## Self-imaging of electromagnetic fields

Optics Express, Vol. 9, Issue 12, pp. 622-630 (2001)

http://dx.doi.org/10.1364/OE.9.000622

Acrobat PDF (486 KB)

### Abstract

The electromagnetic theory of self-imaging fields is considered. Several features are presented, which have no counterparts within the scalar theory of self-imaging. For example, the electromagnetic field self-images at one half of the classical self-imaging distance for scalar fields, the electric and magnetic energy densities can self-image while the scalar field components do not, and the self-imaging distances of the electric and magnetic energy densities can be different. In addition, general expressions for TE and TM polarized fields are presented by using the concept of the angular spectrum of the field.

© Optical Society of America

## 1 Introduction

1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1. [CrossRef]

*z*

_{T}. If the wave vectors are confined to only one ring, the field is a conical wave are thereby its intensity distribution is propagation-invariant [2

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

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

10. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E **54**, R50–R53 (1996). [CrossRef]

13. R. Piestun and J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A **15**, 3039–3044 (1998). [CrossRef]

14. J. Tervo and J. Turunen, “Rotating scale-invariant electromagnetic fields,” Opt. Express **9**, 9–15 (2001), http://www.opticsexpress.org/oearchive/source/33955.htm. [CrossRef] [PubMed]

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

*z*-components of the electric and magnetic fields can self-image at one-half of the classical self-imaging distance

*z*

_{T}applicable to scalar fields (sect. 4), and that the same can be true for the electric and magnetic energy densities (sect. 5). Furthermore, we show (sect. 6) that the self-imaging distances of the electric and magnetic energy densities can differ by a factor of two.

## 2 General TE and TM polarized fields

*z*>0 then the general solution of the Helmholtz wave equation may be expressed as the sum of plane waves known as the angular spectrum representation of the electric field [15]. If we express both the position vector

*r*and the wave vector

*k*in circular cylindrical coordinates, i.e.,

*r*=(

*ρ*,

*ϕ*,

*z*) and

*k*=(

*α*,

*ψ*,

*β*), then the electric field takes the form

*z*=0,

*z*-direction to be the predominant direction of wave propagation, it is a natural choice to express the

*z*-component in terms of the

*x*and

*y*components. By inserting Eq. (1) into Eq. (4), we obtain

*(*

**H***r*) is obtained by using the curl equation

*ω*is the angular frequency and

*µ*

_{0}denotes the magnetic permeability of vacuum. If we denote the angular spectrum of the magnetic field by

*B*(

*α*,

*ψ*) and form an expression of

*(*

**H***r*) analogous to Eq. (1) then, by combining Eqs. (1) and (6), we obtain

*∊*

_{0}is the vacuum permittivity.

*A*

_{α}(

*α*,

*ψ*) and

*A*

_{ψ}(

*α*,

*ψ*) by a rotation operation

*A*

_{α}and

*A*

_{ψ}represent the radial and azimuthal components of the angular spectrum, respectively (see Fig. 1 for the definitions). If the radial component vanishes, we call the angular spectrum azimuthally polarized and

*vice versa*. By examining Eqs. (5) and (9), we immediately notice that if

*A*

_{α}(

*α*,

*ψ*)≡0, the

*z*-component of the electric field vanishes. On the other hand, by Fourier-analysis, it is clear that this conclusion holds also for the opposite and thus we have a simple relation

*z*=constant, the electric field vector is parallel to the interface at each point.

*A*

_{ψ}(

*α*,

*ψ*)≡0, we have a similar relation

16. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A **15**, 1139–1143 (1998). [CrossRef]

## 3 Self-imaging fields

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

*ω*(

*r*,

*t*) is the total energy density of the field,

*t*denotes time, and brackets indicate the time-average. Here

*z*

_{T}is the self-imaging distance, frequently referred to as the Talbot distance because Talbot was the first to observe the phenomenon of self-imaging [17]. The time-averaged energy density is of the form [18]

*ω*

_{e}(

*r*,

*t*) and

*ω*

_{h}(

*r*,

*t*) are the electric and magnetic energy densities, respectively, and the asterisk denotes the complex conjugate.

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

20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. **58**1112–1124 (1968). [CrossRef]

*q*is a natural number or zero. Since we have assumed that only plane-wave components with

*β*>0 exist,

*q*may not assume arbitrarily large values, but there exists an upper limit, which we denote by

*Q*. Clearly, the angular spectrum is confined to a set of concentric rings, known as Montgomery’s rings [19

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

20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. **58**1112–1124 (1968). [CrossRef]

*J*

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

*m*, we obtain the expression for self-imaging electric fields

*b*

_{m,q}are obtained from the angular spectrum vector of the magnetic field by an expression analogous to Eq. (20).

*q*:th Montgomery’s ring is azimuthally polarized if

*q*is even and radially polarized if

*q*is odd. Now the field confined to the

*q*:th ring is of the form

*a*

_{m,q}are constants, if

*q*is even. On the other hand, if

*q*is odd, we obtain

*z*-component of the electric field is present only when

*q*=2

*s*+1, where s is a natural number or zero, we may express Eq. (16) for the

*z*-component in the form

*ξ′*=

*ξ*/2-

*π*. Thus, the

*z*-component of the electric field clearly self-images at

*z*

_{T}/2. A similar conclusion holds naturally also for the

*z*-components of the magnetic field, which is confined to the even-numbered rings.

## 5 Fractional self-imaging of the energy density

22. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. **25**, 785–786 (2000). [CrossRef]

*z*

_{T}/2.

*α*/

*β*≈0. The Fourier-coefficients of the electric end the magnetic fields now reduce to the forms

*φ*is an arbitrary constant, we find, by inserting Eqs. (27) and (29) into Eqs. (19) and (14), that

*z*

_{T}. An example of this kind of field is illustrated in Fig. 3, which is calculated by using the parameters given in Table 2. In that case the ratio of the maximum amplitudes of the

*x*- and

*z*-components is ≈300 and the contribution from the

*z*-component is negligible, although it is taken into account when calculating the energy density.

*z*

_{T}/2, may arise even in the non-paraxial domain. One such solution is obtained by retaining only the zeroth-order mode in Eqs. (22)–(25), i.e., in the case of radially and azimuthally polarized fields. In addition to the property mentioned here, this kind of fields may be self-imaging even in the case that their scalar components are not [9].

## 6 Unequal self-imaging of the electric and magnetic energy densities

*x*- and

*z*-components, with two plane waves having only the

*y*-components as follows:

*a*is real. We immediately notice that the electric energy density self-images at

*z*

_{T}/2, while the magnetic energy density self-images only at

*z*

_{T}. A movie of the energy densities within one self-imaging distance is presented in Fig. 4.

## 7 Conclusions

## Acknowledgments

## References and links

1. | K. Patorski, “The self-imaging phenomenon and its applications,” in Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1. [CrossRef] |

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

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

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

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

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

7. | R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. |

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

9. | J. Tervo, P. Vahimaa, and J. Turunen, “On propagation-invariance and self-imaging of intensity distributions of electromganetic fields,” J. Mod. Opt. (In press). |

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

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

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

13. | R. Piestun and J. Shamir, “Generalized propagation-invariant fields,” J. Opt. Soc. Am. A |

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

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

16. | S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A |

17. | H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. |

18. | M. Born and E. Wolf, |

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

20. | W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. |

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

22. | J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. |

23. | M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. |

**OCIS Codes**

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 1, 2001

Published: December 3, 2001

**Citation**

Jani Tervo and Jari Pekka Turunen, "Self-imaging of electromagnetic fields," Opt. Express **9**, 622-630 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-622

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### References

- K. Patorski, "The self-imaging phenomenon and its applications," in Progr. Opt., Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Chap. 1. [CrossRef]
- J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987). [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. Olivik, "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]
- R. Horak, Z. Bouchal, and J. Bajer, "Nondiffracting stationary electromagnetic field," Opt. Commun. 133, 315-327 (1997). [CrossRef]
- J. Tervo and J. Turunen, "Generation of vectorial propagation-invariant propagation-invariant fields with polarization-grating axicons," Opt. Commun. 192, 13-18 (2001). [CrossRef]
- J. Tervo, P. Vahimaa, and J. Turunen, "On propagation-invariance and self-imaging of intensity distributions of electromganetic fields," J. Mod. Opt. (In press).
- Y. Y. Schechner, R. Piestun, and J. Shamir, "Wave propagation 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 computer-generated holograms," Opt. Commun. 124, 121-130 (1996). [CrossRef]
- R. Piestun and J. Shamir, "Generalized propagation-invariant fields," J. Opt. Soc. Am. A 15, 3039-3044 (1998). [CrossRef]
- J. Tervo and J. Turunen, "Rotating scale-invariant electromagnetic fields," Opt. Express 9, 9-15 (2001), http://www.opticsexpress.org/oearchive/source/33955.htm. [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.
- S. Ruschin and A. Leizer, "Evanescent Bessel beams," J. Opt. Soc. Am. A 15, 1139-1143 (1998). [CrossRef]
- H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
- W. D. Montgomery, "Self-imaging objects of infinite aperture," J. Opt. Soc. Am. 57 772-778 (1967). [CrossRef]
- W. D. Montgomery, "Algebraic formulation of diffraction applied to self imaging," J. Opt. Soc. Am. 58 1112-1124 (1968). [CrossRef]
- G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.
- J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-786 (2000). [CrossRef]
- M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, "Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction," J. Mod. Opt. 47, 2351-2359 (2000).

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