was introduced by Cizmar et al. [31] to describe a focused zero-order Bessel beam generated by an axicon

9

lens, where an aplanatic beam model was applied to the description of the fields in the vicinity of the focal plane formed by the aplanatic axicon lens. The aplanatic model takes into account the position-dependent change in polarization that occurs upon refraction as the beam passes through the lens. This is a very realistic and practical model for the description of a focused laser beam, it has been applied in a number of light scattering investigations, and has also been extended to the case of high-order Bessel beams [33,38,44,45]. In the balance of this section the aplanatic procedure is briefly recounted. It is then applied to generate n-order Bessel beams with the same polarization states as those presented in Section 2. Assuming the center of a spherical coordinate system Oxyz is located at the focal center of a lens, the field components of a focused shaped beam generated by an aplanatic lens can be expressed as [30] Z Z ikf e  ikf αmax 2π EðrÞ ¼ Epw e  ik U r sin αdαdβ 2π α¼0 β¼0 BðrÞ ¼

1 ∇  EðrÞ iω

67 69 71 73 75 77 79 81 83 85

ð45Þ

where Pðα; βÞ describes the incident beam's profile, and Q is the complex polarization vector which determines the polarization of the resulting beam leaving the lens. To illustrate the intrinsic features of the complex polarization vector Q , an example is presented. Assuming that the electric field of an incident beam is linearly polarized along the x axis, one has Einc ¼ Einc ex

65

ð44Þ

where the wave vector is k ¼ ðk sin α cos β; k sin α sin β; kcosαÞ, in which α and β are the polar and azimuthal angles of the wave vector, respectively. The subscript “pw” in Epw indicates plane wave. The vector position is r, and the α integral is performed over the finite range ½0; αmax  due to the finite size of the optical lens. The focal length of the lens is f . The quantity Epw is known as the angular spectrum function [33] Epw ¼ Q Pðα; βÞ

63

ð46Þ

87 89 91 93 95 97 99 101 103

where Einc is the amplitude of the electric field, the superscript “inc” indicates the incident beam. In the cylindrical coordinates ðρ; ϕ; zÞ, Eq. (46) is represented as

105

Einc ¼ Einc ðcosϕeρ  sin ϕeϕ Þ

ð47Þ

109

Considering a focusing effect caused by an aplanatic lens, the unit vector eϕ remains unaffected, while the unit vector eρ is mapped into eθ in spherical coordinates ðr; θ; ϕÞ [30]. Thus, the refracted electric field of the Bessel beam can be expressed in terms of spherical coordinates as

111

Epw ¼ Epw ðα; βÞð cos βeθ  sin βeϕ Þ

107

113 115

ð48Þ

where Epw ðα; βÞ is the amplitude of the component of the plane wave spectrum with the wave vector direction defined by polar angles α and azimuthal angle β. Considering the relationship between spherical coordinates and Cartesian coordinates, the electric field of the resulting beam can be expressed in a Cartesian coordinate system as

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10

1 3 5 7 9 11 13 15

obtain the Bessel beam propagating in the z direction as

the column vector

2

2

3 ð cos α cos 2 β þ sin 2 βÞ 6 7 Epw ¼ Epw ðα; βÞ4 ð1  cos αÞ sin β cos β 5  sin α cos β

ð49Þ

where the first, second, and third entry are the components in the ex , ey , and ez directions in a Cartesian coordinate system. For an incident beam with another polarization, the polarization vector of the resulting beam can be obtained similarly. The complex polarization vector Q of the resulting beam is summarized as [33] 2

px ð cos α cos 2 β þ sin 2 βÞ  py ð1 cos αÞ sin β cos β

7 6 2 2 7 Q ¼6 4  px ð1  cos αÞ sin β cos β þ py ð cos α sin β þ cos βÞ 5

ð50Þ

 px sin α cos β  py sin α sin β

17 19 21 23 25 27

with ðpx ; py Þ being the complex polarization parameters, which determine the polarization character of the resulting beam. Specifically, the values of ðpx ; py Þ with ð1; 0Þ, ð0; 1Þ, ð1; iÞ, ð1;  iÞ, ð cos β; sin βÞ and ð  sin β; cos βÞ correspond to the x, y, left circular, right circular, radial and azimuthal polarization of the incident beam, respectively. Concerning a Bessel beam of n-order, the electric field can be represented as a sum of plane waves all having the same tilt angle α0 with the z axis. The angular spectrum function is expressed as

29

 α0 Þ Epw ¼ Q Pðα; βÞ ¼ Q Epw0 ðα0 ; βÞeinβ δðα sin α0

31

where Epw0 ðα0 ; βÞ is the amplitude of the electric field of the plane waves propagating over a cone with half-cone angle α0 and azimuthal angle β. The Dirac delta function is δð UÞ, which reflects the fact that the angular spectrum of a Bessel beam contains only a single tilt angle of the component plane waves. Substituting Eq. (51) into Eq. (44), the electric field of the Bessel beam can be expressed as one-dimensional integral

33 35 37 39 41

EðrÞ ¼ ¼

43 45

EðrÞ ¼

51 53

Z

2π

β¼0

ð51Þ

Epw0 ðα0 ; βÞQ jα ¼ α0 einβ e  ikr dβ

ð52Þ

ηf 2π

Z

2π

β¼0

Epw0 ðα0 ; βÞQ jα ¼ α0 einβ e  ikz

cos α0

e  ikρ

sin α0 cos ðϕ  βÞ

dβ

ð53Þ

If the incident beam is rotationally symmetric (no dependence on the azimuthal angle β), then the integration in Eq. (53) can be performed analytically (please consult Appendix A for details of the derivation).

55 3.1. Aplanatic Bessel beams with ðpx ; py Þ ¼ ð1; 0Þ 57 59

2 6 6 6 6 6 4

65 67

h i3 ð  iÞn J n ðσÞeinϕ  12P ? ð  iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ þ ð iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 7 h i 7 7 1  P ? 2i ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 7 7 h i 5 n þ 1 iðn þ 1Þϕ n  1 iðn  1Þϕ  P J ð  iÞ e J n þ 1 ðσÞ þ ð iÞ e J n  1 ðσÞ

ð54Þ Bð1;0Þ x

63

69 71 73 75

3

6 ð1;0Þ 7 6 By 7 ¼ BB0 e  ikz z 4 5 Bð1;0Þ z

77

3 h i 1  P ? 2i ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ  ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 6 h i7 7 6 n n þ 2 iðn þ 2Þϕ 6 inϕ 1 e J n þ 2 ðσÞþ ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 7 6 ð iÞ J n ðσÞe þ 2P ? ð  iÞ 7 7 6 h i 5 4 iP J ð  iÞn þ 1 eiðn þ 1Þϕ J n þ 1 ðσÞ  ð iÞn  1 eiðn  1Þϕ J n  1 ðσÞ 2

ð55Þ where EB0 ¼ 12ηf Epw0 ð1 þ cosα0 Þ, BB0 ¼ ωk EB0 , σ ¼ kρ sin α0 , cos α0 sin α0 P ? ¼ 11  þ cos α0 , and P J ¼ 1 þ cos α0 . The expressions in Eqs. (54) and (55) are the same as those given by Mitri et al. [34], except that ð iÞn is included instead of in due to the different time-dependent harmonic factor. It is worth remarking that, to avoid confusion, the resulting Bessel beam described in Eqs. (54) and (55) is not linearly polarized, although it is generated by a linearly polarized incident plane wave [34]. The superscript ð1; 0Þ which is reminiscent of x-polarized is used since the polarization of the Bessel beam described in Eqs. (54) and (55) is predominantly polarized along x axis. For n ¼ 0, the expressions in Eqs. (54) and (55) reduce to those of a zero-order aplanatic Bessel beam with polarization parameters ðpx ; py Þ ¼ ð1; 0Þ, which are the same as those given by Cizmar et al. [31], except that  i is replaced by i due to the different time-dependent harmonic factor.

79 81 83 85 87 89 91 93 95 97 99 101 103 105

where ηf ¼ ikf e  ikf . Considering the equation expð  ik   rÞ ¼ expð  ikz cos α0 Þexp  ik ρ sin α0 cos ðβ  ϕÞ, we then can rewrite Eq. (52) as

47 49

ηf 2π

Ex 6 ð1;0Þ 7 6 Ey 7 ¼ EB0 e  ikz z 4 5 Eð1;0Þ z

2

3

ð1;0Þ 3

For the special case of the polarization parameters ðpx ; py Þ ¼ ð1; 0Þ, using the procedure in Appendix A, we

3.2. Aplanatic Bessel beams with ðpx ; py Þ ¼ ð0; 1Þ

107

For the special case of the polarization parameters ðpx ; py Þ ¼ ð0; 1Þ, we can obtain a Bessel beam propagating in the z direction as 2

Eð0;1Þ x

3

6 ð0;1Þ 7 6 Ey 7 ¼ EB0 e  ikz z 4 5 Eð0;1Þ z 2

h

109 111 113 115

3

i

1  P ? 2i ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ  ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 6 h i7 7 6 n  2 iðn  2Þϕ 6 ð iÞn J ðσÞeinϕ þ 1P ð  iÞn þ 2 eiðn þ 2Þϕ J e J n  2 ðσÞ 7 6 7 n n þ 2 ðσÞþ ð  iÞ 2 ? 7 6 h i 5 4 n þ 1 iðn þ 1Þϕ n  1 iðn  1Þϕ iP J ð  iÞ e J n þ 1 ðσÞ  ð iÞ e J n  1 ðσÞ

ð56Þ

117 119 121 123

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J.J. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 3 5 7 9

2

3 Bxð0;1Þ 6 ð0;1Þ 7 6 By 7 ¼ BB0 e  ikz z 4 5 Bzð0;1Þ 2 h i3  ð iÞn J n ðσÞeinϕ þ 12P ? ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ þ ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 6 7 h i 6 7 6 7 1 P ? 2i ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 6 7 6 7 h i 4 5 n þ 1 iðn þ 1Þϕ n  1 iðn  1Þϕ P J ð  iÞ e J n þ 1 ðσÞ þ ð iÞ e J n  1 ðσÞ

ð57Þ

13

where the superscript ð0; 1Þ which is reminiscent of y-polarized is used since the polarization of the Bessel beam described in Eqs. (56) and (57) is predominantly along the y-axis.

15

3.3. Aplanatic Bessel beams with ðpx ; py Þ ¼ ð1; 7iÞ

17

For the special cases of the polarization parameters ðpx ; py Þ ¼ ð1; 7iÞ, which correspond to the left- and rightcircular polarization of the incident beam, respectively, the field components of the resulting aplanatic Bessel beams can be obtained similarly as those presented in Section 3.1. The explicit expressions can also be obtained by a superposition of the fields with ðpx ; py Þ ¼ ð1; 0Þ and ðpx ; py Þ ¼ ð0; 1Þ as follows 2 ð1;iÞ 3 2 ð1;0Þ 3 2 ð1;iÞ 3 2 ð1;0Þ 3 Bx Ex Ex þ iEð0;1Þ Bx þiBð0;1Þ x x 6 ð1;iÞ 7 6 ð1;0Þ 7 6 7 6 ð0;1Þ 7 6 Ey 7 ¼ 6 Ey þ iEð0;1Þ 7; 6 Bð1;iÞ 7 6 ð1;0Þ 7 y 4 5 4 5 4 y 5 ¼ 4 By þiBy 5 ð1;0Þ ð0;1Þ ð1;0Þ ð0;1Þ ð1;iÞ ð1;iÞ Ez þ iEz Bz þiBz Ez Bz

11

19 21 23 25 27 29

ð58Þ

31

2

33

6 ð1;  iÞ 7 6 ð1;0Þ 7 6 Ey 7 ¼ 6 Ey  iEyð0;1Þ 7; 4 5 4 5 Eð1;0Þ  iEzð0;1Þ Ezð1;  iÞ z

35 37 39 41 43 45 47 49 51 53 55 57 59

Exð1;  iÞ

3

2

Eð1;0Þ  iExð0;1Þ x

3

2

 iÞ Bð1; x

3

2

Bxð1;0Þ iBð0;1Þ x

3

6 ð1;  iÞ 7 6 ð1;0Þ 7 6 By 7 ¼ 6 By iBð0;1Þ 7 y 4 5 4 5 ð1;0Þ ð0;1Þ ð1;  iÞ Bz iBz Bz

4. General description of circularly symmetric Bessel beams

63 65

As shown in Sections 2 and 3, the polarized Bessel beams derived using different methods are presented in different mathematical forms, which are seemingly different answers for the fields. Although each description might have some advantages in dealing with certain problems, this situation casts confusion and sometimes leads to a misuse of Bessel beam expressions. A clear picture of the relationship between different descriptions of Bessel beams is necessary to assist practical analysis where quasiBessel beams are applied, as well as to provide a perspective into the nature of the ideal Bessel beams. Actually, such a relationship becomes apparent now when a comparison is made between the circularly symmetric Davis Bessel beams derived in Section 2 and the aplanatic circularly symmetric Bessel beams in Section 3. The clarification is given as follows.

67 69 71 73 75 77 79 81 83

4.1. Circularly symmetric Bessel beams of polarization (1,0) 85 On one hand, considering the recursion relationship of Bessel functions, and the relationship between Cartesian pffiffiffiffiffiffiffiffiffiffiffiffiffiffi coordinates and cylindrical coordinates ρ ¼ x2 þy2 ,x ¼ ρ cos φ,y ¼ ρ sin φ, kt ¼ k sin α0 , and kz ¼ k cos α0 , the Davis Bessel beam in Eqs. (20)–(25) is rewritten as 2

0Þ Eð1; x

2 6 6 4

2

 i2ϕ 3 0Þ ð1 þcosα0 ÞJ n ðkt ρÞ þ ð1  cosα e J n þ 2 ðkt ρÞ þe  i2ϕ J n  2 ðkt ρÞ 2  7 1 i2ϕ 7 J n þ 2 ðkt ρÞ e  i2ϕ J n  2 ðkt ρÞ 2ið1 cos α0 Þ e 5  iϕ  iϕ i sin α0 e J n þ 1 ðkt ρÞ e J n  1 ðkt ρÞ

0Þ Bð1; x

87 89 91

3

6 ð1; 0Þ 7 6 Ey 7 ¼ k2 E0 ð1þ cosα0 Þð iÞn einϕ e  ikz z 4 5 4 ð1; 0Þ Ez

ð59Þ where the superscripts ð1; iÞ and ð1;  iÞ are reminiscent of left-circularly and right-circularly polarized, respectively. The detailed expressions are not presented here for the sake of conciseness. For the special case of the polarization parameters ðpx ; py Þ ¼ ð cos β; sin βÞ and ð  sin β; cos βÞ, which correspond to the radial and azimuthal polarization of incident beam, respectively, the resulting fields are called transverse magnetic (TM) and transverse electric (TE) fields, respectively. They will be discussed in a separate paper. Additionally, same as the circularly symmetric Davis Bessel beam derived in Sections 2.4–2.6, the fields of the aplanatic Bessel beams presented in this section possess a mirror symmetry with respect to each other that is not possessed by the linearly polarized fields which are given in Sections 2.1 and 2.2. The circularly symmetric Davis Bessel beam and the aplanatic Bessel beams are categorized as axisymmetric beams or circularly symmetric beams [46]. In the expansion procedure of shaped beams in terms of partial waves, which is needed in some scattering theories, such as generalized Lorenz–Mie theory [47], Nullfield theory [48], the evaluation of expansion coefficients of such beams can be greatly simplified [46].

11

93 95 ð60Þ

97 99

3

6 ð1; 0Þ 7 7 ¼ B0 ð1þ cosα0 Þð  iÞn einϕ e  ikz z 6 By 5 4 4 0Þ Bð1; z

101 103

 i2ϕ 3 1 J n þ 2 ðkt ρÞ e  i2ϕ J n  2 ðkt ρÞ 2ið1  cos α0 Þ e  i2ϕ 7 6 ð1  cos α Þ  i2ϕ 0 e J n þ 2 ðkt ρÞ þ e J n  2 ðkt ρÞ 7 6 2 5 4 ð1þ cos α0 ÞJ n ðkt ρÞ  sin α0 eiϕ J n þ 1 ðkt ρÞ þ e  iϕ J n  1 ðkt ρÞ 2

105 ð61Þ

107

On the other hand, based on the ASR method, for an aplanatic Bessel beam with the polarization parameters ðpx ; py Þ ¼ ð1; 0Þ, the expressions in Eqs. (54) and (55) are rewritten as

109

2

113

3 Exð1;0Þ 6 ð1;0Þ 7 6 Ey 7 ¼ EB0 ð  iÞn einϕ e  ikz z 4 5 Ezð1;0Þ  3 J n ðσÞ þ 12P ? ei2ϕ J n þ 2 ðσÞ þ e  i2ϕ J n  2 ðσÞ  6 7 1 i2ϕ 7 J n þ 2 ðσÞ  e  i2ϕ J n  2 ðσÞ 6 2iP ? e 4 5  iϕ  iϕ iP J e J n þ 1 ðσÞ  e J n  1 ðσÞ

111

115 117

2

ð62Þ

119 121 123

61 Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

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12

1 3

2

3 Bð1;0Þ x 6 ð1;0Þ 7 6 By 7 ¼ BB0 ð  iÞn einϕ e  ikz z 4 5 Bð1;0Þ z 3  1 i2ϕ e J n þ 2 ðσÞ e  i2ϕ J n  2 ðσÞ P ? 2i  i2ϕ 7 6 1  i2ϕ J n  2 ðσÞ 7 6 5 4 J n ðσÞ  2P ? e J n þ 2 ðσÞþ e P J eiϕ J n þ 1 ðσÞ þ e  iϕ J n  1 ðσÞ 2

5 7 9 11 13 15 17 19 21 23 25

In this way, it is easy to see that the Davis Bessel beams of Eqs. (60) and (61) and the aplanatic Bessel beams of Eqs. (62) and (63) have the same general functional dependence on the radial distance ρ and azimuthal angle ϕ, but have different half-cone angle α0 -dependent constants. This fact is significant, which hints to the fact that the two different procedures are seeing only two different facets of the same fundamental entity. Thus, a general mathematical expression for the circularly symmetric Bessel beam of polarization ð1; 0Þ is proposed as 2 ð1;0Þ 3 Ex 6 ð1;0Þ 7 6 Ey 7 ¼ E0 gðα0 Þð  iÞn einϕ e  ikz z 4 5 Eð1;0Þ z 2

29

6 6 4

33 35 37

ð63Þ

where EB0 ¼ 12Epw0 ð1 þcosα0 Þ, BB0 ¼ ωk EB0 , σ ¼ kρ sin α0 ¼ kt ρ cos α0 sin α0 , P ? ¼ 11  þ cos α0 , and P J ¼ 1 þ cos α0 .

27

31

symmetric Bessel beam in Eqs. (32)–(37) is rewritten as 2 ð0; 1Þ 3 Ex 6 ð0; 1Þ 7 6 Ey 7 ¼ k2 E0 ð1 þcosα0 Þð  iÞn einϕ e  ikz z 4 5 4 Ezð0; 1Þ

2

 i2ϕ 3 0Þ e J n þ 2 ðkt ρÞþ e  i2ϕ J n  2 ðkt ρÞ ð1þ cosα0 ÞJ n ðkt ρÞþ ð1  cosα 2  7 1 i2ϕ 7 J n þ 2 ðkt ρÞ e  i2ϕ J n  2 ðkt ρÞ 2ið1  cos α0 Þ e 5  iϕ  iϕ i sin α0 e J n þ 1 ðkt ρÞ e J n  1 ðkt ρÞ

Bð1;0Þ x

ð64Þ

 3 cos α0 Þ ei2ϕ J n þ 2 ðkt ρÞ e  i2ϕ J n  2 ðkt ρÞ  i2ϕ 7 6 ð1  cos α Þ  i2ϕ 0 e J n þ 2 ðkt ρÞ þ e J n  2 ðkt ρÞ 7 6 2 5 4 ð1þ cos α0 ÞJ n ðkt ρÞ  sin α0 eiϕ J n þ 1 ðkt ρÞþ e  iϕ J n  1 ðkt ρÞ

51 53

6 6 4

ð0;1Þ 3

77 ð67Þ

2

 1



2

55

3

Ex P ? 2i ei2ϕ J n þ 2 ðσÞ  e  i2ϕ J n  2 ðσÞ  7 6 ð0;1Þ 7 6 6 Ey 7 ¼ EB0 ð iÞn einϕ e  ikz z 6 J n ðσÞ 12P ? ei2ϕ J n þ 2 ðσÞ þ e  i2ϕ J n  2 ðσÞ 7 5 5 4 4  iϕ  iϕ ð0;1Þ P J e J n þ 1 ðσÞþ e J n  1 ðσÞ Ez

3

79 81 83 85 87

ð68Þ  3  J n ðσÞ  12P ? ei2ϕ J n þ 2 ðσÞ þ e  i2ϕ J n  2 ðσÞ  i2ϕ 7 6 ð0;1Þ 7 6 1  i2ϕ n inϕ  ik z z 7 6 By 7 ¼ BB0 ð iÞ e e 6  P ? 2i e J n þ 2 ðσÞ e J n  2 ðσÞ 5 5 4 4  iϕ  iϕ ð0;1Þ P ð  iÞe J ðσÞ þ ie J ðσÞ J Bz nþ1 n1 Bð0;1Þ x

71

75

On the other hand, for an aplanatic Bessel beam with the polarization parameters ðpx ; py Þ ¼ ð0; 1Þ, the expressions in Eqs. (56) and (57) are rewritten as 2

67

73

3

 i2ϕ 3 0Þ e J n þ 2 ðkt ρÞþ e  i2ϕ J n  2 ðkt ρÞ ð1 þcosα0 ÞJ n ðkt ρÞþ ð1  cosα 2  i2ϕ 7 1  i2ϕ 7 J n þ 2 ðkt ρÞ e J n  2 ðkt ρÞ 2ið1  cos α0 Þ e 5  iϕ i sin α0 e J n þ 1 ðkt ρÞ e  iϕ J n  1 ðkt ρÞ

2

89 91

ð69Þ

93

Thus, a general mathematical expression for the circularly symmetric Bessel beam of polarization ð0; 1Þ is proposed as

95

Eð0;1Þ z

When gðα0 Þ ¼ k ð1 þcosα0 Þ=4, the expressions in Eqs. (64) and (65) reduce to the Davis Bessel beam of Eqs. (60) and (61), and when gðα0 Þ ¼ Epw0 ðα0 ; βÞ=2, they reduce to the aplanatic Bessel beam of Eqs. (62) and (63). This generalization of description makes the two versions of the Bessel beams presented in Eqs. (60)–(63) merely the two simplest cases of infinite number of possible n-order circularly symmetric Bessel beams, corresponding to different values of the arbitrary function gðα0 Þ, which is dependent on the optical system being used. Additionally, setting n ¼ 0, the general expressions given in Eqs. (64) and (65) reduce the zero-order case, which are the same as those given by Lock [29].

49

2

65

69

6 ð0; 1Þ 7 6 By 7 ¼ B0 ð1 þ cosα0 Þð  iÞn einϕ e  ikz z 4 5 4 Bzð0; 1Þ

Eð0;1Þ x 6 ð0;1Þ 6 Ey 4

41

47

Bxð0; 1Þ

2

1 2ið1 

ð65Þ

45

2

3

39

43

1 2ið1 

ð66Þ

2

6 ð1;0Þ 7 6 By 7 ¼ B0 gðα0 Þð iÞn einϕ e  ikz z 4 5 Bð1;0Þ z 2

 3 cos α0 Þ ðei2ϕ J n þ 2 ðkt ρÞ e  i2ϕ J n  2 ðkt ρÞÞ  7 6 ð1  cos α0 Þ i2ϕ e J n þ 2 ðkt ρÞ þ e  i2ϕ J n  2 ðkt ρÞ 7 6 5 4 ð1þ cos α0 ÞJ n ðkt ρÞ  iϕ 2  iϕ sin α0 e J n þ 1 ðkt ρÞþ e J n  1 ðkt ρÞ 2

63

97

3 7 7 ¼ E0 gðα0 Þð  iÞn einϕ e  ikz z 5

99 101

 i2ϕ 3 1 J n þ 2 ðkt ρÞ  e  i2ϕ J n  2 ðkt ρÞ 2ið1 cos α0 Þ ðe  7 6 ð1  cos α Þ i2ϕ  i2ϕ 0 e J n þ 2 ðkt ρÞ þ e J n  2 ðkt ρÞ 7 6 5 4 ð1þ cos α0 ÞJ n ðkt ρÞ  iϕ 2  iϕ sin α0 e J n þ 1 ðkt ρÞþ e J n  1 ðkt ρÞ 2

103 ð70Þ

3 Bð0;1Þ x 6 ð0;1Þ 7 6 By 7 ¼ B0 gðα0 Þð  iÞn einϕ e  ikz z 5 4 Bð0;1Þ z 2  i2ϕ 3 0Þ e J n þ 2 ðkt ρÞ þ e  i2ϕ J n  2 ðkt ρÞ  ð1þ cosα0 ÞJ n ðkt ρÞ  ð1  cosα 2  7 6 1 7  2i ð1  cos α0 Þ ei2ϕ J n þ 2 ðkt ρÞ  e  i2ϕ J n  2 ðkt ρÞ 6 5 4  iϕ  iϕ  i sin α0 e J n þ 1 ðkt ρÞ  e J n  1 ðkt ρÞ 2

105 107 109

ð71Þ

2

When gðα0 Þ ¼ k ð1 þcosα0 Þ=4, the expressions in Eqs. (70) and (71) reduce to the Davis Bessel beam of Eqs. (66) and (67), and when gðα0 Þ ¼ Epw0 ðα0 ; βÞ=2, they reduce to the aplanatic Bessel beam of Eqs. (68) and (69).

111 113 115 117

4.2. Circularly symmetric Bessel beams of polarization (0,1) 57 59 61

Similar to the procedure used in Section 4.1, on one hand, considering the recursion relationship of Bessel functions, and the relationship between Cartesian coordinates and cylindrical coordinates, the Davis circularly

4.3. Circularly symmetric Bessel beams of polarization ð1; iÞ and ð1; iÞ

119 121

A general description of a circularly symmetric Bessel beam of polarization ð1; iÞ can be obtained by comparing

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

123

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1 3 5 7 9 11

the expressions of the Davis Bessel beam obtained by a superposition of the fields derived using the vector potentials Πe ¼ Π e ð  ex þiey Þ and Πm ¼ Π m ð  iex þ ey Þ with the aplanatic Bessel beam with parameters ðpx ; py Þ ¼ ð1; iÞ. It can also be obtained by a superposition of the circularly symmetric Bessel beams of polarization ð1; 0Þ and that of ð0; 1Þ 2 ð1; iÞ 3 2 ð1; 0Þ 3 2 ð1; iÞ 3 2 ð1; 0Þ 3 Bx Ex Ex Bx þ iExð0; 1Þ þ iBxð0; 1Þ 6 ð1; iÞ 7 6 ð1; 0Þ 7 6 ð1; iÞ 7 6 ð1; 0Þ 7 6 Ey 7 ¼ 6 Ey 6 7 6 þ iEyð0; 1Þ 7 þ iByð0; 1Þ 7 4 5 4 5; 4 B y 5 ¼ 4 B y 5 Ezð1; 0Þ þ iEzð0; 1Þ Bzð1; 0Þ þ iBzð0; 1Þ Ezð1; iÞ Bzð1; iÞ ð72Þ

13 The explicit expressions are 15 17 19

iÞ 3 Eð1; x 6 ð1; iÞ 7 6 Ey 7 ¼ E0 gðα0 Þð  iÞn einϕ e  ikz z 5 4 iÞ Eð1; z 2 3 ð1 þcosα0 ÞJ n ðkt ρÞ þ ð1 cosα0 Þei2ϕ J n þ 2 ðkt ρÞ 6 7 4 ið1 þcosα0 ÞJ n ðkt ρÞ  ið1 cosα0 Þei2ϕ J n þ 2 ðkt ρÞ 5

2

2

23

iÞ Bð1; x 6 ð1; iÞ 6 By 4 iÞ Bð1; z 2

25

29 31 33 35 37 39 41 43 45

2

Eð1; x 6 ð1; 6 Ey 4 Eð1; z

57 59 61

 ið1þ cosα0 ÞJ n ðkt ρÞ  ið1  cosα0 Þei2ϕ J n þ 2 ðkt ρÞ

2 ð1; 0Þ 1Þ 3 Ex  iEð0; x 7 6 ð1; 0Þ ð0; 1Þ 7 7 ¼ 6 Ey 7;  iEy 5 4 5 0Þ 1Þ  iÞ Eð1;  iEð0; z z  iÞ

3

 iÞ

3

ð74Þ

2

Bð1; x 6 ð1; 6 By 4 Bð1; z

2 ð1; 0Þ 1Þ 3 Bx  iBð0; x 7 6 ð1; 0Þ ð0; 1Þ 7 7 ¼ 6 By 7  iBy 5 4 5 0Þ 1Þ  iÞ Bð1;  iBð0; z z  iÞ

3

 iÞ

ð75Þ

The explicit expressions are  iÞ Eð1; x 6 ð1;  iÞ 6 Ey 4  iÞ

3 7 7 ¼ E0 gðα0 Þð iÞn einϕ e  ikz z 5

3 ð1 þ cosα0 ÞJ n ðkt ρÞþ ð1  cosα0 Þe  i2ϕ J n  2 ðkt ρÞ 6  ið1þ cosα ÞJ ðk ρÞ  ið1  cosα Þe  i2ϕ J 7 4 0 n t 0 n  2 ðkt ρÞ 5  2i sin α0 e  iϕ J n  1 ðkt ρÞ

49

55

7 7 ¼ B0 gðα0 Þð  iÞn einϕ e  ikz z 5

2

47

53

3

Similarly, a general description of the circularly symmetric Bessel beam of polarization ð1;  iÞ can also be proposed, by comparing the expressions of Davis Bessel beams obtained by a superposition of the fields derived using the vector potentials Πe ¼  Π e ðex þ iey Þ and Πm ¼ Π m ðiex þ ey Þ with the aplanatic Bessel beam with parameters ðpx ; py Þ ¼ ð1;  iÞ. It can also be obtained by a subtraction of the circularly symmetric Bessel beams of polarization ð1; 0Þ and that of ð0; 1Þ

Eð1; z

51

ð73Þ

7 6 4 ð1 þ cosα0 ÞJ n ðkt ρÞ ð1  cosα0 Þei2ϕ J n þ 2 ðkt ρÞ 5 2 sin α0 eiϕ J n þ 1 ðkt ρÞ

27

2

Bð1; x 6 ð1; 6 By 4 Bð1; z

2

 iÞ  iÞ  iÞ

2

and the aplanatic Bessel beam which are derived in Sections 2 and 3, respectively. In this way, the two versions of the Bessel beams in Sections 2 and 3 are merely the two simplest cases of an infinite number of possible n-order circularly symmetric Bessel beams, corresponding to different values of the arbitrary function gðα0 Þ. Four specific polarization states are presented for demonstration, including ð1; 0Þ which is reminiscent of x-polarization (see Eqs. (64) and (65)), ð0; 1Þ which is reminiscent of ypolarization (see Eqs. (70) and (71)), ð1; iÞ which is reminiscent of left-circular polarization (see Eqs. (73) and (74)), ð1;  iÞ which is reminiscent of right-circular polarization (see Eqs. (76) and (77)). Additionally, if the timedependent harmonic factor expð iωtÞ were used, the expressions of Bessel beams should be revised a little bit by replacing ð  iÞn and e  ikz z with in and eikz z , respectively.

63 65 67 69 71 73 75 77 79

i2 sin α0 eiϕ J n þ 1 ðkt ρÞ

21

13

ð76Þ

3 7 7 ¼ B0 gðα0 Þð iÞn einϕ e  ikz z 5 3

ið1 þcosα0 ÞJ n ðkt ρÞ þ ið1 cosα0 Þe  i2ϕ J n  2 ðkt ρÞ 6 7 4 ð1 þcosα0 ÞJ n ðkt ρÞ  ð1 cosα0 Þe  i2ϕ J n  2 ðkt ρÞ 5 2 sin α0 e  iϕ J n  1 ðkt ρÞ

ð77Þ

In this section, general descriptions of circularly symmetric Bessel beams of arbitrary order were derived for four specific polarization states. This is achieved by an analysis of the relationship between the Davis Bessel beam

5. Conclusions

81

The accurate description of shaped beams plays a significant role in the analysis of beam properties, beam propagation as well as light-matter interactions. For instance, explicit mathematical expressions for electromagnetic (EM) field components are used directly in several scattering computational methods for the prediction of scattering properties of scatterers, such as the discrete dipole approximation (DDA), the finite-difference timedomain (FDTD) technique, the Method of Moments (MOM), the Multiple Multipole (MMP) method and others. Further expansion of the mathematical expressions into partial waves or plane waves is required in analytical methods, e.g. GLMT, or semi-analytical methods, e.g. the Null-field method. In the description of an ideal Bessel beam, two different procedures are commonly applied to obtain the fields of an n-order Bessel beam: (a) the ASR procedure which obtains the fields by a superposition of partial plane waves, and (b) the Davis procedure which obtains the fields from a polarized vector potential. This two different procedures give two seemingly different answers for the fields. Nevertheless, in this paper, by deriving a class of circularly symmetric Bessel beam using the Hertz potential vector, and making comparisons with the circularly symmetric Bessel beams derived using ASR procedure, it reveals that the functional dependence of the two answers is identical for the circularly symmetric Bessel beams. In this way, the gap between different descriptions of polarized Bessel beams derived using different approaches is bridged. This fact is significant, and hints at some fundamental unification that the two different descriptions are seeing only two different facets of the same entity. Thus, a general description for circularly symmetric Bessel beams are proposed in this paper, which makes the Davis type Bessel beam in Section 2 and the ASR type Bessel beam in Section 3 merely the two simplest cases of infinite number of possible circularly symmetric Bessel beams, corresponding to different values of the arbitrary function gðα0 Þ. The unification or generalized of different descriptions is important for the analysis of the properties of beams as well as further applications. To demonstrate the properties

83

Please cite this article as: Wang JJ, et al. General description of circularly symmetric Bessel beams of arbitrary order. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.07.011i

85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123

J.J. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

14

1 3 5 7

of polarized Bessel beams, magnitude distributions of the fields, energy density and Poynting vectors are plotted for both paraxial and nonparaxial cases. The code for producing the distribution patterns is available upon request. The results presented in this paper provide a new perspective on the description of Bessel beams and can be very useful in the light scattering-related problems where Bessel beams are applied.

1 2π 1 2π

Z

2π

einβ e  iρ

cos ðϕ  βÞ

0

Z

2π

" einβ

0

cos ðlβÞ

Acknowledgments

13 Q4 Q5

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). 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 grant of the China Scholarship Council. The authors also thanks a lot for the constructive suggestions from the reviewers of this paper.

15 17 19 21 23

sin ðlβÞ

Appendix A

27

To perform the integration over Z 2π ηf Q jα ¼ α0 einβ e  iσ EðrÞ ¼ Epw0 e  ikz cos α0 2π β¼0

29 31 33 35

cos ðϕ  βÞ

dβ

ðA  1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 where σ ¼ kρ sin α0 , ρ ¼ x þ y , ϕ ¼ tan ðy=xÞ, and 2

2

px ð cos α cos β þ sin βÞ  py ð1 cos αÞ sin β cos β 2

3

7 6 2 2 7 Q ¼6 4  px ð1  cos αÞ sin β cos β þ py ð cos α sin β þ cos βÞ 5;

ðA  2Þ

63 65

e  iρ

cos ½β  ϕ

dβ

67

" # ð  iÞn þ l eiðn þ lÞϕ J n þ l ðρÞ þ ð  iÞn  l eiðn  lÞϕ J n  l ðρÞ 1 ¼ : 2 i U ð  iÞn þ l eiðn þ lÞϕ J n þ l ðρÞ þ iU ð  iÞn  l eiðn  lÞϕ J n  l ðρÞ

69

ðA  10Þ

71

For a Bessel beam with the polarization parameters ðpx ; py Þ ¼ ð0; 1Þ, the complex polarization vector is 2 3  ð1  cos αÞ sin β cos β 6 ð cos α sin 2 β þ cos 2 βÞ 7 Q ¼4 5  sin α sin β 2 1 3  2ð1  cos αÞ sin 2β 6 7 ¼ 4 12½ð1 þ cos αÞ þ ð1  cos αÞ cos 2β 5; ðA  11Þ

73

 sin α sin β where cos 2 β ¼ 12ð1 þcos2βÞ and sin 2 β ¼ 12ð1 cos2βÞ are used. Performing the Integration in Eq. (A-1), a Bessel beam propagating in the z direction is obtained 2

25

ðA  9Þ

#

9 11

dβ ¼ ð  iÞn J n ðρÞeinϕ

ð0;1Þ 3

Ex 6 ð0;1Þ 7 6 Ey 7 ¼ EB0 e  ikz z 5 4 Eð0;1Þ z 3 2 h i 1  P ? 2i ð iÞn þ 2 eiðn þ 2Þϕ J n þ 2 ðσÞ  ð iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 6 h i7 7 6 n n þ 2 iðn þ 2Þϕ 6 1 inϕ e J n þ 2 ðσÞþ ð  iÞn  2 eiðn  2Þϕ J n  2 ðσÞ 7 6 ð iÞ J n ðσÞe þ 2P ? ð  iÞ 7: 7 6 h i 5 4 iP J ð  iÞn þ 1 eiðn þ 1Þϕ J n þ 1 ðσÞ ð  iÞn  1 eiðn  1Þϕ J n  1 ðσÞ

75 77 79 81 83 85 87 89 91 93

ðA  12Þ

95

Bessel beams with other polarization parameters ðpx ; py Þ can be implemented similarly.

97

 px sin α cos β  py sin α sin β

99

37 39 41 43 45 47 49

the following mathematical relationships are applied Z 2π 1 cos nβeiρ cos ðβ  φÞ dβ ¼ in J n ðρÞ cos nφ ðA  3Þ 2π 0 1 2π 1 2π 1 2π

51 53 55 57

2π

sin nβeiρ

cos ðβ  φÞ

0

Z

2π

einβ eiρ

cos ðβ  φÞ

0

Z

2π 0

" einβ

cos ðlβÞ sin ðlβÞ

dβ ¼ in J n ðρÞsinnφ π

dβ ¼ einφ J n ðρÞein2 ¼ in J n ðρÞeinφ

ðA  4Þ

ðA  5Þ

# eiρ

cos ½β  φ

dβ

2 3 in þ l eiðn þ lÞφ J n þ l ðρÞ þ in  l eiðn  lÞφ J n  l ðρÞ 14 5: ¼ 2  iU in þ l eiðn þ lÞφ J n þ l ðρÞ þi Uin  l eiðn  lÞφ J n  l ðρÞ ðA  6Þ

Assuming φ ¼ π þ ϕ, then we have Z 2π 1 cos nβe  iρ cos ðβ  ϕÞ dβ ¼ ð  iÞn J n ðρÞ cos nϕ 2π 0

59 61

Z

1 2π

Z

2π 0

sin nβe  iρ

cos ðβ  ϕÞ

dβ ¼ ð iÞn J n ðρÞsin nφ

ðA  7Þ

ðA  8Þ

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