Self-imaging of electromagnetic fields

Jani Tervo; Jari Turunen

doi:10.1364/OE.9.000622

1 Introduction

The phenomenon of self-imaging is well-understood within the scalar theory [1]: it is known that all wave vectors contained by a self-imaging field must be confined on a set of concentric rings with interdependent radii, which are intimately connected to the self-imaging distance 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]. The electromagnetic theory of these propagation-invariant fields is fairly well understood [3–9], and several phenomena have been found, which have no counterparts in scalar optics. Furthermore, the scalar theory of rotating fields [10–13], a class of fields falling between the classes of propagation-invariant and self-imaging fields, was recently shown to give an inaccurate prediction of the behavior of the rotating electromagnetic fields [14].

Relatively little is known about the electromagnetic theory of self-imaging fields [4,9]. In this paper we present several features of self-imaging, which can only be predicted by means of electromagnetic theory. We start by presenting a general definition of TE and TM polarized electromagnetic fields (sect. 2) and by presenting the expressions for electromagnetic self-imaging fields (sect. 3). We then proceed to show that under certain conditions the 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

Let us consider a monochromatic electromagnetic field propagating in free space. If we assume that no sources exist in a half-space 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

E (ρ, ϕ, z) = \int_{0}^{\infty} \int_{0}^{2 π} A (α, ψ) \exp [i α ρ \cos (ϕ - ψ) + i β z] α d α d ψ,

where

β = {\begin{matrix} \sqrt{k^{2} - α^{2}} & when α < k \\ i \sqrt{α^{2} - k^{2}} & otherwise \end{matrix}

and, by Fourier inversion at z=0,

A (α, ψ) = \frac{1}{{(2 π)}^{2}} \int_{0}^{\infty} \int_{0}^{2 π} E (ρ, ϕ, 0) \exp [- i α ρ \cos (ϕ - ψ)] ρ d ρ d ϕ .

According to Maxwell’s divergence equation

\nabla \cdot E (r) = 0

the three scalar components of the field are connected. However, two of the components may be assumed to be independent and the third is then obtained by using Eq. (4). Since we have chosen the positive 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

A_{z} (α, ψ) = - \frac{α}{β} [A_{x} (α, ψ) \cos ψ + A_{y} (α, ψ) \sin ψ],

where the subscripts denote the angular spectrum components of the corresponding cartesian components of the field.

The magnetic field H (r) is obtained by using the curl equation

\nabla \times E (r) = i ω μ_{0} H (r),

where 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

B_{x} (α, ψ) = \frac{- 1}{k β} \sqrt{\frac{ε_{0}}{μ_{0}}} [A_{x} (α, ψ) α^{2} \sin ψ \cos ψ + A_{y} (α, ψ) (α^{2} \sin^{2} ψ + β^{2})],

B_{y} (α, ψ) = \frac{1}{k β} \sqrt{\frac{ε_{0}}{μ_{0}}} [A_{x} (α, ψ) (α^{2} \cos^{2} ψ + β^{2}) + A_{y} (α, ψ) α^{2} \sin ψ \cos ψ],

B_{z} (α, ψ) = \frac{α}{k} \sqrt{\frac{ε_{0}}{μ_{0}}} [- A_{x} (α, ψ) \sin ψ + A_{y} (α, ψ) \cos ψ],

where ∊ ₀ is the vacuum permittivity.

Let us introduce the angular spectrum components A_α (α, ψ) and A_ψ (α, ψ) by a rotation operation

{\begin{matrix} A_{α} (α, ψ) = & A_{x} (α, ψ) \cos ψ + A_{y} (α, ψ) \sin ψ \\ A_{ψ} (α, ψ) = & - A_{x} (α, ψ) \sin ψ + A_{y} (α, ψ) \cos ψ \end{matrix} .

Clearly, 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

E_{z} (ρ, ϕ, z) = 0 \Leftrightarrow A_{α} (α, ψ) = 0,

which may be used as a general definition for TE polarized fields of arbitrary form in the sense that if a planar interface is located at a plane z=constant, the electric field vector is parallel to the interface at each point.

Furthermore, if A_ψ (α, ψ)≡0, we have a similar relation

H_{z} (ρ, ϕ, z) \equiv 0 \Leftrightarrow A_{ψ} (α, ψ) \equiv 0,

i.e., the angular spectrum is radially polarized if, and only if, the field is TM polarized in the sense that the magnetic field vector is parallel to the interface.

It was remarked by Ruschin and Leizer that radially or azimuthally polarized fields may be understood to be TM or TEp olarized, respectively [16]. However, the formalism introduced here clearly shows that this conclusion is dependent on the polarization state of the angular spectrum rather than of the actual field. At this point it is important to remember that, for example, an azimuthally polarized field must have azimuthally polarized angular spectrum, but a field having azimuthally polarized angular spectrum does not have to be similarly polarized itself.

Fig. 1. The x/y decompositions of the radial (left) and azimuthal (right) components of the angular spectrum vector in two different points of the Fourier-plane.

Download Full Size | PDF

3 Self-imaging fields

Let us next assume that the field satisfies the self-imaging condition [4, 9]

〈 w (ρ, ϕ, z + z_{T}, t) 〉 = 〈 w (ρ, ϕ, z, t) 〉,

where 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]

〈 w (r, t) 〉 = 〈 w_{e} (r, t) 〉 + 〈 w_{h} (r, t) 〉 = \frac{1}{4} ε_{0} E (r) \cdot E^{*} (r) + \frac{1}{4} μ_{0} H (r) \cdot H^{*} (r),

where ω _e (r, t) and ω _h (r, t) are the electric and magnetic energy densities, respectively, and the asterisk denotes the complex conjugate.

Unlike in the scalar case [19, 20], the general solution of Eq. (13) of is not known. However, it is clear that if each scalar component of the electric field satisfies simultaneously the scalar self-imaging condition

∣ E_{j} (ρ, ϕ, z + z_{T}) ∣ = ∣ E_{j} (ρ, ϕ, z) ∣,

where j=x, y, or z, then the electric energy density must be self-imaging. A similar conclusion holds naturally also for the magnetic energy density. The general solution of Eq. (15) is obtained when the z-components of the angular spectrum components are related by

β_{q} = \frac{ξ - q 2 π}{z_{T}},

where 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, 20]:

A (α, ψ) = \sum_{q = 0}^{Q} δ (α - α_{q}) A_{q} (ψ),

where δ is the (radial) Dirac delta and the constants α_q are related to β_q by Eq. (2). By inserting Eq. (16) into Eq. (1) and using the Jacobi-Anger expansion [21]

\exp (i ϱ \cos σ) = \sum_{m = - \infty}^{\infty} i^{m} J_{m} (ϱ) \exp (i m σ)

where J_m is a Bessel function of the first kind and order m, we obtain the expression for self-imaging electric fields

E (ρ, ϕ, z) = \sum_{q = 0}^{Q} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q} J_{m} (α_{q} ρ) \exp (i m ϕ),

where we have defined

a_{m, q} = i^{m} \int_{0}^{2 π} A (α_{q}, ψ) \exp (- i m ψ) d ψ .

From the expressions (7)–(9) we immediately notice that when the electric field is of the form (19), also the magnetic field must be self-imaging and is hence of the form

H (ρ, ϕ, z) = \sum_{q = 0}^{Q} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} b_{m, q} J_{m} (α_{q} ρ) \exp (i m ϕ),

where the coefficients b_m,q are obtained from the angular spectrum vector of the magnetic field by an expression analogous to Eq. (20).

4 Fractional self-imaging of the z-component

Let us assume that the electric field is of the form (19) such that the angular spectrum of the field confined on 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

E_{q} (r) = \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q} {[J_{m - 1} (α_{q} ρ) \exp (- i ϕ) + J_{m + 1} (α_{q} ρ) \exp (i ϕ)] \hat{x}

and

H_{q} (r) = - \frac{i}{k} \sqrt{\frac{ε_{0}}{μ_{0}}} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q} \exp (i m ϕ)

where a_m,q are constants, if q is even. On the other hand, if q is odd, we obtain

E_{q} (r) = \frac{1}{β_{q}} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q} \exp (i m ϕ) {- 2 i α_{q} J_{m} (α_{q} ρ)} \hat{z}

and

H_{q} (r) = - i \frac{k}{β_{q}} \sqrt{\frac{ε_{0}}{μ_{0}}} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q} \exp (i m ϕ)

Since the z-component of the electric field is present only when q=2s+1, where s is a natural number or zero, we may express Eq. (16) for the z-component in the form

β_{q} = β_{s} = \frac{ξ - (2 s + 1) 2 π}{z_{T}} = \frac{ξ' - s 2 π}{z_{T} ⁄ 2},

where 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.

Table 1. The parameters assumed in Fig. 2.

View Table | View all tables in this article

Fig. 2. (1.35 MB) Movie of the squared absolute value of the z-component of the electric field and the time-averaged electric energy density within one self-imaging distance. The field is calculated by using the parameters given in Table 1.

Download Full Size | PDF

5 Fractional self-imaging of the energy density

It is clear that, since we are dealing with a vector space, it is possible to find a pair of vectors, which are orthogonal at every point of space. Actually in three-dimensional space it is, at least in principle, possible to find three orthogonal vectors. However, in the case of a general electromagnetic field, finding even two orthogonal vectors is not a straightforward task, at least if the properties of the vectors must be somehow bounded. An example of such a task is the theory of 100% efficient beam splitters, in which case we have to find a pair of orthogonal paraxial fields whose combined energy density is constant in one plane, but may be expanded as a finite vectorial Fourier-series such that the absolute values of the (vectorial) Fourier-coefficients are equal [22, 23]. In the following, we examine paraxial orthogonal fields, which may be combined such that the energy density self-images at z _T/2.

Let us assume that only the paraxial plane-wave components have significant amplitudes, i.e., α/β≈0. The Fourier-coefficients of the electric end the magnetic fields now reduce to the forms

a_{m, q} = a_{m, q}^{x} \hat{x} + a_{m, q}^{y} \hat{y}

and

b_{m, q} = \sqrt{\frac{ε_{0}}{μ_{0}}} [- a_{m, y}^{y} \hat{x} + a_{m, y}^{x} \hat{y}] .

We immediately notice that, as in the case of a plane wave, the electric and the magnetic energy densities are equal and thus it is sufficient to examine the properties of the electric field only. If we now assume that

a_{m, y}^{y} = \exp (i β_{q} z_{T} ⁄ 2 + i φ) a_{m, q}^{x},

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

〈 w_{e} (r, t) 〉 = {∣ \sum_{q = 0}^{Q} \exp (i β_{q} z) \sum_{m = - \infty}^{\infty} a_{m, q}^{x} J_{m} (α_{q} ρ) \exp (i m ϕ) ∣}^{2}

The electric energy density clearly satisfies the relation

〈 w_{e} (ρ, ϕ, z + z_{T} ⁄ 2, t) 〉 = 〈 w_{e} (ρ, ϕ, z, t) 〉

and hence the energy density self-images at one half of the conventional self-imaging distance, although the cartesian components self-image at 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.

A similar effect, in which the energy density self-images at 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].

Table 2. The parameters assumed in Fig. 3.

View Table | View all tables in this article

Fig. 3. (740 kB) Movie of the squared absolute values of the cartesian components of the electric field and the electric density within one self-imaging distance. The field is calculated by using the values given in Table 2.

Download Full Size | PDF

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

We saw above that in the paraxial-domain the self-imaging properties of the electric and the magnetic energy densities are equivalent. However, there exist cases, in which they differ radically from each other and, in fact, it is possible that their self-imaging distances are unequal.

To prove that, we shall consider the following simple example. Let us assume that the electric field is constructed by superposing a fundamental Bessel field-mode, which has only the x- and z-components, with two plane waves having only the y-components as follows:

E (ρ, ϕ, z) = \frac{a}{2} J_{0} (α_{1} ρ) \exp (i β_{1} z) \hat{x} + \frac{a}{2} {\exp (i k z) + \exp [i (α_{2} ρ \cos ϕ + β_{2} z)]} \hat{y}

where β ₁ and β ₂ are related to k=β ₀ by Eq. (16). Then the magnetic field takes the following form:

H (ρ, ϕ, z) = a \sqrt{\frac{ε_{0}}{μ_{0}}} {- \frac{1}{2} \exp (i k z) - \frac{β_{2}}{2 k} \exp [i (α_{2} ρ \cos ϕ + β_{2} z)]

The time-averaged electric and magnetic energy densities are then obtained from Eq. (14) by a straightforward calculation. The results are

〈 w_{e} (r, t) 〉 = \frac{a^{2} ε_{0}}{4} [\frac{1}{2} + \frac{1}{2} \cos (4 π z ⁄ z_{T} - α_{2} ρ \cos ϕ) + J_{0}^{2} (α_{1} ρ) + \frac{α_{1}^{2}}{β_{1}^{2}} \cos^{2} ϕ J_{1}^{2} (α_{1} ρ)]

and

〈 w_{h} (r, t) 〉 = a^{2} ε_{0} {\frac{1}{8} + \frac{α_{1}^{4}}{16 k^{2} β_{1}^{2}} J_{2}^{2} (α_{1} ρ) + \frac{α_{1}^{2}}{4 k^{2}} J_{1}^{2} (α_{1} ρ) \sin^{2} ϕ

Here we have assumed that the amplitude 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.

Fig. 4. (1.87 MB) Movie of the electric and the magnetic energy densities given by Eqs. (34) and (35) within one self-imaging distance.

Download Full Size | PDF

The example presented here may be easily extended to propagation-invariant fields, since if only the Bessel field-mode and the paraxial plane wave are present, we obtain a field whose electric energy density is propagation-invariant but whose magnetic energy density is self-imaging.

7 Conclusions

We have introduced several self-imaging phenomena for electromagnetic fields, which have no counterparts within scalar optics. It is appears likely that a considerable number of other such phenomena can be found. Unfortunately, because of the definition of self-imaging with the aid of energy density of a vector field, it appears difficult to develop a formalism that would cover all these cases.

Acknowledgments

Jani Tervo thanks the Finnish Academy of Science and Letters and the Finnish Cultural Foundation for partial financial support. Jari Turunen acknowledges the sabbatical leave provided by the Academy of Finland.

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 4, 651–654 (1987). [CrossRef]

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

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

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

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

7. R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997). [CrossRef]

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

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 54, R50–R53 (1996). [CrossRef]

11. S. Chávez-Cerda, G. S. McDonald, and G. H. S. New, “Nondiffracting Beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996). [CrossRef]

12. C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (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]

15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995), sect. 3.2.

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

17. H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

18. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

19. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57772–778 (1967). [CrossRef]

20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 581112–1124 (1968). [CrossRef]

21. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 2001), p. 681.

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

23. 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).

q	m	β_q	$a_{q, m}^{x}$
0	-3	1.75π/λ	1
1	-2	1.50π/λ	1
2	3	1.25π/λ	i
3	2	1.00π/λ	-1
4	1	0.75π/λ	i
5	4	0.50π/λ	1

q	m	β_q	$a_{q, m}^{x}$	$a_{q, m}^{y}$
0	0	6.28λ	i	i
1	-1	6.27λ	2	-2
2	0	6.26λ	i	i

q	m	β_q	$a_{q, m}^{x}$
0	-3	1.75π/λ	1
1	-2	1.50π/λ	1
2	3	1.25π/λ	i
3	2	1.00π/λ	-1
4	1	0.75π/λ	i
5	4	0.50π/λ	1

q	m	β_q	$a_{q, m}^{x}$	$a_{q, m}^{y}$
0	0	6.28λ	i	i
1	-1	6.27λ	2	-2
2	0	6.26λ	i	i

Self-imaging of electromagnetic fields

Abstract

1 Introduction

2 General TE and TM polarized fields

3 Self-imaging fields

5 Fractional self-imaging of the energy density

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

7 Conclusions

Acknowledgments

References and links

Supplementary Material (3)

Cited By

Figures (4)

Tables (2)

Equations (52)

Optics Express