## General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields

Optics Express, Vol. 10, Issue 18, pp. 949-959 (2002)

http://dx.doi.org/10.1364/OE.10.000949

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### Abstract

A novel decomposition of the transversal part of the electric field vector of a general non-paraxial electromagnetic field is presented, which is an extension of the radial/aximuthal decomposition and is known as γζ decomposition. Purely γ and ζ polarized fields are examined and the decomposition is applied to propagation-invariant, rotating, and self-imaging electromagnetic fields. An experimental example on the effect of state of polarization in the propagation characteristics of the field: its is shown that a simple modification of the polarization conditions of the angular spectrum converts a self-imaging field into a propagation-invariant field.

© 2002 Optical Society of America

## 1 Introduction

8. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. **85**, 159–161 (1991). [CrossRef]

9. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. **2**, 51–60 (1993). [CrossRef]

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

11. Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. **43**, 1905–1920 (1996). [CrossRef]

9. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. **2**, 51–60 (1993). [CrossRef]

12. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express **9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

13. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. **46**, 2355–2369 (1998). [CrossRef]

14. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A **18**, 1612–1617 (2001). [CrossRef]

## 2 The γζ decomposition of the transverse electric field

*= (*

**r***x*,

*y*,

*z*) and the wave vector

*= (*

**k***k*

_{x},

*k*

_{y},

*k*

_{z}) in circular cylindrical coordinates by using the relations

*k*= |

*| is the wave number. The angular spectrum*

**k****(**

*A**α*,

*Ψ*) is obtained by Fourier-inversion at

*z*= 0:

*may be assumed to be independent. The third is obtained from the Maxwell’s divergence equation*

**E***E*

_{x}

*x̂*+

*E*

_{y}

*ŷ*into two components by the operation

*q*is an integer. The longitudinal component of the field vector is retained as original. Equation (7) clearly defines also a new pair of unit vectors, denoted by

*ϕ*coordinate.

*E*

_{ζ}

^{(q)}are equivalent to the basis vectors discussed recently in Ref. [14

14. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A **18**, 1612–1617 (2001). [CrossRef]

*q*, since they experience a rotation of

*q*2π within 0 ≤

*ϕ*≤ 2π. In the case

*q*= 1 the basis vectors are rotationally symmetric and are customarily called the radial and azimuthal components denoted by

*E*

_{ρ}and

*E*

_{ϕ}, respectively. For values

*q*> 1 the basis vectors experience several full rotations in the counterclockwise direction. On the other hand, for negative values of

*q*the direction of rotation of the basis vectors is reversed. For example, if

*q*= -2 the basis vectors experience a rotation of - 4

*π*radians in the clockwise direction when 0 ≤

*ϕ*≤ 2

*π*. In the case

*q*= 0 the basis vectors represent unmodified cartesian vectors. An example illustrating the directions of the basis vectors with values

*q*= 1 and

*q*= -2 is given in Fig. 1.

8. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. **85**, 159–161 (1991). [CrossRef]

9. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. **2**, 51–60 (1993). [CrossRef]

*J*

_{m}denotes the Bessel function of the first kind and of order

*m*, and defined the functions

*j*=

*x*or

*y*.

## 3 Purely γ- or ζ-polarized fields

17. R. H. Jordan and D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A **12**, 84–94 (1995). [CrossRef]

18. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. **49**, 1537–1543 (2002). [CrossRef]

*q*≠ 0, since in the case

*q*= 0 the field representation is clearly given by Eq. (3).

*ρ*and

*z*, we arrive at the requirement that the integrand must vanish identically:

*m*. However, because the functions

*J*

_{m-q}and

*J*

_{m+q}are linearly independent for all

*m*≠ 0, Eq. (13) implies that only the functions

*m*= ±

*q*may have non-zero values. In addition, these functions are connected by the relations

*q*= 1. This is seen also by inserting Eqs. (3) and (14) into Eq. (6), which yields an propagation formula for the longitudinal component of the field:

*ρ*,

*ϕ*,

*z*) clearly depends on the

*ϕ*-coordinate for all

*q*= 1.

*E*

_{γ}(

*ρ*,

*ϕ*,

*z*) ≡ 0 and then repeating similar steps as above. The results are

*J*

_{1}subfields with different radii. This is a generalization of the result given by Jordan and Hall [17

17. R. H. Jordan and D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A **12**, 84–94 (1995). [CrossRef]

*q*= 1 assuming that the azimuthally polarized field is rotationally symmetric. The derivation introduced above shows that the rotationally symmetric case examined by Jordan and Hall is the only possible one, which explains the results reported recently by Lapucci and Ciofini [19

19. A. Lapucci and M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express **9**, 603–609 (2001),http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-603. [CrossRef] [PubMed]

*γ*or the ζ component of the field vanishes. In all other cases both components are superpositions of Bessel functions with different orders. However, the linearity of Maxwell’s equations implies that the results hold also for all linear combinations of

*ρ*,

*ϕ*,

*z*) and

*ρ*,

*ϕ*,

*z*).

## 4 Paraxial propagation-invariant electromagnetic fields

*w*(

*r*,

*t*)〉 remains exactly the same in every transversal plane

*z*= constant [9

**2**, 51–60 (1993). [CrossRef]

18. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. **49**, 1537–1543 (2002). [CrossRef]

*z*. The condition (21) may also be expressed in the form

*x*- and

*y*-components may be constructed rather easily [9

**2**, 51–60 (1993). [CrossRef]

18. J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. **49**, 1537–1543 (2002). [CrossRef]

*z*-component, the total energy density of this kind of field must remain constant upon propagation [18

**49**, 1537–1543 (2002). [CrossRef]

*z*-component. However, as can be easily verified by inserting Eq. (6) into (3), the condition that the

*z*-component disappears leads immediately to an azimuthally polarized angular spectrum, i.e.,

*A*

_{x}(

*ρ*,

*Ψ*) cos

*Ψ*+

*A*

_{y}(

*ρ*,

*Ψ*) sin

*Ψ*= 0 [20

20. J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express **9**, 622–630 (2001), http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622. [CrossRef] [PubMed]

**49**, 1537–1543 (2002). [CrossRef]

21. J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. **192**, 13–18 (2001). [CrossRef]

*γ*and

*ζ*components of the field are propagation-invariant, i.e.,

*ξ*and

*ν*are as-yet arbitrary real functions. In view of Eqs. (15) and (18), the field expressions are of the form

*and*

*c*_{γ}*c*

_{ζ}are arbitrary complex constants and

*α*

_{Γ}and

*α*

_{ζ}denote the constant

*α*

_{0}for the

*γ*and

*ζ*components, respectively.

*z*-component of the field is obtained straightforwardly by using Eqs. (17) and (19) and it takes the form

_{γ}≠

*α*

_{ζ}the

*z*-component is modified upon propagation in almost all cases, although both

*γ*and

*ζ*components are propagation-invariant. Thus, the electric energy density

*w*

_{e}

*∝*∥

**E**(

*ρ*,

*ϕ*,

*z*)∥

^{2}(and also the magnetic energy density) is not generally propagation-invariant, but instead self-imaging. Although there exist several special cases, in which the modifications of the electric and magnetic energy densities cancel each other [9

**2**, 51–60 (1993). [CrossRef]

20. J. Tervo and J. Turunen, “Self-imaging of electromagnetic fields,” Opt. Express **9**, 622–630 (2001), http: //www. opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622. [CrossRef] [PubMed]

*α*

_{γ}or

*α*

_{ζ}is equal to zero [9

**2**, 51–60 (1993). [CrossRef]

*q*= 1, i.e., when

*E*

_{γ}and

*E*

_{ζ}represent radial and azimuthal components, respectively [18

**49**, 1537–1543 (2002). [CrossRef]

*α*

_{j}/

*β*

_{j}→ 0 the contribution to the energy density from the

*z*-component of the field may be neglected and the field becomes propagation-invariant.

## 5 Experimental results on propagation-invariance

1. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. **57**, 772–778 (1967). [CrossRef]

*θ*, directly affects the polarization state of the field. If a linearly polarized input field is used, the angle

*θ*=

*π*/4 converts an

*x*-polarized wave into a

*y*-polarized wave and

*vice versa*. On the other hand, if the optical axis is parallel (or perpendicular) to the direction of polarization, the polarization state is not modified. These cases are illustrated in Fig. 3.

*f*= 1000 mm.

## 6 Rotating intensity distributions

*γ*and

*ζ*polarized fields may be used to produce not only propagation-invariant fields, but rotating fields as well. The condition for rotation of an electromagnetic field can be defined as [12

12. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express **9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

*z*. Here

*η*is a constant that defines both the direction of propagation and the self-imaging distance. Contrary to the case of propagation-invariant fields, the energy-density distributions of electromagnetic extensions of rotating scalar fields do not generally rotate [12

12. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express **9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

13. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. **46**, 2355–2369 (1998). [CrossRef]

*x*- and

*y*- components of the field to form rotating intensity distributions. However, the task of finding that kind of fields directly from Eq. (28) appears to be a very demanding task and hence some other method must be used. In the following, we use the

*γζ*decomposition to find a class of such solutions.

*ρ*,

*ϕ*, Δ

*z*) is an arbitrary real function, rotates. Inserting this requirement into Eqs. (8) and (9) yields, similarly to the case of exactly rotating fields [12

**9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

*s*is a sign function which can assume values

*s*= -1 or

*s*= +1 for each

*m*, and

*β*

_{0}is a constant. Naturally, in the cases

*q*= 0 or

*q*= 1, we obtain the rotation conditions for scalar or electromagnetic fields, respectively [12

**9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

*m*leading to real values of

*β*

_{ms}, in view of Eqs. (4) and (31). The coefficients

*a*

_{ms}are defined by

*a*

_{ms}=

_{ms})/

*α*

_{ms}. An example of the field satisfying Eqs. (30) and (31) is illustrated in Fig. 4. It can be clearly seen that the

*x*-component of the field is not rotating. However, the

*γ*-component, as well as the

*ζ*-component (which is not shown), do rotate. This means that the contribution to the intensity form the transversal part of the field rotates.

*q*= 1, which provides also rotating

*z*-components, of course assuming that both conditions (30) and (31) are met. Now the

*z*-component of the electric field takes the form [12

**9**, 9–15 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9. [CrossRef] [PubMed]

*m*in Eqs. (32) and (33), we find that the transversal part of the field takes the form

## 7 Conclusions

## Acknowledgements

## References and links

1. | W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. |

2. | K. Patorski, “The self-imaging phenomenon and its applications,” in |

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

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

5. | Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagating with rotating intensity distributions,” Phys. Rev. E |

6. | S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. |

7. | C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. |

8. | S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. |

9. | J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. |

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

11. | Z. Bouchal, R. Horák, and J. Wagner, “Propagation-invariant electromagnetic fields,” J. Mod. Opt. |

12. | J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express |

13. | P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotl-yar, V. A. Soifer, and A. T. Friberg, “Rotating optical fields: experimental demonstration with diffractive optics,” J. Mod. Opt. |

14. | F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A |

15. | L. Mandel and E. Wolf, |

16. | G. B. Arfken and H. J. Weber, |

17. | R. H. Jordan and D. G. Hall, “Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,” J. Opt. Soc. Am. A |

18. | J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. |

19. | A. Lapucci and M. Ciofini, “Polarization state modifications in the propagation of high azimuthal order annular beams,” Opt. Express |

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

21. | J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. |

**OCIS Codes**

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 31, 2002

Revised Manuscript: August 30, 2002

Published: September 9, 2002

**Citation**

Pertti Paakkonen, Jani Tervo, Pasi Vahimaa, Jari Turunen, and Franco Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express **10**, 949-959 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-18-949

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### References

- W. D. Montgomery, ???Self-imaging objects of in.nite aperture,??? J. Opt. Soc. Am. 57, 772???778 (1967). [CrossRef]
- K. Patorski, ???The self-imaging phenomenon and its applications,??? in Progress in Optics Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1.
- J. Durnin, ???Exact solutions for nondifiracting beams. I. The scalar theory,??? J. Opt. Soc. Am. A 4, 651???654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, ???Diffraction-free beams,??? Phys. Rev. Lett. 58, 1499???1501 (1987). [CrossRef] [PubMed]
- Y. Y. Schechner, R. Piestun, and J. Shamir, ???Wave propagating with rotating intensity distributions,??? Phys. Rev. E 54, R50???R53 (1996). [CrossRef]
- S. Chavez-Cerda, G. S. McDonald, and G. H. S. New, ???Nondiffracting Beams: travelling, standing, rotating and spiral waves,??? Opt. Commun. 123, 225???233 (1996). [CrossRef]
- C. Paterson and R. Smith, ???Higher-order Bessel waves produced by axicon-type computergenerated holograms,??? Opt. Commun. 124, 121???130 (1996). [CrossRef]
- S. R. Mishra, ???A vector wave analysis of a Bessel beam,??? Opt. Commun. 85, 159???161 (1991). [CrossRef]
- J. Turunen and A. T. Friberg, ???Self-imaging and propagation-invariance in electromagnetic fields,??? Pure Appl. Opt. 2, 51???60 (1993). [CrossRef]
- Z. Bouchal and M. Olivýk, ???Non-diffractive vector Bessel beams,??? J. Mod. Opt. 42, 1555???1566 (1995). [CrossRef]
- Z. Bouchal, R. Horak and J. Wagner, ???Propagation-invariant electromagnetic fields,??? J. Mod. Opt. 43, 1905???1920 (1996). [CrossRef]
- J. Tervo and J. Turunen, ???Rotating scale-invariant electromagnetic fields,??? Opt. Express 9, 9???15 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-1-9</a>. [CrossRef] [PubMed]
- P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer and A. T. Friberg, ???Rotating optical fields: experimental demonstration with diffractive optics,??? J. Mod. Opt. 46, 2355???2369 (1998). [CrossRef]
- F. Gori, ???Polarization basis for vortex beams,??? J. Opt. Soc. Am. A 18, 1612???1617 (2001). [CrossRef]
- L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), Sect. 3.2.
- G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
- R. H. Jordan and D. G. Hall, ???Highly directional surface emission from concentric-circle gratings on planar optical waveguides: the field-expansion method,??? J. Opt. Soc. Am. A 12, 84???94 (1995). [CrossRef]
- J. Tervo, P. Vahimaa, and J. Turunen, ???On propagation-invariant and self-imaging intensity distributions of electromagnetic fields,??? J. Mod. Opt. 49, 1537???1543 (2002). [CrossRef]
- A. Lapucci and M. Ciofini, ???Polarization state modifications in the propagation of high azimuthal order annular beams,??? Opt. Express 9, 603???609 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-12-603">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-12-603</a>. [CrossRef] [PubMed]
- J. Tervo and J. Turunen, ???Self-imaging of electromagnetic fields,??? Opt. Express 9, 622???630 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-12-622</a>. [CrossRef] [PubMed]
- J. Tervo and J. Turunen, ???Generation of vectorial propagation-invariant fields by polarizationgrating axicons,??? Opt. Commun. 192, 13???18 (2001). [CrossRef]

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