## Evanescent field of vectorial highly non-paraxial beams

Optics Express, Vol. 16, Issue 5, pp. 2845-2858 (2008)

http://dx.doi.org/10.1364/OE.16.002845

Acrobat PDF (344 KB)

### Abstract

In terms of the Fourier spectrum, a simple but general analytical expression is given for the evanescent field associated to a certain kind of non-paraxial exact solutions of the Maxwell equations. This expression enables one to compare the relative weight of the evanescent wave with regard to the propagating field. In addition, in those cases in which the evanescent term is significant, the magnitude of the field components across the transverse profile (including the evanescent features) can be determined. These results are applied to some illustrative examples.

© 2008 Optical Society of America

## 1. Introduction

^{1–27}In fact, non-paraxial fields have revealed to be very useful in Optics, as occurs, for example, in high-resolution microscopy, particle trapping, high-density recording and tomography.

^{24}At the waist plane, the transverse size of such strongly-focused beams is smaller than the wavelength. Usually, the field amplitude is evaluated far enough from the waist in order to neglect the contribution of the evanescent waves. However, this kind of waves can be of considerable interest because of the possibility of subwavelength resolution, beyond the diffraction limit.

^{3,6,12}or by using the complex-source-point mode

^{10}). Among them, several kind of representations of an electromagnetic beam based on its angular plane-wave spectrum have been reported in recent years.

^{13,19,23,25,26,27}Such kind of decomposition is particularly useful because it enables to separate the contribution of the propagating and evanescent waves

^{8,9,20,21,26}(also called homogeneous and inhomogeneous parts of the spectrum

^{28}). When the contribution of the evanescent waves is negligible, it has been shown

^{13,19,25}that the electric field solution can be written as the sum of two terms: one of them is transverse to the propagation direction; another one exhibits a non-zero longitudinal component and its associated magnetic field is also transverse. This analytical structure differs from alternative proposals also based on the angular spectrum (see, for example, Ref. 26

26. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A **23**, 3197–3202 (2006). [CrossRef]

^{26}same section. Finally, the main conclusions are summarized in Section 5.

## 2. Formalism and key definitions

*and*

**E***represent the spatial structure of the electric and magnetic fields, respectively,*

**H***k*=

*nω*/

*c*is the wavenumber,

*n*the refractive index of the medium,

*ω*the angular frequency of the radiation and

*c*the speed of light in vacuum. As is well known, the fields

*and*

**E***can be expressed in terms of their angular plane-wave spectrum,*

**H***. In Eq. (2)*

**H***denotes the spatial Fourier transform of*

**Ẽ***(the same would apply for*

**E***). Although*

**H̃***x*,

*y*and

*z*should be considered, in principle, as equivalent directions, for simplicity we will choose

*z*as the direction of propagation of the beam. Moreover, in the present paper we will be concerned with the values of the field at planes transverse to this axis.

*≡(i*

**L***ku*, i

*kv*, ∂/∂z). Instead of using Cartesian coordinates, for the sake of convenience we will next handle cylindrical coordinates

*R*,

*θ*and

*z*, along with polar coordinates,

*ρ*and

*ϕ*for the transverse Fourier-transform variables u, v. Thus we have

**σ**=(

*σ*,

_{x}*σ*,

_{y}*σ*)=(

_{z}*ρ*cos

*ϕ*,

*ρ*sin

*ϕ*,

*ξ*). It is clear from Eq. (8.b) that the third component of

*would be, in general, a complex number whose physical meaning will be apparent later. Note also that condition (6) is a direct consequence of the Maxwell equation*

**σ****∇**·

*=0.*

**E***should fulfill condition (6).*

**Ẽ**_{0}## 3. Field decomposition into propagating and evanescent waves

**, given by Eq. (9), as the sum of two terms**

*E*(*R*,*θ*,*z*)^{13,19,25}and write the field

*(*

**E***R*,

*θ*,

*z*), given by Eq. (10), in the alternative equivalent form

*. Note that the vectors*

**E͂**_{0}*and*

**s****are connected through the Maxwell’s divergence equation (condition (6)) in the interval**

*E͂*_{0}*ρ*∈[0, 1]. The vectors

**e**_{1}and

**e**_{2}can be considered as reference axes with respect to which the components of

**can be given.**

*E͂*_{0}

**s**_{ev}(see Eq. (14.b) assures the fulfilment of condition (6) when

*ρ*∈[1,∞]. In addition, for this range of values of

*ρ*, vectors

*e*

_{1}and

*e*

_{ev}constitute now the two-dimensional reference basis. It should be noted that the choice of the above reference vectors, namely,

*e*

_{1},

*e*

_{2}and

*e*

_{ev}, characterizes the formalism handled in the present paper. A final remark should be added with regard to the notation used to define the inner product: In the present paper,

*·*

**a***should be understood as*

**b***a*

_{x}b_{x}^{*}+

*a*

_{y}b_{y}^{*}+

*a*

_{z}b_{z}^{*}, where * means conjugation.

*a*(

*ρ*,

*ϕ*),

*b*(

*ρ*,

*ϕ*) and

*b*

_{ev}(

*ρ*,

*ϕ*) contain the complete information about the Fourier-transform spectrum of the global field (cf. Eq. (5)): In fact, these functions can be inferred from

**as follows**

*E͂*_{0}^{13,19,25}the first term that appears in the right-hand side of Eq. (12) represents a superposition of plane waves that generate a pure transverse field,

**E**_{TE}, without longitudinal component (

*E*

_{z}=0). The second term,

**E**_{TM}, gives

^{13}an electric field (perpendicular to

**E**_{TE}at the far field) whose associated magnetic field,

**H**_{TM}, is also orthogonal to the propagation axis

*z*. This structure of the propagating field essentially follows our particular choice of the orthogonal reference axes,

**e**_{1},

**e**_{2}and

*. In the present work the novelty arises from the third term, which contains the evanescent part, understood as a superposition of inhomogeneous waves whose constant phase surfaces are planes orthogonal to the transverse (non-unitary) vector*

**s**

**s**_{0}, namely,

*b*(

*ρ*,

*ϕ*) and

*b*

_{ev}(

*ρ*,

*ϕ*) vanish.

*at some initial plane. Furthermore, as we will see in the next section, the field magnitude and polarization features of the transverse profile of any vectorial beam can be computed in a direct way from Eq. (12). In particular, the third term of this equation provides a simple but general procedure to obtain the evanescent field associated to the exact solution given by Eq. (9). In addition, the separate contribution of TE-, TM- and evanescent parts of the electric field allows one to compare the weight of these terms (for example, by comparing the respective square modulus, integrated throughout the beam profile). Consequently, one can determine the ranges of the propagation distance and of the beam size for which the evanescent waves are not negligible. It should also be noted that the evanescent term involves the sum (*

**E͂**_{0}**·**

*E͂*_{0}**)**

*e*_{1}*+(*

**e**_{1}*·*

**E͂**_{0}*)*

**e**_{ev}*, which can be understood as the projection vector*

**e**_{ev}*onto the subspace generated by*

**E͂**_{0}

**e**_{1}and

**e**_{ev}. A similar interpretation led to the concept of “closest” field to a given vector-field solution, introduced not long ago

_{13}for the propagating term

**E**

_{pr}of the general solution.

## 4. Numerical examples

*=0, so it cannot represent an exact solution of the Maxwell equations. In such a model, the vector plane-wave spectrum is represented by the function*

**E***C*is a normalization constant,

*D*=1/

*kω*

_{0}is a constant proportional to the beam divergence at the far field, and

*ω*

_{0}is the 1/e intensity beam radius of the Gaussian factor at the near field. To derive an exact non-paraxial solution based on the paraxial Gaussian model, one can follow alternative procedures. On the one hand (see, for example, Ref. 26

26. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A **23**, 3197–3202 (2006). [CrossRef]

*z*=0 have a Gaussian structure. The angular spectrum of each transverse component would then be obtained by using the inverse Fourier transform. In addition, the plane-wave spectrum associated to the longitudinal component

*E*

_{z}is analytically inferred from Eq. (6). The electric field vector

**(**

*E**)=(*

**r***E*

_{x},

*E*

_{y},

*E*

_{z}) (including its evanescent part) would finally evolve in the

*z*>0 half-space according with well-known integral expressions.

^{26}The transverse components of this solution would therefore exhibit a Gaussian profile across the plane z=0.

*=*

**f͂·e**_{1}*a*(

*ρ*,

*ϕ*);

*f͂*·*e*_{2}=

*b*(

*ρ*,

*ϕ*); and

*f*͂ ·*e*_{ev}=

*b*

_{ev}(

*ρ*,

*ϕ*). We get

*a*,

*b*and

*b*

_{ev}into Eq. (12). This solution should be considered as the generalization (which now includes the evanescent part) of the so-called “closest” field associated to the (paraxial) Gaussian model. Note that our solution should be considered as the field closest (in an algebraic sense) to the Gaussian model (in this connection, recall that functions

*a*,

*b*and

*b*

_{ev}give the projections of

*onto*

**f͂**

**e**_{1},

**e**_{2}and

**e**_{ev}, respectively). Note also that, although the above discussion has considered the Gaussian case, it can immediately be extended to a general function

**.**

*f͂*(*ρ*,*ϕ*)

**E**_{TE}and

**E**_{TM}, associated to the propagating waves, were studied elsewhere:

^{13,19,25}. It was shown that the transverse part (

*E*

_{x},

*E*

_{y}) of the propagating field closely resembles both, the profile irradiance and the polarization distribution of a conventional (paraxial) Gaussian beam (of course, the presence of a longitudinal z-component involves a conceptually important difference with regard to the paraxial model). It should be pointed out that, for the beam size and the propagation distances considered in those papers, the influence of the evanescent waves was negligible. In the present paper, however, we are basically interested on the evanescent field (cf. Eq. 11.b)). Accordingly, we have to choose parametric values for which the evanescent behaviour becomes significant.

**I**

_{pr}-

**I**

_{ev}|/

**I**

_{pr}for

*ω*

_{0}=0.1λ (highly non-paraxial case) at different distances

*z*from the initial plane

*z*=0, where

*I*

_{pr}and

*I*

_{ev}compute the (integrated) squared modulus of the propagating and evanescent fields, respectively. Thus the ratio

*Δ*provides direct information about the importance of the propagating-field solution:

*Δ*ranges from zero (the field is purely evanescent) to 1 (the evanescent term vanishes). For the field we are considering, the evanescent waves are significant enough for propagation distances not longer than, say,

*λ*. Of course, if the waist size

*ω*

_{0}increases, the relative weight of

**E**_{ev}would drastically reduce, as it should be expected. For example, when

*ω*

_{0}≈0.2λ, the distance for noticeable evanescent waves does not exceed 0.1

*λ*.

*ω*

_{0}and

*z*, we will show the evanescent structure involved in the closest-field solution associated to a Gaussian beam.

*ω*

_{0}=0.1

*λ*and

*z*=0.1

*λ*(recall that the angular spectrum is defined by Eqs. (18)). In all the figures, abcises and ordinates correspond to the conventional transverse Cartesian axes x and y, respectively. The length of the side of each square frame is 8

*λ*. Within such frames, the red colour has been associated to the peak value. Furthermore, integration over the entire transverse plane has been normalised to 1 in Fig. 4(d). The rest of the plots are then referred to this value. For example, in Fig. 3(d) the integration gives the value 0.19. This means that the global evanescent field contributes to the overall combined field (propagating + evanescent waves) in a percentage of 19 % (and analogously for the other figures).

*E*|

_{x}^{2})

_{pr}is significant on a region whose transverse size along

*x*is nearly

*λ*. This value reduces to 0.5

*λ*for (|

*E*|

_{x}^{2})

_{ev}in Fig. 3(a). Accordingly, it should be expected that the size of the region over which is significant enough the squared modulus of the x-component of the combined field would closely take the intermediate value ≈0.7λ, in agreement with Fig. 4(a). A similar interpretation applies for the rest of the figures.

*(*

**f͂***π*,

*ϕ*) now reads

*ω*

_{0}and

*z*take the same values as before. The characteristics of the associated closest-field solution are reported in Figs. 5, 6 and 7. As expected, all the graphics (c) and (d) show rotational symmetry. Note also the different spatial structure exhibited by the evanescent wave with regard to the former case (see Figs. 3 and 6). It should also be mentioned the similar contributions (percentages) involved in the linear and circular cases, associated to the z-components (figures (c)) and to the global waves (figures (d)).

## 5. Conclusions

26. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A **23**, 3197–3202 (2006). [CrossRef]

## Acknowledgements

## References and Links

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

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

3. | P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. |

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

5. | S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A |

6. | P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. |

7. | S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beam,” J. Opt. Soc. Am. A |

8. | T. Setala, A. T. Friberg, and M. Kaivola, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E |

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

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

11. | S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A |

12. | C. J. R. Sheppard, “Polarization of almost-planes waves,” J. Opt. Soc. Am. A |

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

14. | J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express |

15. | P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express |

16. | C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilman, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximation,” J. Opt. Soc. Am. A |

17. | R. Borghi, A. Ciattoni, and M. Santarsiero, “Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions,” J. Opt. Soc. Am. A |

18. | A Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. |

19. | P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Progr. Quantum Electron. |

20. | H. F. Arnoldus and J. T. Foley, “Traveling and evanescent fields of an electric point dipole,” J. Opt. Soc. Am. A |

21. | N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. A |

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

23. | G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. |

24. | K. Belkebir, P. C. Chaumet, and A. Sentetac, “Influence of multiple scatering on three-dimensional imaging with optical diffraction tomography,” J. Opt. Soc. Am. A |

25. | R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. |

26. | P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A |

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

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

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 29, 2007

Revised Manuscript: January 10, 2008

Manuscript Accepted: January 24, 2008

Published: February 15, 2008

**Citation**

R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, "Evanescent field of vectorial highly non-paraxial beams," Opt. Express **16**, 2845-2858 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2845

Sort: Year | Journal | Reset

### References

- G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983). [CrossRef]
- D. G. Hall, "Vector-beam solutions of Maxwell’s wave equations," Opt. Lett. 21, 9-11 (1996). [CrossRef] [PubMed]
- P. Varga and P. Török, "Exact and approximate solutions of Maxwell’s equations for a confocal cavity," Opt. Lett. 21, 1523-1525 (1996). [CrossRef] [PubMed]
- A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997) [CrossRef]
- S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998). [CrossRef]
- P. Varga and P. Török, "The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation," Opt. Commun. 152, 108-118 (1998). [CrossRef]
- S. R. Seshadri, "Partially coherent Gaussian Schell-model electromagnetic beam," J. Opt. Soc. Am. A 16, 1373-1380 (1999). [CrossRef]
- T. Setala, A. T. Friberg, and M. Kaivola, "Decomposition of the point-dipole field into homogeneous and evanescent parts," Phys. Rev. E 59, 1200-1206 (1999). [CrossRef]
- 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]
- C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999). [CrossRef]
- S. R. Seshadri, "Average characteristics of partially coherent electromagnetic beams," J. Opt. Soc. Am. A 17, 780-789 (2000). [CrossRef]
- C. J. R. Sheppard, "Polarization of almost-planes waves," J. Opt. Soc. Am. A 17, 335-341 (2000). [CrossRef]
- R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001). [CrossRef]
- J. Tervo and J. Turunen, "Self-imaging of electromagnetic fields," Opt. Express 9, 622-630 (2001). [CrossRef] [PubMed]
- P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen and F. Gori, General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002). [PubMed]
- C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilman and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximation," J. Opt. Soc. Am. A 19, 404-412 (2002). [CrossRef]
- R. Borghi, A. Ciattoni and M. Santarsiero, "Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions," J. Opt. Soc. Am. A 19, 1207-1211 (2002). [CrossRef]
- A Ciattoni, B. Crosignani and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002). [CrossRef]
- P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002). [CrossRef]
- H. F. Arnoldus and J. T. Foley, "Traveling and evanescent fields of an electric point dipole," J. Opt. Soc. Am. A 19, 1701-1711 (2002). [CrossRef]
- N. I. Petrov, "Evanescent and propagating fields of a strongly focused beam, " J. Opt. Soc. Am. A 20, 2385-2389 (2003). [CrossRef]
- K. Duan and B. Lü, "Polarization properties of vectorial nonparaxial Gaussian beam in the far field," Opt. Lett. 30, 308-310 (2005). [CrossRef] [PubMed]
- G. Zhou, X. Chu and L. Zhao, "Propagation characteristics of TM Gaussian beam," Opt. Laser Technol. 37, 470-474 (2005). [CrossRef]
- K. Belkebir, P. C. Chaumet and A. Sentetac, "Influence of multiple scatering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006). [CrossRef]
- R. Martínez-Herrero, P. M. Mejías, S. Bosch and A. Carnicer, "Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams," J. Opt. A: Pure Appl. Opt. 8, 524-530 (2006). [CrossRef]
- P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006). [CrossRef]
- H. Guo, J. Chen and S. Zhuang, "Vector plane spectrum of an arbitrary polarized electromagnetic wave," Opt. Express 14, 2095-2100 (2006). [CrossRef] [PubMed]
- G. C. Sherman, J. J. Stamnes and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760-776 (1976). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.