1. Introduction

It is well kown that radially polarized fields (RPFs), as well as azimuthally polarized beams, have attained considerable attention during the last decade. [1–12] Such interest arises from their multiple applications, for example, in high-resolution microscopy, particle trapping, and material processing, to mention only some of them. In the present work we are interested in the analysis of the vectorial structure of these two types of fields when they freely propagates. In this sense, note that a beam could exhibit a radial (or azimuthal) polarization distribution throughout the initial cross-section, but it does not retain, in general, its character (radial or azimuthal) upon propagation.

Among the treatments used in the literature to investigate the vectorial structure of electromagnetic beams, [13–16] the so-called angular spectrum representation has revealed to be particularly useful. [17–22] In particular, when the contribution of the evanescent waves is negligible, it has been proposed [17, 18, 20] an exact electric field solution 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 structure of the field follows from the particular choice of a triad of mutually orthogonal reference axes. The present work takes advantage of such formalism in order to obtain general analytical expressions and properties of free-propagating fields with radial or azimuthal polarization at some transverse plane.

The paper is arranged as follows. In the next section, the formalism and the key definitions are introduced. In Section 3 the propagation of radially polarized fields into free space is analysed, along with several general properties. Section 4 is devoted to the study of the propagation of azimuthally polarized beams. Fields that retain the type of polarization (radial or azimuthal) at any transverse plane are also considered in the above sections. The results are applied to illustrative examples in Section 5. Finally, the main conclusions are summarized in Section 6.

2. Formalism and key definitions

Let us consider time-harmonic electric and magnetic fields, E and H, respectively, fulfilling the Maxwell equations in a homogeneous isotropic medium. We assume the beam propagates essentially along the z-axis. In terms of the angular spectrum, E reads

and a similar expression for H. In Eq. (1) (x, y, z) are the Cartesian variables, k is the wavenumber of the light, and Ẽ represents the spatial Fourier transform of E. For the sake of convenience, we will use from now on planar polar coordinates (ρ,ϕ), related to the Cartesian Fourier-transform variables u and ν through the equations u=ρ cosϕ, ν=ρ sinϕ. For propagation distances long enough to neglect the contribution of the evanescent waves, we can consider that ρ takes values on the interval [0, 1], and the Fourier-transform solution of Eq. (1) would be written in the form [17]

with

whereH̃ is the spatial Fourier transform of H, Ẽ ₀·s=0 (the symbol “·” denotes the inner product), and s is the unit vector in the direction of each plane wave, i.e., $s(\rho ,\varphi )=(\rho \cos \varphi ,\rho \sin \varphi ,\sqrt{1-{\rho }^2})$ . On the basis of previous results reported in the literature,[17] the electric field solution of the Maxwell equations can be expressed in cylindrical coordinates, R,θ, z, as follows:

where a(ρ,ϕ)=Ẽ ₀·e ₁ and b(ρ,ϕ)=Ẽ ₀·e ₂ are integrable functions that define the spectrum of the field, e ₁=(sinϕ, -cosϕ, 0) and ${\mathit{e}}_2=(\sqrt{1-{\rho }^2}\cos \varphi ,\sqrt{1-{\rho }^2}\sin \varphi ,-\rho )$ are unitary vectors, and x=R cosθ; y=R sinθ. Recall that s, e ₁ and e ₂ form a triad of mutually orthogonal system of unit vectors. Note also that the case b(ρ,ϕ)=0 in Eq. (4) represents a field transverse to the z-axis under free propagation. We refer to it as the TE-field, E _TE. On the other hand, the magnetic field associated to the case a(ρ,ϕ)=0 is transverse to the z direction. Accordingly, we denote this electric-field term in Eq. (4) as E _TM. The general case will be the combination of these two terms, which are mutually orthogonal at the far field. [17,18] At each point of a transverse plane z=constant, instead of using the Cartesian components (E _x, E _y) of the field (defined with respect to some laboratory axes), we will handle its radial and azimuthal components, E_R and E _θ, namely,

Note that the longitudinal component E _z remains unchanged. Then, by using Eqs. (4) and (5) we obtain the following general expressions for the radial and azimuthal components of any free-propagating field:

where $\tilde{b}(\rho ,\varphi )\equiv b(\rho ,\varphi )\sqrt{1-{\rho }^2}$ . Taking these equations into account, a radially (azimuthally) polarized field at some transverse plane would then be defined as that field whose E _θ (E _R) component equals zero at such plane. We will next analyse these two cases separately.

3. Propagation of radially polarized fields

In this section we will study the free-space propagation of a field with radially polarized structure at the transverse plane z=0. To begin with let us first note that the azimuthal component E _θ of the RPF should vanish at z=0, so that (see Eq. (6.b))

This equation can also be written in the form

Let us now consider the first term of Eq. (8) (we call it I ₁). After elementary integration, this term becomes

In a general case, a(ρ=1,ϕ) could differ from zero. However, it was pointed out in Section 2 that here we are considering fields for which the contribution of the evanescent waves [21, 22] is negligible. Consequently, in the present work the function a(ρ=1,ϕ) should be taken equal to zero, for any ϕ. Otherwise, the limit ρ=1 of the integration interval for the evanescent field would contribute in a significant way to the global field. Equation (9) reduces to

In a similar way, the second term, I ₂, of the left-hand side of Eq. (8) reads

and Eq. (8) takes the form

Accordingly, Eq. (12) would be satisfied provided functions a(ρ,ϕ) and b̃(ρ,ϕ) fulfil

The general solution of this equation is then given by writing functions a and b̃ in the form

where F can be any mathematically well-behaved function in the domain we are considering. Recall that a(1,ϕ)=0, and b̃(1,ϕ)=0. Taking Eqs. (4) and (14) into account, we finally obtain a general propagation expression for a field with radially polarized structure at plane z=0:

To get deeper insight into the physical consequences of this equation, let us consider a Fourier series expansion of function F(ρ,ϕ), namely,

where

From Eqs. (15) and (16), it can be shown after some algebra that the field E _rad reads

where

J _m being the Bessel function of order m, and the prime denoting derivation with respect to ρ. In Eq. (18) u _R and u _θ are unitary vectors in the radial and azimuthal directions, i.e.,

Equation (18) is one of the main results of the present work. It shows how propagates into free space a field whose vectorial structure is radially polarized at some initial transverse plane z (in this sense, note that B_m(R, z=0)=0, for any m). Since B _m depends, in general, on the propagation distance z, the azimuthal term does not vanish, and the field will not, in general, remain radially polarized after propagation.

Several properties of this class of fields can be obtained immediately from Eq. (18). More specifically, on the propagation axis z (R=0), the field components reduce to

By taking these equations into account, the following conclusions are inferred at once:

Let us finally consider the special but important case of RPFs whose function F(ρ,ϕ) does not depend on the coordinate ϕ. This implies a(ρ,ϕ)=0 (see Eq. (14.a)), along with f_m(ρ)=δ _m0 f ₀(ρ)(δ _m0 denoting the Kronecker delta). Moreover, from Eq. (19.b) we have B ₀(R,z)=0 for such fields. Taking the above properties into account, we conclude that this kind of fields are radially polarized not only at the initial plane z=0 but also at any other transverse plane. In other words, these fields remain radially polarized under free propagation. Moreover, since a(ρ,ϕ) equals zero, they are pure TM fields, in the sense described in Section 2. In addition, function b̃ does not depend on ϕ, so that the angular spectrum of these fields is independent of ϕ as well. It should be remarked that the present work recovers early results of Refs. 5 and 7. Note finally that the longitudinal component of these beams differs from zero all along the z-axis, in contrast with the fields fulfilling f _m=0=0 and behaving as radially polarized only at plane z=0, for which E _z vanishes on the z axis.

4. Propagation of azimuthally polarized fields (APFs)

Let us now analyse the free-space propagation of a field E _azim with azimuthally-polarized structure at some initial plane z=0. In such a case, the radial component E _R (see Eq. (6.a)) should be zero, and we have

From this expression, by following a similar method to that used for RPFs (cf. Eqs. (8)–(13)), we get after some algebra (compare with Eq. (13))

The solution of this equation can be written in the form

where G(ρ,ϕ) represents any function with the appropriate mathematical requirements. After substituting Eqs. (24) into Eq. (4), we obtain the propagation law

Following a parallel procedure to that used in the radially polarized case, we writeG(ρ,ϕ) in series form, namely,

and Eq. (25) becomes

where

the prime denoting again derivation with respect to ρ. Equation (27) is the other main result of this work. It provides a general expression for the vector E associated to an APF at a certain transverse plane. Note that, in spite of U_m(R,0)=0, the field E _azim does not remain, in general, azimuthally polarized upon propagation: Function U _m(R,z) differs from zero (when z≠0), and a radial field-component would also contribute.

Several properties follow at once from Eq. (27). They are the azimuthal analogy of the properties exhibited by RPFs. On the z axis

Equation (29.c) explicitly shows that, for any APF, the longitudinal component vanishes on the propagation axis. Furthermore, Eqs. (29.a) and (29.b) indicate that any APF is zero along the z-axis if g ₁=g _-1=0. In addition, when g _m=1=0 and g _m≠-1=0 (or vice versa), the APF exhibits circular polarization on the z axis.

Let us finally study those fields that remain azimuthally polarized at any transverse plane. When G(ρ,ϕ) does not depend on ϕ, it can be shown that the field E _azim reads

where

We see at once from Eq. (30) that this kind of field remains azimuthally polarized under free propagation. Moreover, it is a pure TE-field (the z-component is zero everywhere), and the function a(ρ,ϕ) that characterizes its angular plane-wave spectrum is independent of ϕ. These results are also in agreement with those obtained in earlier works (see, for example, Ref. 7 and references therein).

It should finally be noted (see Eq. (27)) that the longitudinal components of the fields behaving as azimuthally polarized only at a plane (say, z=0) do not vanish when R≠0. This shows an important difference with regard to the APFs given by Eq. (30).

5. Application to an example

We will next apply the results derived in the above sections to a certain set of fields, defined by a function F (see Eqs. (14)) given by

where δ denotes here the Dirac delta function, a is a dimensionless constant, and p=1, 2, 3,… Different values of these two parameters generate the beams belonging to this family, which describes radially polarized fields at the initial plane z=0 (see Eq. (15)). Due to the special form of F(ρ,ϕ), the radial and azimuthal field components at any transverse plane are given by simple analytical functions. From the substitution of Eq. (32) into the propagation equation (15), and after some calculations, we obtain the amplitude of the radial and azimuthal components of this field at a transverse plane z:

where ∝ means proportionality (both expressions (33.a) and (33.b) involve the same value of the proportionality constant), and R̃ and z̃ are dimensionless variables,

It follows from Eqs. (21) that the value p=1 gives a purely-transverse circularly polarized field on the z-axis. On the other hand, when p≠1, the transverse components E _x and E _y vanish, in accordance with Eqs. (21). The field is purely longitudinal on the propagation axis. It should finally be remarked that, on the z-axis, to speak about radial or azimuthal components of a field does not make sense.

Figure 1 shows the squared modulus of this kind of fields at the initial plane z=0 for the values p=1, 2 and 3. Note that, at this plane, |E_R|² does not explicitly depend on parameter a, on the contrary to that occurs when the field propagates, as it will next be apparent.

It also follows at once from Eqs. (33) that these fields do not retain the radially-polarized character upon free propagation. However, the existence of zeros of function J_p(R̃) involves the appearance of circular rings around the z-axis, in which the azimuthal field component vanishes.

 

Fig. 1. Squared modulus of the fields defined by Eqs. (32) and (33) in terms of R̃ at the initial plane z=0. Continuous line: p=1; dotted line: p=2; dashed line: p=3. Ordinates are given in arbitrary units. Note that, at plane z=0, we have E _θ=0.

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Figures 2 and 3 illustrate the magnitude of the radial and azimuthal-components at a transverse plane for two distances of propagation: z=λ and z=20 λ from the initial plane. In both cases a=½ and p=1. Ordinates are given in arbitrary units. One sees that the propagated field behaves essentially as radially polarized except for a region around the z-axis whose radius is nearly λ. Within such a region, the magnitude of the azimuthal component is similar to that of the radial one (Fig. 2) or even higher (cf. Fig. 3, for longer propagation distances). In addition, Fig. 3 also exhibits the appearance of circular rings in which the radial field component vanishes.

 

Fig. 2. Squared modulus of the radial field component, |E_R|² (dotted line), and azimuthal component, |E_θ|² (continuous line) in terms of R̃, at the transverse plane z=λ. The curves are plotted for p=1. Remember that, on the propagation axis, the field is circularly polarized.

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Fig. 3. The same as in Fig. 2, but now calculated at the transverse plane z=20 λ.

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This structure of the spatial polarization distribution is more clearly illustrated in Fig. 4, which plots the ratio

far away from the initial plane (in other words, z→∞). There is again no explicit dependence on the parameter a. In addition, note that 0≤q≤1.

Peaks show the presence of a pure azimuthally-polarized field at certain circumferences, whereas, in the valleys (q=0), the field becomes radially polarized. Note also that the size of all the aforementioned rings increases for higher values of parameter p.

 

Fig. 4. Ordinates give the ratio q (defined by Eq. (35)) in terms of R̃ in the limit z→∞. The curves correspond to the cases p=1 (continuous line); p=2 (dotted line); and p=3 (dashed line).

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For the sake of completeness, Figs. 5 and 6 refer to the radial and azimuthal components at the same transverse plane to that of Figs. 1 and 2, but now for the values p=2. The main qualitative difference with regard to the case p=1 appears on the z-axis (R̃=0): the field is now purely longitudinal, as it should be expected.

 

Fig. 5. The same as in Fig. 2 but now with p=2. On the axis (R̃=0) the transverse field components vanish, and the field only exhibits a longitudinal component.

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Fig. 6. The same as in Fig. 5 but now calculated at the transverse plane z=20 λ. Again, the transverse components of the field vanish on the axis.

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6. Conclusions

Within the framework of the angular spectrum representation, on the basis of a particular choice of a triad of mutually orthogonal reference axes, general expressions have been obtained to analytically describe the free-space propagation of electromagnetic beams whose vectorial structure, at a transverse plane, exhibits radial or azimuthal polarization. Furthermore, analytical expressions have been given for the transverse distributions upon propagation of the radial, azimuthal and longitudinal components of such fields. These results allow to get at once the polarization behaviour of both types of fields along the propagation axis. In addition, the particular but important case of those fields that retain their polarization state (radial or azimuthal) under propagation has been studied as a direct consequence of the above general results. Finally, as an illustrative example, the transverse polarization structure of a certain set of fields has been discussed.