## Vector analyses of linearly and circularly polarized Bessel beams using Hertz vector potentials |

Optics Express, Vol. 22, Issue 7, pp. 7821-7830 (2014)

http://dx.doi.org/10.1364/OE.22.007821

Acrobat PDF (4640 KB)

### 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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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

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]

*et al.*developed a paraxial wave equation for the radially and azimuthally polarized Bessel-Gauss beam propagating in free space [15

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] [PubMed]

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]

*et al.*have derived solutions and experimentally verified the radially polarized Bessel-Gauss beams by superimposing decentered Gaussian beams with differing polarization states [19

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] [PubMed]

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] [PubMed]

_{n}and TE

_{n}Bessel beams of order one have been reported recently, which offers potential applications in imaging and optical micromanipulation systems [21

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] [PubMed]

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

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] [PubMed]

## 2. Linearly and circularly polarized Bessel beams

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

*exp(jωt)*time harmonic dependence is assumed. Subscripts

*e*and

*m*refer to the electric and magnetic cases, respectively. Both

*J*(

_{n}*k*) denotes the

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

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

*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}*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.

*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

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

*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.

*n*= 0 gives rise to the lowest-order circularly polarized Bessel beams. The one as stated in [23

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

*n*= 1 in Eqs. (9) in this paper.

## 3. Discussions

*n*and the ratio of wave number over its transverse component

*k*/

*k*. 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

_{t}*k*/

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

_{t}### 3.1 Polarization properties

*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

*k*/

*k*, are compared and discussed.

_{t}*|E*and

_{z}|*|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}*k*/

*k*≈6.17). Similar behavior for the field components is observed except for the transverse components of magnetic fields

_{t}*|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

*k*/

*k*.

_{t}*k*/

*k*is far greater than one, whereas for

_{t}*k*/

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

_{t}*E*and

_{x}*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

*k*/

*k*. 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.

_{t}## Acknowledgments

## References and links

1. | J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

2. | A. J. Cox and D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. |

3. | Y. Z. Yu and W. B. Dou, “Generation of pseudo-Bessel beams at THz frequencies by use of binary axicons,” Opt. Express |

4. | J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. |

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 |

6. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

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. |

8. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

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. |

10. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. |

11. | A. J. Cox and J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. |

12. | Z. P. Jiang, Q. S. Lu, and Z. J. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. |

13. | R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D |

14. | S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. |

15. | R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. |

16. | D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. |

17. | P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A |

18. | P. L. Greene and D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A |

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 |

20. | C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. |

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. |

22. | A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. |

23. | Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. |

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. |

25. | E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. |

26. | Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. |

27. | M. Ornigotti and A. Aiello, “Radially and azimuthally polarized nonparaxial Bessel beams made simple,” Opt. Express |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(260.0260) Physical optics : Physical optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 22, 2014

Revised Manuscript: March 19, 2014

Manuscript Accepted: March 19, 2014

Published: March 27, 2014

**Citation**

Yanxun Wang, Wenbin Dou, and Hongfu Meng, "Vector analyses of linearly and circularly polarized Bessel beams using Hertz vector potentials," Opt. Express **22**, 7821-7830 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7821

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### References

- J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
- A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30(11), 1330–1332 (1991). [CrossRef] [PubMed]
- Y. Z. Yu, W. B. Dou, “Generation of pseudo-Bessel beams at THz frequencies by use of binary axicons,” Opt. Express 17(2), 888–893 (2009). [CrossRef] [PubMed]
- J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]
- P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14(8), 1817–1824 (1997). [CrossRef]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef] [PubMed]
- C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, 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]
- M. A. Bandres, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001). [CrossRef]
- J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988). [CrossRef] [PubMed]
- A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17(4), 232–234 (1992). [CrossRef] [PubMed]
- Z. P. Jiang, Q. S. Lu, Z. J. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34(31), 7183–7185 (1995). [CrossRef] [PubMed]
- R. D. Romea, W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D 42(5), 1807–1818 (1990). [CrossRef]
- S. C. Tidwell, D. H. Ford, W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992). [CrossRef]
- R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994). [CrossRef] [PubMed]
- D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21(1), 9–11 (1996). [CrossRef] [PubMed]
- P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A 13(5), 962–966 (1996). [CrossRef]
- P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15(12), 3020–3027 (1998). [CrossRef]
- D. N. Schimpf, W. P. Putnam, M. D. W. Grogan, S. Ramachandran, F. X. Kärtner, “Radially polarized Bessel-Gauss beams: decentered Gaussian beam analysis and experimental verification,” Opt. Express 21(15), 18469–18483 (2013). [CrossRef] [PubMed]
- C. J. R. Sheppard, S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24(22), 1543–1545 (1999). [CrossRef] [PubMed]
- A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, 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] [PubMed]
- A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008). [CrossRef] [PubMed]
- Z. Bouchal, M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995). [CrossRef]
- K. Volke-Sepulvea, 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]
- E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45(11), 1099–1101 (1977). [CrossRef]
- Y. Z. Yu, W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. 5, 57–71 (2008). [CrossRef]
- M. Ornigotti, A. Aiello, “Radially and azimuthally polarized nonparaxial Bessel beams made simple,” Opt. Express 21(13), 15530–15537 (2013). [CrossRef] [PubMed]

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