## The evanescent wavefield part of a cylindrical vector beam |

Optics Express, Vol. 21, Issue 19, pp. 22246-22254 (2013)

http://dx.doi.org/10.1364/OE.21.022246

Acrobat PDF (1861 KB)

### Abstract

The evanescent wave of the cylindrical vector field is analyzed using the vector angular spectrum of the electromagnetic beam. Comparison between the contributions of the TE and TM terms of both the propagating and the evanescent waves associated with the cylindrical vector field in free space is demonstrated. The physical pictures of the evanescent wave and the propagating wave are well illustrated from the vectorial structure, which provides a new approach to manipulating laser beams by choosing the states of polarization in the cross-section of the field.

© 2013 OSA

## 1. Introduction

1. Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**(1), 1–57 (2009). [CrossRef]

3. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**(1), 149–157 (2012). [CrossRef] [PubMed]

4. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express **10**(7), 324–331 (2002). [CrossRef] [PubMed]

5. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [CrossRef] [PubMed]

6. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. **105**(25), 253602 (2010). [CrossRef] [PubMed]

7. G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. **17**(5), 760–776 (1976). [CrossRef]

13. A. V. Shchegrov and P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A **16**, 2583–2584 (1999). [CrossRef]

14. M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. **60**(7), 3155–3164 (2012). [CrossRef]

16. M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. **95**(11), 111107 (2009). [CrossRef]

17. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science **317**(5840), 927–929 (2007). [CrossRef] [PubMed]

18. A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science **320**(5875), 511–513 (2008). [CrossRef] [PubMed]

19. L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A **78**(1), 013819 (2008). [CrossRef]

21. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express **16**(5), 2845–2858 (2008). [CrossRef] [PubMed]

24. B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express **20**(16), 17684–17694 (2012). [CrossRef] [PubMed]

## 2. Theoretical formulation

*z*-axis is taken to be the direction of wave propagation. A cylindrical optical vector field can be described as [1

1. Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**(1), 1–57 (2009). [CrossRef]

3. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**(1), 149–157 (2012). [CrossRef] [PubMed]

*m*is the topological charge, and θ

_{0}is the initial phase.

*A*(r) represents the amplitude distribution in the cross-section of the cylindrical vector field. The time dependent factor

*m*= 1 with θ

_{0}= 0 and π/2, the vector fields correspond to the radially and azimuthally polarized vector fields, respectively. When

*m*= 0, Eq. (1) degenerates to the horizontal (for θ

_{0}= 0) and vertical (for θ

_{0}= π/2) linearly polarized fields, respectively.

*I*

_{pr}and

*I*

_{ev}, are calculated respectively [19

19. L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A **78**(1), 013819 (2008). [CrossRef]

21. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express **16**(5), 2845–2858 (2008). [CrossRef] [PubMed]

*ø*from zero to 2π yield a single integration of ρ. Then, the corresponding results can be obtained by performing numerical integration over ρ. The ratio

*δ*= (

*I*

_{pr}-

*I*

_{ev})/

*I*

_{pr}provides the direct information about the propagating and the evanescent components of the field.

*δ*= (

*I*

_{pr}-

*I*

_{ev})/

*I*

_{pr}for w = 0.1λ (highly nonparaxial case) and w = 0.5λ cases as a function of different distances

*z*from the initial plane

*z*= 0 are shown in Fig. 1. It can be clearly seen that the evanescent field dominates near the source plane and the relative weight of

*I*_{ev}would drastically decrease with the increase of the propagation distance. It becomes negligible in the propagation distance z = 0.5λ as shown in Fig. 1. Comparing Figs. 1(a) with 1(b), one can recognize that the relative weight of

*I*_{ev}would reduce when the waist size

*w*increases, i.e., the relative weight of the evanescent wave component is increasing with increasing nonparaxial feature as the waist size w decreases. As it is expected, bigger values of the waist size

*w*leads to reduced nonparaxial behavior. The transverse and the longitudinal components of the evanescent wave become negligible with larger waist size

*w*and hence the relative weight of

*I*_{ev}would be smaller. Furthermore, it can also be found from Fig. 1 that the relative weight of

*I*_{ev}would increase with the increase of the topological charge. It can be explained by the fact that the field distribution will increasingly diverge and extend from the center of beam with the increasing topological charge. As a result, the relative weight of

*I*

_{ev}would increase with the increasing topological charge under the same conditions and beam parameters. The calculation results indicate that the evanescent wave component with different initial phase θ

_{0}for any topological charge (except m = 1) is the same as shown in Fig. 1. For m = 1 case with different initial phase θ

_{0}, the relative weight of

*I*_{ev}of the azimuthal polarization (m = 1, θ = π/2) is the maximum and the relative weight of

*I*_{ev}of the radial polarization (m = 1, θ = 0) is the minimum as shown in Fig. 1. The physical explanation for the exception m = 1 is that the values of the z-components of either the propagation wave or the evanescent wave are different for different initial phase θ

_{0}, and especially the values of the z-components of either the propagation wave or the evanescent wave are the maximum for θ

_{0}= π/2 (i.e., radial polarization) whereas the z-components of either propagation wave or evanescent wave are zero for θ

_{0}= 0 (i.e., azimuthal polarization), which can also be recognized from Eqs. (5)-(8).

*w*and the propagation distance

*z*, we will intuitively show the contribution of the evanescent and the propagation waves to the optical field. Examples of the intensity distributions of the TE and TM components for the evanescent and the propagation waves are depicted in Figs. 2, 3, and 4. The value of

*w*is set to be 0.1λ for the highly nonparaxial case. The plane z = 0.2λ is selected as the reference plane. θ is considered to be π/4 for m = 0, 1, and 2, respectively. Apparently, the TE and TM terms are orthogonal to each other in the field. The intensity distribution profiles of the TE and TM components of the evanescent and the propagation terms are similar, but the magnitude of intensity for the propagation term is smaller than that of the evanescent term. This verifies that the evanescent field dominates near the source plane and the contribution of the evanescent wave to the cylindrical vector field is considerable in magnitude. From Fig. 2 and Fig. 4, for m = 0 and m = 2 with w = 0.1λ and propagation distance z = 0.2λ, it can be found that the intensity distribution profiles of both the evanescent and the propagation waves are composed of a pattern with two peaks and some pairs of side lobes. Moreover, the intensities of the side lobes are much lower than those of the peaks. In these cases, the magnitude of the TM term is greater than that of the TE term. As analyzed above, the magnitude of the intensities of the evanescent wave and its two terms (TE and TM) decrease with the increase of the waist width

*w*and the propagation distance z. The two peaks will superimpose as a Gaussian distribution. The numerical results indicate that the two peaks will superimpose as a Gaussian distribution for m = 0 and the propagation distance z = 0.2λ when w = 0.5λ as described in Ref [30

30. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express **16**(6), 3504–3514 (2008). [CrossRef] [PubMed]

_{0}= π/4, w = 0.1λ, z = 0.2λ. These patterns resemble the doughnut shape with a dark spot in the center and some bright rings around it. Such a field has important applications in optical trapping and manipulating of nano-objects [3

3. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**(1), 149–157 (2012). [CrossRef] [PubMed]

## 3. Conclusion

*w*and drastically decrease with the increasing propagation distance. The intensity distribution profiles of the evanescent term are similar to those of the propagating term, but the magnitude of the intensities of the propagating term is different from that of the evanescent term. The intensity distribution is different with the increasing waist width w or propagation distance z.

## Acknowledgments

## References and links

1. | Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

2. | X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express |

3. | B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express |

4. | Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express |

5. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. |

6. | X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. |

7. | G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. |

8. | F. De Fornel, |

9. | G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A |

10. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

11. | A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. |

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

13. | A. V. Shchegrov and P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A |

14. | M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag. |

15. | A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE |

16. | M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett. |

17. | R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science |

18. | A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science |

19. | L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A |

20. | R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B |

21. | R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express |

22. | J. W. Goodman, |

23. | K. E. Okan, |

24. | B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express |

25. | R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A |

26. | H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express |

27. | G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. |

28. | K. Duan and B. Lü, “Polarization properties of vectorial nonparaxial Gaussian beams in the far field,” Opt. Lett. |

29. | R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams |

30. | G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(260.0260) Physical optics : Physical optics

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 10, 2013

Revised Manuscript: August 19, 2013

Manuscript Accepted: September 5, 2013

Published: September 13, 2013

**Citation**

Rui-Pin Chen and Guoqiang Li, "The evanescent wavefield part of a cylindrical vector beam," Opt. Express **21**, 22246-22254 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22246

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### References

- Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
- X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express18(10), 10786–10795 (2010). [CrossRef] [PubMed]
- B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express20(1), 149–157 (2012). [CrossRef] [PubMed]
- Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
- X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010). [CrossRef] [PubMed]
- G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys.17(5), 760–776 (1976). [CrossRef]
- F. De Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics (Springer. 2001).
- G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A27(3), 1693–1695 (1983). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A16(6), 1381–1386 (1999). [CrossRef]
- A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun.136(1-2), 114–124 (1997). [CrossRef]
- D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett.21(1), 9–11 (1996). [CrossRef] [PubMed]
- A. V. Shchegrov and P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A16, 2583–2584 (1999). [CrossRef]
- M. F. Imani and A. Grbic, “Generating evanescent Bessel beams using near-field plates,” IEEE Trans. Antenn. Propag.60(7), 3155–3164 (2012). [CrossRef]
- A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE99(10), 1806–1815 (2011). [CrossRef]
- M. F. Imani and A. Grbic, “Tailoring near-field patterns with concentrically corrugated plates,” Appl. Phys. Lett.95(11), 111107 (2009). [CrossRef]
- R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science317(5840), 927–929 (2007). [CrossRef] [PubMed]
- A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffraction focusing with patterned surfaces,” Science320(5875), 511–513 (2008). [CrossRef] [PubMed]
- L. E. Helseth, “Radiationless electromagnetic interference shaping of evanescent cylindrical vector waves,” Phys. Rev. A78(1), 013819 (2008). [CrossRef]
- R. Martínez-Herrero, P. M. Mejías, I. Juvells, and A. Carnicer, “Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields,” Appl. Phys. B106(1), 151–159 (2012). [CrossRef]
- R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express16(5), 2845–2858 (2008). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics. (Greenwood Village: Roberts and Company, 2004).
- K. E. Okan, Diffraction, Fourier Optics and Imaging. (Wiley & Sons, 2007).
- B. Gu and Y. Cui, “Nonparaxial and paraxial focusing of azimuthal-variant vector beams,” Opt. Express20(16), 17684–17694 (2012). [CrossRef] [PubMed]
- R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A18(7), 1678–1680 (2001). [CrossRef] [PubMed]
- H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express14(6), 2095–2100 (2006). [CrossRef] [PubMed]
- G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett.31(17), 2616–2618 (2006). [CrossRef] [PubMed]
- K. Duan and B. Lü, “Polarization properties of vectorial nonparaxial Gaussian beams in the far field,” Opt. Lett.30(3), 308–310 (2005). [CrossRef] [PubMed]
- R. P. Chen and K. H. Chew, “Far field properties of a vortex Airy beam,” Laser and Particle Beams31(01), 9–15 (2013). [CrossRef]
- G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express16(6), 3504–3514 (2008). [CrossRef] [PubMed]

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