## Abstract

Using the transverse Hertz vector potentials, vector analyses of linearly and circularly polarized Bessel beams of arbitrary orders are presented in this paper. Expressions for the electric and magnetic fields of vector Bessel beams in free space that are rigorous solutions to the vector Helmholtz equation are derived. Their respective time averaged energy density and Poynting vector are also obtained, in order to exhibit their non-diffracting properties. Polarization patterns and magnitude profiles with different parameters are displayed. Particular emphasis is placed on the cases where the ratio of wave number over its transverse component *k*/*k _{t}* approximately equals to one and largely exceeds it, which corresponding to the nonparaxial and paraxial condition, respectively. These results allow us to recognize that the vector Bessel beams exhibit new and important features, compared with the scalar fields.

© 2014 Optical Society of America

## 1. Introduction

The concept of the so-called “non-diffracting Bessel beams” was put forward for the first time by Durnin and associates in 1987 [1]. They also provided a method for generating the zero-order Bessel beam by illuminating an angular slit placed in the focal plane of a lens with a plane wave. After that, various techniques, such as the holographic optical elements or axicons, have been presented for generating Bessel beams, including the higher-order ones [2–5]. Apart from Bessel beams, which are the exact solutions to the scalar Helmholtz wave equation, other non-diffracting beams known as Mathieu beams or Weber beams are also obtained for free-space propagation [6–9].

In many earlier studies, Bessel beams have been described by the scalar wave theory, which simply provides satisfactory results under paraxial conditions, namely, the size of the central spot of the beam is much bigger than the wavelength [10–12]. It is necessary to carry out vector analyses of Bessel beams thoroughly in order to reveal many other important and interesting features for further applications. Several vector solutions to the Maxwell’s equations are constructed, which are identified as transverse electric (TE) and transverse magnetic (TM) modes, or polarization states such as radial, azimuthal, linear and circular polarizations. Some dynamic and propagation invariance properties are analyzed from different viewpoints.

Radially polarized Bessel beams have been explored when researchers concentrated on the work of laser electron acceleration [13]. A Bessel beam of order one was produced by focusing a radially polarized laser beam with an axicon and the transverse intensity distributions agreed well with the theory [14]. Hall *et al.* developed a paraxial wave equation for the radially and azimuthally polarized Bessel-Gauss beam propagating in free space [15–18]. Several free parameters determine the form and behavior of these beams. In the paraxial approximation, Schimpf *et al.* have derived solutions and experimentally verified the radially polarized Bessel-Gauss beams by superimposing decentered Gaussian beams with differing polarization states [19].

TM and TE modes have been reported from a wide range of laser types. Sheppard considered the TM and TE beam modes based on the complex dipole sources theory [20]. The experimental realizations of TM_{n} and TE_{n} Bessel beams of order one have been reported recently, which offers potential applications in imaging and optical micromanipulation systems [21]. April has obtained closed-form expressions for the electric and magnetic fields of TM and TE mode laser beams. The fields are exact solutions to the Helmholtz equation, spherical Bessel and associated Legendre functions included [22].

In [23], Bouchal and Olivík derived expressions for the solutions to the vector Helmholtz equation, which corresponding to the non-diffracting vector Bessel beams of arbitrary orders. Based on the angular spectrum theory, four simplest polarization states were obtained, namely radial, azimuthal, linear and circular polarizations. However, the circularly polarized Bessel beam mentioned in that paper is not the simplest one, which we will demonstrate next. General constructions and connections of the vector non-diffracting beams in different coordinates, including Bessel, Mathieu and Weber beams were presented in [24]. The TM and TE modes and polarization states were evaluated in detail.

The Hertz vector potentials can be used to solve radiation and propagation problems [25]. The chosen of Hertz electric and magnetic vector potentials oriented along the propagation axis gives rise to the TM and TE mode Bessel beams, respectively [26]. Ornigotti and Aiello presented a method for the realization of radially and azimuthally polarized nonparaxial Bessel beams using Hertz vector potentials, the spatial and polarization patterns were obtained and the applications of these beams were discussed [27].

## 2. Linearly and circularly polarized Bessel beams

As shown in [25], by using the Hertz electric and magnetic potentials ${\prod}_{e}$ and ${\prod}_{m}$, the electric field $E$ and magnetic field $H$ can be expressed as

respectively, where an*exp(jωt)*time harmonic dependence is assumed. Subscripts

*e*and

*m*refer to the electric and magnetic cases, respectively. Both ${\prod}_{e}$ and ${\prod}_{m}$ satisfy the homogeneous vector Helmholtz equations in source-free regions, namely

The choice of ${\prod}_{e}={\prod}_{e}\widehat{z}$ and ${\prod}_{m}={\prod}_{m}\widehat{z}$, oriented along the propagation direction in circular cylindrical coordinates, gives rise to the TM and TE mode Bessel beams, respectively, if ${\prod}_{e}$ and ${\prod}_{m}$ take the form

where*J*(

_{n}*k*) denotes the

_{t}ρ*n*-order Bessel functions of the first kind [26]. The parameters

*k*,

*k*and

_{t}*k*are the wave number and its transverse and longitudinal components, respectively, associated by the relation

_{z}*k*+

_{t}^{2}*k*=

_{z}^{2}*k*.

^{2}One can obtain azimuthally and radially polarized Bessel beams only if *n* = 0 in Eq. (3a) and Eq. (3b), respectively. It is intriguing to point out that when the transverse component of electric field is azimuthally polarized, the transverse component of magnetic field is radially polarized in the meantime, and vice versa.

In this paper, we consider the case for ${\prod}_{e}$ and ${\prod}_{e}$ orienting along the direction perpendicular to the propagation axis in Cartesian coordinates.

The linearly polarized Bessel beams of arbitrary orders are generated for the selection:

By setting *n* = 0, we obtain the lowest-order linearly polarized Bessel beams, and the expressions of the fields are accordance with the one presented in [23], section 2 example 4.

Using the above equations, the time averaged energy density $<{W}_{m}>$ and Poynting vector $<{S}_{m}>$ can be expressed as

It can be seen from Eqs. (6) and (7) that the time averaged energy density and Poynting vector remain unchanged in any plane perpendicular to the propagation axis (i.e. *z*-axis). This is the characteristic of the so-called “non-diffracting Bessel beams”. As is obvious, all the transverse components of the time averaged Poynting vector are zero for lowest-order linearly polarized Bessel beams.

Circular polarization states can be constructed by superposing two orthogonal linear polarizations with quadrature phase and equal magnitude. The circularly polarized Bessel beams can be obtained by choosing

The substitution of Eq. (8) into Eq. (2b) results in the electric and magnetic components of circularly polarized Bessel beams.

Setting *n* = 0 gives rise to the lowest-order circularly polarized Bessel beams. The one as stated in [23], section 2 example 3, corresponds to *n* = 1 in Eqs. (9) in this paper.

The time averaged energy density $<{W}_{m}>$ and Poynting vector $<{S}_{m}>$ are given by Eqs. (10) and (11), respectively.

In contrast with the linearly polarized Bessel beam, both the transverse and the longitudinal components of the time averaged Poynting vector exist even for the lowest-order circularly polarized Bessel beams.

The expressions for the electric and magnetic field as well as the time averaged energy densitiy and Poynting vector for choosing ${\prod}_{e}={\prod}_{e}\widehat{y}$ and ${\prod}_{e}={\prod}_{e}(\widehat{x}+j\widehat{y})$can be easily obtained from Eqs. (5)-(7), and (9)-(11) by means of the duality transformation $E\to H$, $H\to -E$, and $\mu \to \epsilon $, respectively. It is convenient to stress at this point that the choice of ${\prod}_{e}$ yields the magnetic field $H$is linearly or circularly polarized. The related formulas for these Bessel beams are not written here for the sake of space.

## 3. Discussions

In this section, by choosing ${\prod}_{m}={\prod}_{m}\widehat{y}$ and ${\prod}_{m}={\prod}_{m}(\widehat{x}+j\widehat{y})$, linearly and circularly polarized Bessel beams with different parameters are discussed in detail, including the order of Bessel function *n* and the ratio of wave number over its transverse component *k*/*k _{t}*. The polarization patterns as well as the electric and magnetic fields distributions are displayed. Magnitude profiles for the time averaged energy density and Poynting vector are also plotted. Computed results demonstrate significant differences for these beams with different parameters, especially for the cases where

*k*/

*k*approximately equals to one and largely exceeds it, which leading to the nonparaxial and paraxial condition, respectively.

_{t}#### 3.1 Polarization properties

The magnitude profiles and vector diagrams of the transverse components of the electric fields for linearly polarized and circularly polarized Bessel beams with different orders *n* are illustrated in Figs. 1 and 2, respectively. For linearly polarized Bessel beams with any order *n*, the field orientation remains constant at each point while its magnitude oscillates in time. For circularly polarized Bessel beams, the rotations in time of the transverse electric field vector become apparent, and it is interesting to point out that the transverse components of the electric fields at any point are in the same or opposite directions only if *n* = 0.

#### 3.2 The electric and magnetic components

In this subsection, the electric and magnetic fields distributions of vector Bessel beams, associated with different parameters *k*/*k _{t}*, are compared and discussed.

As illustrated in Figs. 3 and 4, for linearly polarized Bessel beams, the amplitude distributions of *|E _{z}|* and

*|H*are similar but shifted by Δφ = π / 2. Notice that when

_{z}|*k*is much greater than

*k*(i.e.

_{t}*k*/

*k*≈7.84), corresponding to the paraxial condition, the longitudinal components of the electric fields become negligible. However, when

_{t}*k*approaches

_{t}*k*(i.e.

*k*/

*k*≈1.02), leading to the nonparaxial condition, the longitudinal components dominate the total fields.

_{t}Visual comparison of the plots in Figs. 5 and 6 shows that, for circularly polarized Bessel beams, all the transverse and longitudinal components of the electromagnetic fields are circular symmetrical under the paraxial condition (i.e. *k*/*k _{t}* ≈6.17). Similar behavior for the field components is observed except for the transverse components of magnetic fields

*|H*and

_{x}|*|H*for the case

_{y}|*k*/

*k*≈1.01. The proportions of the longitudinal components of the electric fields under nonparaxial or paraxial condition are just the same as those for linearly polarized Bessel beams.

_{t}#### 3.3 The time averaged Poynting vector and energy density

As mentioned above, the time averaged Poynting vector and energy density of linearly polarized and circularly polarized Bessel beams represent non-diffracting properties. We show a comparison of the amplitude patterns for different parameters *k*/*k _{t}*.

As illustrated in Fig. 7 and 8, it is recognized that the longitudinal component of the time averaged Poynting vector accounts for the major part when *k*/*k _{t}* is far greater than one, whereas for

*k*/

*k*approximately equals to one, the transverse component dominates the total component.

_{t}The time averaged energy density will always keep circular symmetry except for the linearly polarized Bessel beam under the nonparaxial condition, as observed in Fig. 9. It can be understood from Fig. 4 that the major electric and magnetic components are *E _{x}* and

*H*, respectively, the former is circularly-symmetric while the latter is non circularly-symmetric. However, for linearly polarized Bessel beams under the paraxial condition, all the major electric and magnetic components (

_{y}*E*and

_{x}*H*in Fig. 3) are circularly-symmetric. The circular-symmetry property of the energy density can be easily obtained from Eq. (10) for circularly polarized Bessel beams.

_{y}## 4. Conclusions

Hertz vector potentials orienting along the propagation axis in circular cylindrical coordinates give rise to the TM and TE modes as well as the azimuthally and radially polarized Bessel beams. While Hertz vector potentials orienting along the direction perpendicular to the propagation axis in Cartesian coordinates lead to the linearly and circularly polarized beams of arbitrary orders, which we discussed in detail in this paper. The electric and magnetic fields of linearly and circularly polarization states are derived. Their respective time averaged energy density and Poynting vector are also obtained. We concentrate our attention on different parameters to analyze these beams, such as the polarization states, the order of Bessel functions and the ratio of wave number over its transverse component *k*/*k _{t}*. It is straightforward to demonstrate that the scalar theory is not applicable to the nonparaxial condition. All these results provide new insight into the properties of the non-diffracting beams.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under grants 61171025, 61101020, and the Specialized Research Fund for the Doctoral Program of Higher Education under grant 20110092120012.

## References and links

**1. **J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651–654 (1987). [CrossRef]

**2. **A. J. Cox and D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. **30**(11), 1330–1332 (1991). [CrossRef]

**3. **Y. Z. Yu and W. B. Dou, “Generation of pseudo-Bessel beams at THz frequencies by use of binary axicons,” Opt. Express **17**(2), 888–893 (2009). [CrossRef]

**4. **J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**(1-6), 297–301 (2000). [CrossRef]

**5. **P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, and A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A **14**(8), 1817–1824 (1997). [CrossRef]

**6. **J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**(20), 1493–1495 (2000). [CrossRef]

**7. **C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. **222**(1-6), 75–80 (2003). [CrossRef]

**8. **M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**(1), 44–46 (2004). [CrossRef]

**9. **J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. **195**(1-4), 35–40 (2001). [CrossRef]

**10. **J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. **13**(2), 79–80 (1988). [CrossRef]

**11. **A. J. Cox and J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. **17**(4), 232–234 (1992). [CrossRef]

**12. **Z. P. Jiang, Q. S. Lu, and Z. J. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. **34**(31), 7183–7185 (1995). [CrossRef]

**13. **R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D **42**(5), 1807–1818 (1990). [CrossRef]

**14. **S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. **31**(7), 1527–1531 (1992). [CrossRef]

**15. **R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. **19**(7), 427–429 (1994). [CrossRef]

**16. **D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. **21**(1), 9–11 (1996). [CrossRef]

**17. **P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A **13**(5), 962–966 (1996). [CrossRef]

**18. **P. L. Greene and D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A **15**(12), 3020–3027 (1998). [CrossRef]

**19. **D. N. Schimpf, W. P. Putnam, M. D. W. Grogan, S. Ramachandran, and F. X. Kärtner, “Radially polarized Bessel-Gauss beams: decentered Gaussian beam analysis and experimental verification,” Opt. Express **21**(15), 18469–18483 (2013). [CrossRef]

**20. **C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. **24**(22), 1543–1545 (1999). [CrossRef]

**21. **A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. **31**(11), 1732–1734 (2006). [CrossRef]

**22. **A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. **33**(14), 1563–1565 (2008). [CrossRef]

**23. **Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. **42**(8), 1555–1566 (1995). [CrossRef]

**24. **K. Volke-Sepulvea and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. **8**(10), 867–877 (2006). [CrossRef]

**25. **E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. **45**(11), 1099–1101 (1977). [CrossRef]

**26. **Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. **5**, 57–71 (2008). [CrossRef]

**27. **M. Ornigotti and A. Aiello, “Radially and azimuthally polarized nonparaxial Bessel beams made simple,” Opt. Express **21**(13), 15530–15537 (2013). [CrossRef]