1 Introduction

In recent years, strongly focused light beams have attracted the attention of multiple researchers. Applications of such fields range from particle trapping or high-resolution microscopy to material processing, among others. In particular, when the beam is focused in a spot smaller than the wavelength, the paraxial approach cannot be used and a non-paraxial treatment is required [1–9].

Among the different formulations for vectorial non-paraxial electromagnetic fields available in the literature [10–14], the description of such beams using the plane-wave angular spectrum framework has been proved particularly suitable [5, 15–17], since this approach enables us to separate the propagating and evanescent contributions of the electromagnetic field [1, 5, 18–20]. Moreover, the propagating and evanescent waves can be described as the sum of two terms: a first one, transverse to the direction of propagation z, and a second one that is a non-zero longitudinal component along the z axis.

In this paper, attention is focused on beams whose transverse component at some plane is azimuthally polarized. The objective of this article is to develop analytical expressions for computing both the spatial shape of transverse azimuthally polarized (TAP) fields and the ratio between the different components; the analysis is carried out within the framework of the plane-wave angular spectrum approach.

This paper is organized as follows. In Sect. 2, the formalism and the key definitions are introduced. General equations for the propagating and evanescent waves are provided. In Sect. 3, TAP fields are introduced and in Sect. 4 an illustrative example of such beams is discussed. In particular, the relative contribution of the transverse and longitudinal terms of both the propagating and the evanescent parts is considered. Finally, the conclusions are presented in Sect. 5.

2 Formalism and basic definitions

Let us consider a monochromatic electromagnetic beam whose propagation is described by the Maxwell equations (for simplicity, we will consider free-space propagation). As is well known, the electric and magnetic fields, E and H, can be expressed in terms of their plane-wave angular spectrum [17],

$$ \varvec{E}(x,y,z) = \int\limits_{ - \infty }^{ + \infty } {\tilde{\varvec{E}}(u,v,z)\exp [ik(xu + yv)]\;{\text{d}}u\;{\text{d}}v} $$
(1a)
$$ \varvec{H}(x,y,z) = \int\limits_{ - \infty }^{ + \infty } {\tilde{\varvec{H}}(u,v,z)\exp [ik(xu + yv)]\;{\text{d}}u\;{\text{d}}v,} $$
(1b)

where k is the wavenumber, and, for simplicity, the time-dependent factor \( \exp ( - i\omega t) \) has been omitted. In these equations, \( \tilde{\varvec{E}} \) and \( \tilde{\varvec{H}} \) denote the spatial Fourier transform of E and H, respectively. For simplicity, we will choose z as the direction of propagation of the beam. For the sake of convenience, we will next use cylindrical coordinates R, θ, and z, i.e., x = Rcosθ and y = Rsinθ, along with polar coordinates, ρ and ϕ, related to the transverse Cartesian Fourier-transform variables u, v by the expressions u = ρcosϕ and v = ρsinϕ. In terms of these variables, a general solution of the Maxwell equations in the Fourier space can formally be written as follows

$$ \tilde{\varvec{E}}(\rho ,\phi ,z) = \tilde{\varvec{E}}_{0} (\rho ,\phi )\exp (ikz\xi ), $$
(2)

with

$$ \begin{aligned} \xi &= (1 - \rho^{2} )^{1/2} \quad \rho \le 1 \hfill \\ \xi &= \pm i(\rho^{2} - 1)^{1/2} \quad \rho > 1, \hfill \\ \end{aligned} $$
(3)

where the signs + and − correspond, respectively, to positive (z > 0) and negative (z < 0) values of the variable z. In the present work, attention will be focused on the region \( z \ge 0 \).

Since the electric field should obey the Maxwell equation \( \nabla \cdot \varvec{E} = 0 \), the Cartesian components of \( \tilde{\varvec{E}}_{0} \) should satisfy the condition

$$ \tilde{E}_{0x} u + \tilde{E}_{0y} v + \tilde{E}_{0z} \xi = 0. $$
(4)

The vector \( \tilde{\varvec{H}} \) is then obtained from \( {\tilde{\mathbf{E}}}_{0} \) in the form

$$ {\tilde{\mathbf{H}}}(\rho ,\phi ,z) = ({\varvec{\sigma}} \times {\tilde{\mathbf{E}}}_{0} )\exp (ikz\xi ), $$
(5)

where the so-called Gaussian system of units is used and

$$ {\varvec{\sigma}} = (u,v,\xi ) = (\rho \cos \phi ,\rho \sin \phi ,\xi ). $$
(6)

We see from Eq. (6) that the third component of vector \( {\varvec{\sigma}} \) could be a complex number whose presence is closely related with the appearance of evanescent waves (see below). On the basis of the plane-wave spectrum given by Eq. (2), the electric-field solution of the Maxwell equations at any transverse plane z reads

$$ \varvec{E}(R,\theta ,z) = \int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {\tilde{\varvec{E}}_{0} (\rho ,\phi )\exp [ik\rho R\cos (\theta - \phi )]\exp (ikz\xi )\rho \;{\text{d}}\rho \;{\text{d}}\phi ,} } $$
(7)

where \( \tilde{\varvec{E}}_{0} \) should fulfill Eq. (4). For our purposes, let us now split this field \( \varvec{E}(R,\theta ,z) \) as the sum of two terms [19, 20]:

$$ \varvec{E}(R,q,z) = \varvec{E}_{\text{pr}} (R,q,z) + \varvec{E}_{\text{ev}} (R,q,z), $$
(8)

where

$$ \varvec{E}_{\text{pr}} (R,\theta ,z) = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {\tilde{\varvec{E}}_{0} (\rho ,\phi )\exp [ik\rho R\cos (\theta - \phi )]\exp (ikz\xi )\rho \;{\text{d}}\rho \;{\text{d}}\phi ,} } $$
(9a)
$$ \varvec{E}_{\text{ev}} (R,\theta ,z) = \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\tilde{\varvec{E}}_{0} (\rho ,\phi )\exp [ik\rho R\cos (\theta - \phi )]\exp (ikz\xi )\rho \;{\text{d}}\rho \;{\text{d}}\phi .} } $$
(9b)

The first term, E pr, involves a superposition of plane waves, and represents the contribution of the propagating waves The second term of Eq. (8), E ev, should be understood as a superposition of inhomogeneous waves that decay at different rates along the propagation axis.

Let us now choose a reference system formed by the orthogonal unitary vectors e 1 and e 2, namely [17]

$$ \varvec{e}_{1} = ({ \sin }\phi ,- { \cos }\phi ,0) \, $$
(10a)
$$ \varvec{e}_{2} = \left( {\sqrt {1 - \rho^{2} } \cos \phi ,\sqrt {1 - \rho^{2} } \sin \phi , - \rho } \right). $$
(10b)

The propagating field can then be written in the form

$$ \varvec{E}_{\text{pr}} (\varvec{r}) = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {a(\rho ,\phi )\varvec{e}_{1} (\phi )\exp (ik\varvec{s} \cdot \varvec{r})\rho \;{\text{d}}\rho \;{\text{d}}\phi + \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {b(\rho ,\phi )\varvec{e}_{2} (\rho ,\phi )\exp (ik\varvec{s} \cdot \varvec{r})\rho \;{\text{d}}\rho \;{\text{d}}\phi } } } } , $$
(11)

where \( \varvec{r} \) denotes the position vector \( \varvec{r} = (x,y,z) \), functions \( a(\rho ,\phi ) \)and \( b(\rho ,\phi ) \) give the projection of \( \tilde{\varvec{E}}_{0} \) onto the vectors e 1 and e 2, respectively, i.e., \( a(\rho ,\phi ) = \tilde{\varvec{E}}_{0} \cdot \varvec{e}_{1} \); \( b(\rho ,\phi ) = \tilde{\varvec{E}}_{0} \cdot \varvec{e}_{2} \), the dot symbolizing the inner product. It is interesting to remark that, for the propagating field E pr, condition \( \nabla \cdot \varvec{E} = 0 \) reduces to \( \tilde{\varvec{E}}_{0} (\rho ,\phi ) \cdot \varvec{s}(\rho ,\phi ) = 0 \), where \( \varvec{s}(\rho ,\phi ) = (\rho { \cos }\phi ,\rho { \sin }\phi ,\sqrt {1 - \rho^{2} } ) \), with \( \rho \in [0,1] \), is a unitary vector associated to each plan wave. Thus, the triad \( \varvec{s} , { }\varvec{e}_{1} {\text{ and }}\varvec{e}_{2} \) constitute a mutually orthogonal system of unit vectors.

Let us finally consider the field solution associated with the evanescent term [cf. Eq. (8) and (9b)]. In a similar way to that used for the propagating term, we can formally write the evanescent part of the field in the form [19, 20]

$$ \varvec{E}_{\text{ev}} = \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\left[ {a(\rho ,\phi )\varvec{e}_{1} (\phi ) + \hat{b}(\rho ,\phi )\varvec{e}_{\text{ev}} (\rho ,\phi )} \right]\exp [ik(\varvec{s}_{\text{ev}} \cdot \varvec{r})]\rho \;{\text{d}}\rho \;{\text{d}}\phi ,} } $$
(12)

where the unit vector e 1 was defined before,

$$ \varvec{e}_{\text{ev}} = \frac{1}{{\sqrt {2\rho^{2} - 1} }}( \pm i(\rho^{2} - 1)^{1/2} \cos \phi \pm i(\rho^{2} - 1)^{1/2} \sin \phi , - \rho ) $$
(13a)
$$ \varvec{s}_{\text{ev}} = \left( {\rho \cos \phi ,\rho \sin \phi , \pm i\sqrt {(\rho^{2} - 1)} } \right),\quad {\text{with}},\;\rho \in [0,1] $$
(13b)

are vectors with complex components (again, the double sign + and − refers to z > 0 and z < 0, respectively) and

$$ \hat{b}(\rho ,\phi ) = \tilde{\varvec{E}}_{0} \cdot \varvec{e}_{\text{ev}} . $$
(14)

In Eq. (14), the product \( \varvec{a} \cdot \varvec{b} \equiv a_{x} b_{x}^{ * } + a_{y} b_{y}^{ * } + a_{z} b_{z}^{ * } \). Thus, note that \( \varvec{e}_{\text{ev}} \cdot \varvec{s}_{\text{ev}} = 0 \) for any z. In addition, the field \( \varvec{E}_{\text{ev}} \) fulfills condition (4) when \( \rho \in \left[ {1,\infty } \right] \). It should be remarked that the separate contribution of the evanescent part allows one to compare the relative weight of this term with regard to the propagating field by calculating the respective square modulus, integrated throughout the beam profile.

Note also that this evanescent part can be thought of as a superposition of inhomogeneous waves whose constant phase surfaces are planes orthogonal to the (non-unitary) transverse vector s 0, defined in the form \( \varvec{s}_{0} = (\rho { \cos }\phi ,\rho { \sin }\phi , 0 ) \), and whose constant-amplitude planes are perpendicular to the propagation axis z.

The global field \( \varvec{E}(R,\theta ,z) \) would finally read [see Eq. (8)]

$$ \begin{aligned} \varvec{E}(R,\theta ,z) & = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {a(\rho ,\phi )\varvec{e}_{1} (\phi )\exp (ik\varvec{s} \cdot \varvec{r})} } \rho \,{\text{d}}\rho \,{\text{d}}\phi \\ & \quad + \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {b(\rho ,\phi )e_{2} (\rho ,\phi )\exp } } (iks \cdot r)\rho\,{\text{d}}\rho\,{\text{d}}\phi \\ & \quad + \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\left[ {a(\rho ,\phi )\varvec{e}_{1} (\phi ) + \hat{b}(\rho ,\phi )\varvec{e}_{\text{ev}} (\rho ,\phi )} \right]\exp (ik\varvec{s}_{\text{ev}} \cdot \varvec{r})\rho\,{\text{d}}\rho\,{\text{d}}\phi,} } \\ \end{aligned} $$
(15)

where the angular spectrum \( \tilde{\varvec{E}}_{{\mathbf{0}}} \) at some initial plane would characterize each particular beam.

In this work, we will focus our attention in the radial and azimuthal components, \( E_{R} \) and \( E_{\theta } \)

$$ E_{R} = \cos \theta E_{x} (R,\theta ,z) + \sin \theta E_{y} (R,\theta ,z) $$
(16a)
$$ E_{\theta } = - \sin \theta E_{x} (R,\theta ,z) + \cos \theta E_{y} (R,\theta ,z). $$
(16b)

Using Eqs. (11) and (12), general expressions for the radial and azimuthal components of propagating and evanescent field can be obtained, namely

$$ \begin{aligned} E_{R} (R,\theta ,z) &= \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {\left[ { - a(\rho ,\phi )\sin (\theta - \phi ) + \sqrt {1 - \rho^{2} } b(\rho ,\phi )\cos (\theta - \phi )} \right]\exp (ik\varvec{s} \cdot \varvec{r})\rho\,{\text{d}}\rho\,{\text{d}}\phi}}\\ &\quad + \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\left[ {- a(\rho ,\phi )\sin (\theta - \phi ) + \hat{b}(\rho ,\phi )\frac{{i\sqrt {\rho^{2} - 1} }}{{\sqrt {2\rho^{2} - 1} }}\cos (\theta - \phi )} \right]\exp (ik\varvec{s}_{\text{ev}} \cdot \varvec{r})\rho\,{\text{d}}\rho\,{\text{d}}\phi,}} \\ \end{aligned} $$
(17a)
$$ \begin{aligned} E_{\theta } (R,\theta ,z) & = - \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {\left[ {a(\rho ,\phi )\cos (\theta - \phi ) + \sqrt {1 - \rho^{2} } b(\rho ,\phi )\sin (\theta - \phi )} \right]\exp (ik\varvec{s} \cdot \varvec{r})\rho\,{\text{d}}\rho{\text{d}}\phi}}\\ & \quad - \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\left[ {a(\rho ,\phi )\cos (\theta - \phi ) + \frac{{i\sqrt {\rho^{2} - 1} }}{{\sqrt {2\rho^{2} - 1} }}\hat{b}(\rho ,\phi )\sin (\theta - \phi )} \right]} \exp (ik\varvec{s}_{\text{ev}} \cdot \varvec{r})\rho\,{\text{d}}\rho\,{\text{d}}\phi.}\\ \end{aligned} $$
(17b)

Azimuthally polarized fields at the transversal plane z = 0 are obtained imposing that the radial component \( E_{R} (R,\theta ,0) \) should vanish, i.e.

$$ \begin{aligned} E_{R} (R,\theta ,0) & = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {\left[ { - a(\rho ,\phi )\sin (\theta - \phi ) + \sqrt {1 - \rho^{2} } b(\rho ,\phi )\cos (\theta - \phi )} \right]\exp (ikR\rho \cos (\theta - \phi ))\rho \,{\text{d}}\rho \,{\text{d}}\phi } } \\ & \quad + \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\left[ { - a(\rho ,\phi )\sin (\theta - \phi ) + \hat{b}(\rho ,\phi )\frac{{i\sqrt {\rho^{2} - 1} }}{{\sqrt {2\rho^{2} - 1} }}\cos (\theta - \phi )} \right]} \exp (ikR\rho \cos (\theta - \phi ))\,\rho \,{\text{d}}\rho \,{\text{d}}\phi = 0.} \\ \end{aligned} $$
(18)

Equation (18) leads to the condition

$$ \frac{1}{R}\frac{\partial A}{\partial \theta } + \frac{\partial B}{\partial R} = 0, $$
(19)

where

$$ A(R,\theta ) = \int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {a(\rho ,\phi )e^{ikR\rho \cos (\theta - \phi )} } \,{\text{d}}\rho \,{\text{d}}\phi } $$
(20a)

and

$$ B(R,\theta ) = \int\limits_{0}^{1} {\int\limits_{0}^{2\pi } {\sqrt {1 - \rho^{2} } b(\rho ,\phi )e^{ikR\rho \cos (\theta - \phi )} \,{\text{d}}\rho \,{\text{d}}\phi + \int\limits_{1}^{\infty } {\int\limits_{0}^{2\pi } {\frac{{i\sqrt {\rho^{2} - 1} }}{{\sqrt {2\rho^{2} - 1} }}} } \widehat{b}(\rho ,\phi )e^{ikR\rho \cos (\theta - \phi )} \,{\text{d}}\rho \,{\text{d}}\phi } } . $$
(20b)

3 Transverse azimuthally polarized fields

Let us now consider a non-paraxial field whose transverse component at plane z = 0 is azimuthally polarized. By applying the condition presented in Eq. (19), the Fourier series expansion (spiral spectrum) of functions \( a(\rho ,\phi ) \), \( b(\rho ,\phi ) \) and \( \hat{b}(\rho ,\phi ) \) can be expressed as

$$ a(\rho ,\phi ) = \left\{ {\begin{array}{*{20}c} {a_{0} (\rho ) + \sum\limits_{m \ne 0} {\frac{ - i}{m}\frac{d}{d\rho }\left[ {\rho \sqrt {1 - \rho^{2} } b_{m} (\rho )} \right]\exp (im\phi ),\quad \rho \le 1} } \\ {a_{0} (\rho ) + \sum\limits_{m \ne 0} {\frac{ - i}{m}\frac{d}{d\rho }\left[ {\frac{{i\rho \sqrt {\rho^{2} - 1} }}{{\sqrt {2\rho^{2} - 1} }}\hat{b}_{m} (\rho )} \right]\exp (im\phi ),\quad \rho > 1} } \\ \end{array} } \right. $$
(21a)
$$ b(\rho ,\phi ) = \sum\limits_{m \ne 0} {b_{m} (\rho )\exp (im\phi ),\quad \rho \le 1} $$
(21b)
$$ \hat{b}(\rho ,\phi ) = \sum\limits_{m \ne 0} {\hat{b}_{m} (\rho )\exp (im\phi ),\quad \rho > 1} . $$
(21c)

These relations are the main results of this work (see the “Appendix” for a more detailed discussion). We thus conclude that both, the propagating and the evanescent waves of a TAP field, are given as a sum of two parts:

  • One of them remains azimuthally polarized at transverse planes z other than the initial one z = 0, and its longitudinal component is zero. This corresponds to the plane-wave angular spectrum without topological charge (m = 0).

  • The other part (whose angular spectrum exhibits topological charge, m ≠ 0) is no longer azimuthally polarized upon free propagation, and its longitudinal component differs from zero even at the plane z = 0.

The significance of the evanescent waves can be evaluated by means of the ratio I e /I , being I e and I the integrated squared modulus of the evanescent and total electric fields, respectively

$$ I_{e} /I = \frac{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {\varvec{E}_{\text{ev}} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {\varvec{E}_{\text{pr}} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } + \int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {\varvec{E}_{\text{ev}} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}. $$
(22)

Let us now focus our attention on the longitudinal component of TAP fields. Associated with the propagating and evanescent waves, the z component is analytically determined by the following expressions:

$$ \left( {E_{\text{pr}} } \right)_{z} (R,\theta ,z) = - \sum\limits_{m \ne 0} {i^{m} 2\pi \exp (im\phi )\int\limits_{0}^{1} {b_{m} } } (\rho )J_{m} (kR\rho )\exp \left( {ikz\sqrt {1 - \rho^{2} } } \right)\rho \,{\text{d}}\rho $$
(23a)
$$ \left( {E_{\text{ev}} } \right)_{z} (R,\theta ,z) = - \sum\limits_{m \ne 0} {i^{m} 2\pi \exp (im\phi )\int\limits_{1}^{\infty } {\frac{\rho }{{\sqrt {2\rho^{2} - 1} }}\hat{b}_{m} (\rho )J_{m} (kR\rho )\exp \left( { - kz\sqrt {\rho^{2} - 1} } \right)\rho \,{\text{d}}\rho ,} } $$
(23b)

where \( J_{m} \) is the Bessel functions of order m. It is important to note that \( (E_{\text{pr}} )_{z} (0,\theta ,z) = (E_{\text{ev}} )_{z} (0,\theta ,z) = 0 , { } \) for any z.

The relative contributions of the transverse and longitudinal terms of the propagating field can be compared to each other through the ratio

$$ \eta_{\text{pr}} = \frac{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{pr}} )_{z} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{pr}} )_{T} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}, $$
(24)

where the subscripts T and z refer to the transverse and longitudinal components of E pr, respectively, and the integrals involve the squared modulus of such components, integrated over the whole transverse plane. Since the components of E pr are written in terms of their plane-wave spectrum, it follows from the application of the Parseval theorem, that they do not change upon free propagation. Consequently, it would suffice to calculate the above ratio at the initial plane z = 0.

With regard to the evanescent wave associated with TAP fields, the relative weight of its longitudinal and transverse components can be compared at z = 0 using the ratio

$$ \eta_{\text{ev}} = \frac{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{ev}} )_{z} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{ev}} )_{T} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}, $$
(25)

where again the integrals represent the squared modulus (integrated over the whole plane) of the longitudinal and transverse components associated with the evanescent part of the field. Finally, in a similar way, the relative importance of the total longitudinal component I z /I can be evaluated from the ratio

$$ I_{z} /I = \frac{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{pr}} )_{z} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } + \int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {(E_{\text{ev}} )_{z} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}{{\int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {E_{\text{pr}} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } + \int\nolimits_{0}^{\infty } {\int\nolimits_{0}^{2\pi } {\left| {E_{\text{ev}} } \right|^{2} R\,{\text{d}}R\,{\text{d}}\theta } } }}. $$
(26)

It is apparent from Eq. (21) that the Fourier spectrum of \( a,b \) and \( \hat{b} \) play a fundamental role in the behavior of the preceding parameters. In the next section, we will apply these parameters, namely, \( \eta_{\text{pr}} , \, \eta_{\text{ev}} , \, I_{z} /I, \, I_{e} /I \), to an example.

4 Example

To illustrate the behavior of the propagating and evanescent components of a TAP field with respect to the topological charge, let us consider a kind of donut-shape field characterized by the following angular spectrum [see Eqs. (15) and (21)]:

$$ \begin{array}{l} {a(\rho ,\phi ) = \frac{ - i}{m}\frac{\text{d}}{{{\text{d}}\rho }}\left[ {\rho^{n + 1} (1 - \rho^{2} )\exp \left( { - \frac{{\rho^{2} }}{{D^{2} }}} \right)} \right]\exp (im\phi ),\quad \rho \le 1} \\ {a(\rho ,\phi ) = \frac{i}{m}\frac{\text{d}}{{{\text{d}}\rho }}\left[ {\frac{{\rho^{n + 1} (\rho^{2} - 1)}}{{\sqrt {2\rho^{2} - 1} }}\exp \left( { - \frac{{\rho^{2} }}{{D^{2} }}} \right)} \right]\exp (im\phi ),\quad \rho > 1} \\ \end{array} $$
(27a)
$$ b(\rho ,\phi ) = \rho^{n} \sqrt {1 - \rho^{2} } \exp \left( { - \frac{{\rho^{2} }}{{D^{2} }}} \right)\exp (im\phi ) \, ,\quad \rho \le 1 $$
(27b)
$$ \hat{b}(\rho ,\phi ) = i\rho^{n} \sqrt {\rho^{2} - 1} \exp \left( { - \frac{{\rho^{2} }}{{D^{2} }}} \right)\exp (im\phi ),\quad \rho > 1, $$
(27c)

where m, n = 1, 2, 3,…, and \( D = 1/kw_{0} \). Note that here \( w_{0} \) denotes a parameter closely connected with the transverse size of the field at the beam waist. Moreover, this field exhibits a topological charge m; the variation of n provides information about the relative importance of the evanescent component.

Figure 1 plots the parameters η pr, η ev, I z /I and I e /I at plane z = 0 in terms of m and n for w 0 = 0.2λ. We observe from the figures that, η pr, η ev and I z /I are increasing functions of topological charge m (for fixed n). For η pr and I z /I, the growth is more pronounced for higher n, opposite occurs for η ev. Moreover, the longitudinal component as well as the evanescent wave is particularly significant for n ≥ 4. Figure 2 shows the same parameters at the plane z = 0 in terms of m and k for w 0 = 0.5λ. The behavior of the parameters η pr, η ev, I z /I and I e /I is similar to the previous case, but now η ev increases with n for fixed m. A significant decrease of the evanescent component is another consequence of w 0 enlargement. Finally, Fig. 3 represents, at the plane z = 0, the global (propagating + evanescent) contributions (squared modulus of the electric field) of the transverse and longitudinal components of the beams considered in the example, for w 0 = 0.2λ and w 0 = 0.5λ. In all cases, the figures exhibit donut-shape profiles, and are equal to zero on the propagation axis z (x = y = 0), as expected.

Fig. 1
figure 1

Parameters η pr, η ev, I e /I and I z /I in terms of m and different values of k for w 0 = 0.2λ

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Fig. 2
figure 2

Parameters η pr, η ev, I e /I and I z /I in terms of m and different values of k for w 0 = 0.5λ

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Fig. 3
figure 3

Squared modulus (at z = 0) of the electric field associated to the global (propagating + evanescent) contributions of the total (transverse + longitudinal), the transverse and the longitudinal components of the beams considered in the example for m = k=1 and m = k=10. The left and right columns show the results obtained for w 0 = 0.2λ and w 0 = 0.5λ, respectively. All the curves have been normalized to the global case

Full size image

Finally, to provide a more visual presentation of the information contained in the previous figure, Fig. 4 displays the 2D distributions of the square module of global field distributions at z = 0. It is apparent that the beams tend to be less focused for larger values of m and k.

Fig. 4
figure 4

2D representation of the total field at z = 0 (see Fig. 3, first row). The images are presented in false color using the jet colormap and normalized to its maximum value

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5 Conclusions

In terms of the plane-wave angular spectrum of a light beam, the analytical structure is shown for both the propagating and the evanescent parts of non-paraxial fields whose transverse component at some plane is azimuthally polarized (TAP fields). Such beams have been split into transverse and longitudinal components. The analytical expressions allow us to compute both the spatial shape of such components and the ratio between their respective contributions, integrated over the whole transverse plane. In the examples, a kind of donut-like TAP fields has been investigated in terms of their transverse beam size for different values of their topological charge. The example shows that in the non-paraxial regime, the contribution of both the longitudinal and evanescent components of these TAP fields can be significant.