## Generation of non-diffractive Bessel beams by inward cylindrical traveling wave aperture distributions |

Optics Express, Vol. 22, Issue 15, pp. 18354-18364 (2014)

http://dx.doi.org/10.1364/OE.22.018354

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### Abstract

The focusing capabilities of an inward cylindrical traveling wave aperture distribution and the non-diffractive behaviour of its radiated field are analyzed. The wave dynamics of the infinite aperture radiated field is clearly unveiled by means of closed form expressions, based on incomplete Hankel functions, and their ray interpretation. The non-diffractive behaviour is also confirmed for finite apertures up to a defined limited range. A radial waveguide made by metallic gratings over a ground plane and fed by a coaxial feed is used to validate numerically the analytical results. The proposed system and accurate analysis of non-diffractive Bessel beams launched by inward waves opens new opportunities for planar, low profile beam generators at microwaves, Terahertz and optics.

© 2014 Optical Society of America

## 1. Introduction

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

2. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499 (1987). [CrossRef] [PubMed]

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

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

16. S. Chávez-Cerda and G. H. C. New, “Evaluation of focused Hankel waves and Bessel beams,” Opt. Commun. **181**, 369–377 (2000). [CrossRef]

12. M. Ettorre and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes,” IEEE Trans. Antennas Propag. **60**, 3605–3613 (2012). [CrossRef]

11. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Near field diffraction of Hankel beams,” in *Frontiers in Optics*, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JSuA36. [CrossRef]

6. Z. Li, K. B. Alici, H. Caglayan, and E. Ozbay, “Generation of an axially asymmetric Bessel-like beam from a metallic subwavelength aperture,” Phys. Rev. Lett. **102**, 143901 (2009). [CrossRef] [PubMed]

5. R. M. Herman and T. A. Wiggins, “Production and uses of diffracionless beams,” J. Opt. Soc. Am. A **8**, 932–942 (1991). [CrossRef]

17. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. **109**, 093904 (2012). [CrossRef] [PubMed]

## 2. Analytical derivation: infinite case

*z*-axis normal to the aperture. For the sake of convenience, the observation point is expressed either in cylindrical (

*ρ*,

*ϕ*,

*z*) or in spherical (

*r*,

*θ*,

*ϕ*) coordinates; vectors are bold and a hat denotes a unit vector. In the following discussion, Transverse Magnetic (TM) modes with respect to the

*z*-direction will only be considered. However, a similar procedure can be applied to Transverse Electric (TE) modes. An inward cylindrical wave is assumed for the tangential electric field on the radiating aperture:

*e*,

^{jωt}*ω*= 2

*πf*being the angular frequency, is assumed and suppressed. The electric field radiated by the aperture is given by [18

18. L. B. Felsen and N. Marcuvitz, *Radiation and Scattering of Waves*, Series on Electromagnetic Wave Theory (IEEE Press, 1994). [CrossRef]

*k*, and

_{ρ}*k*is the wavenumber in free space, whereas

*J*() and

_{n}*n*-th order Bessel and Hankel functions of the

*i*-th kind, respectively. Equation (2) is the Hankel-transform of the tangential field distribution over the aperture and, for the assumed inward cylindrical wave distribution, it is given by After some mathematical manipulations (refer to the Appendix for more details), the radiated field is exactly expressed as the superposition of the GO and SW contributions with

*n*-th order incomplete Hankel functions of

*i*-th kind with the second argument

19. R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag. **52**, 3373–3389 (2004). [CrossRef]

*θ*<

*θ*where

_{a}*U*(

*θ*−

_{a}*θ*) = 1, whereas becomes an inward Hankel beam outside such a cone where

*U*(

*θ*−

_{a}*θ*) = 0. By considering the Bessel function as the superposition of two Hankel functions

*θ*<

*θ*(Fig. 2(a)). At the GO discontinuity cone

_{a}*θ*=

*θ*, the SW contribution (Eq. (6)) exhibits an opposite abrupt discontinuity which renders the total field smooth and continuous. Such a transitional behaviour is described by the sign and incomplete Hankel functions when

_{a}19. R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag. **52**, 3373–3389 (2004). [CrossRef]

**polarized) spherical wave associated to a ray launched at the aperture center (origin of the reference system in Fig. 1), with a radiation null on the aperture symmetry axis**

*θ̂**θ*= 0.

*z*(1

^{st}column) and

*ρ*(2

^{nd}column) components and the total electric field (3

^{rd}column) radiated by an inward cylindrical wave (2

^{nd}row) with

*k*= 0.6

_{ρa}*k*in the vertical

*z*−

*ρ*plane. As clear from the previous results, the non diffractive behaviour of the radiated field can be appreciated within the cone with angle

*θ*≈ 37° (dotted line) where the various components of the electric field recover the respective components of a standard Bessel beam (1

_{a}^{st}row).

*θ*<

*θ*. Its ray interpretation is shown in Fig. 2(b). Again, the SW contribution discontinuity perfectly matches the GO jump at the shadow boundary cone, thus providing a smooth continuous total field. Therefore, it is apparent that an outward Hankel distribution cannot produce a Bessel beam, as also clear from Fig. 3(c).

_{a}## 3. Finite case

*a*= 7

*λ*(refer to Fig. 1) is shown in Fig. 4. Analogously to Fig. 3,

*z*,

*ρ*components and total electric field are arranged in the 1

^{st}, 2

^{nd}, and 3

^{rd}column, respectively; while the results for a Bessel, and inward Hankel aperture distributions are arranged in the 1

^{st}, 2

^{nd}row, respectively. In both cases

*k*is equal to 0.6

_{a}*k*. In the present finite case additional GO shadow boundaries (dotted lines) arise from the aperture rim and limit the GO contribution in a conical region above the aperture with vertex on the z-axis at a distance

*z*=

*a*cot

*θ*≃ 9.33

_{a}*λ*, as shown in Fig. 5; hence the non-diffractive behaviour is limited up to this distance for both the reference Bessel [1

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

## 4. Proposed structure: radial waveguide loaded with metallic gratings

20. M. Albani, A. Mazzinghi, and A. Freni, “Automatic design of CP-RLSA antennas,” IEEE Trans. Antennas Propag. **60**, 5538–5547 (2012). [CrossRef]

21. M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. C. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antennas Propag. **62**, 1991–1999 (2014). [CrossRef]

*ρ*and sizes

_{i}*w*of the circular slots (refer to Fig. 6). Indeed, slots are positioned at those points where the phase of the outward cylindrical feeding wave inside the PPW matches that of the target inward cylindrical aperture distribution, according to an holographic criterion. The slot width is used to modulate the amplitude of the aperture distribution. In addition the design procedure assures that the total energy launched by the coaxial feed in the PPW is radiated, thus avoiding any spurious radiation by the edges of the structure. The geometrical details of the structure are provided in Table 1.

_{i}*E*component of the electric field at the distance

_{z}*z*= 4.667

*λ*from the aperture, corresponding to the maximum transverse extension of the rhomboidal region where the Bessel beam is created (dashed line in Fig. 7), compared to the ideal truncated (

*a*= 7

*λ*) inward Hankel aperture distribution. The good agreement reveals the accuracy of the launcher design. It is worth noting that the traveling nature of the synthesized aperture distribution guarantees a wide-band operation as a difference with resonant Bessel designs [2

2. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499 (1987). [CrossRef] [PubMed]

13. M. Ettorre, S. M. Rudolph, and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes: experimental validation,” IEEE Trans. Antennas Propag. **60**, 2645–2653 (2012). [CrossRef]

## 5. Conclusion

## Appendix

*Ẽ*(

_{t}*k*) in an integral form after interchanging the order of the integrals, the spectral

_{ρ}*k*-integral is closed analytically by means of resulting in the spatial representation Finally, the change of integration variable is used with the upper/lower sign in the first/second integral of Eq. (13), so that the TM scalar potential reduces to namely the sum of two zero order, second kind incomplete Hankel functions [19

_{ρ}19. R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag. **52**, 3373–3389 (2004). [CrossRef]

*θ*=

_{a}*k*and cos

_{ρa}/k*θ*=

_{a}*k*. In the case of an inward cylindrical wave distribution the tangential aperture field can be represented as

_{za}/k*θ*,

*θ*∈ [0,

_{a}*π*/2], then

*θ*≤

*θ*and

_{a}*θ*>

*θ*, then

_{a}**52**, 3373–3389 (2004). [CrossRef]

*S*as the sum of a GO and a SW contributions with It is worth noting that although Eq. (18) is already an exact compact closed form for the potential, its exact rearrangement in Eq. (20) better highlights the wave constituents of the total field. Indeed the GO contribution is asymptotically dominated only by the saddle point, whereas the SW is asymptotically dominated only by the end point at any observation aspect. As a matter of fact, the non-uniform asymptotic expression [19

**52**, 3373–3389 (2004). [CrossRef]

*θ*=

*θ*is present, where the non-uniform asymptotic expression (24) fails. By differentiating the TM potential according to Eq. (9), the electric field expressions Eqs. (4)–(6) are finally obtained. The same calculations can be repeated for the field radiated by an infinite aperture distribution taking on an outward Hankel function using the scalar potential in Eq. (15), obtaining the result in Eq. (7).

_{a}## Acknowledgments

## References and links

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

2. | J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

3. | M. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. |

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

5. | R. M. Herman and T. A. Wiggins, “Production and uses of diffracionless beams,” J. Opt. Soc. Am. A |

6. | Z. Li, K. B. Alici, H. Caglayan, and E. Ozbay, “Generation of an axially asymmetric Bessel-like beam from a metallic subwavelength aperture,” Phys. Rev. Lett. |

7. | W. B. Williams and J. B. Pendry, “Generating Bessel beams by use of localized modes,” J. Opt. Soc. Am. A |

8. | A. Mazzinghi, M. Balma, D. Devona, G. Guarnieri, G. Mauriello, M. Albani, and A. Freni, “Large depth of field pseudo-Bessel beam generation with a RLSA antenna,” IEEE Trans. Antennas Propag. (to be published). |

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

10. | J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. A |

11. | M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Near field diffraction of Hankel beams,” in |

12. | M. Ettorre and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes,” IEEE Trans. Antennas Propag. |

13. | M. Ettorre, S. M. Rudolph, and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes: experimental validation,” IEEE Trans. Antennas Propag. |

14. | M. F. Inami and A. Grbic, “Generating Bessel beams using an electrically-large annular slot,” in |

15. | S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. |

16. | S. Chávez-Cerda and G. H. C. New, “Evaluation of focused Hankel waves and Bessel beams,” Opt. Commun. |

17. | J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. |

18. | L. B. Felsen and N. Marcuvitz, |

19. | R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag. |

20. | M. Albani, A. Mazzinghi, and A. Freni, “Automatic design of CP-RLSA antennas,” IEEE Trans. Antennas Propag. |

21. | M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. C. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antennas Propag. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1960) Diffraction and gratings : Diffraction theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 14, 2014

Revised Manuscript: June 12, 2014

Manuscript Accepted: June 17, 2014

Published: July 22, 2014

**Citation**

M. Albani, S. C. Pavone, M. Casaletti, and M. Ettorre, "Generation of non-diffractive Bessel beams by inward cylindrical traveling wave aperture distributions," Opt. Express **22**, 18354-18364 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18354

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### References

- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499 (1987). [CrossRef] [PubMed]
- M. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol.24, 315–321 (1992). [CrossRef]
- J. Arlt and K. Dholakia, “Generation of high-order Bessel-beams by use of an axicon,” Opt. Commun.177, 297–301 (2000). [CrossRef]
- R. M. Herman and T. A. Wiggins, “Production and uses of diffracionless beams,” J. Opt. Soc. Am. A8, 932–942 (1991). [CrossRef]
- Z. Li, K. B. Alici, H. Caglayan, and E. Ozbay, “Generation of an axially asymmetric Bessel-like beam from a metallic subwavelength aperture,” Phys. Rev. Lett.102, 143901 (2009). [CrossRef] [PubMed]
- W. B. Williams and J. B. Pendry, “Generating Bessel beams by use of localized modes,” J. Opt. Soc. Am. A22, 992–997 (2005). [CrossRef]
- A. Mazzinghi, M. Balma, D. Devona, G. Guarnieri, G. Mauriello, M. Albani, and A. Freni, “Large depth of field pseudo-Bessel beam generation with a RLSA antenna,” IEEE Trans. Antennas Propag. (to be published).
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25, 1493–1495 (2000). [CrossRef]
- J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. A62, 4261 (2000).
- M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Near field diffraction of Hankel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JSuA36. [CrossRef]
- M. Ettorre and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes,” IEEE Trans. Antennas Propag.60, 3605–3613 (2012). [CrossRef]
- M. Ettorre, S. M. Rudolph, and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes: experimental validation,” IEEE Trans. Antennas Propag.60, 2645–2653 (2012). [CrossRef]
- M. F. Inami and A. Grbic, “Generating Bessel beams using an electrically-large annular slot,” in Proceedings of IEEE AP-S/URSI-USNC Symposium (IEEE2013).
- S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt.46, 923–930 (1999).
- S. Chávez-Cerda and G. H. C. New, “Evaluation of focused Hankel waves and Bessel beams,” Opt. Commun.181, 369–377 (2000). [CrossRef]
- J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett.109, 093904 (2012). [CrossRef] [PubMed]
- L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Series on Electromagnetic Wave Theory (IEEE Press, 1994). [CrossRef]
- R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag.52, 3373–3389 (2004). [CrossRef]
- M. Albani, A. Mazzinghi, and A. Freni, “Automatic design of CP-RLSA antennas,” IEEE Trans. Antennas Propag.60, 5538–5547 (2012). [CrossRef]
- M. Ettorre, M. Casaletti, G. Valerio, R. Sauleau, L. Le Coq, S. C. Pavone, and M. Albani, “On the near-field shaping and focusing capability of a radial line slot array,” IEEE Trans. Antennas Propag.62, 1991–1999 (2014). [CrossRef]

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