Transverse and longitudinal components of the propagating and evanescent waves associated to radially polarized nonparaxial fields

A comparison is established between the contributions of transverse and longitudinal components of both the propagating and the evanescent waves associated to freely propagating radially polarized nonparaxial beams. Attention is focused on those fields that remain radially polarized upon propagation. In terms of the plane-wave angular spectrum of these fields, analytical expressions are given for determining both the spatial shape of the above components and their relative weight integrated over the whole transverse plane. The results are applied to two kinds of doughnut-like beams with radial polarization, and we compare the behavior of such fields at two transverse planes.


Introduction
As is well known, the longitudinal component (along the propagation direction z) of a light beam is negligible in the paraxial approximation. Consequently, the electric field vector is assumed to be transverse to the z-axis and represented by means of two components. This provides a considerable simplification in the calculations. However, in a number of applications (for instance, particle trapping, high-density recording, and high-resolution microscopy, to mention only some of them), the light beam is strongly focused and raises spot sizes smaller than the wavelength. In such cases, the paraxial approach is no longer valid, and a nonparaxial treatment is required. This is a topic of active research, which has been extensively studied in the last years [1][2][3][4][5][6][7][8][9].
So far, different vectorial formulations of nonparaxial electromagnetic fields have been investigated in the literature (see, for example, [10][11][12][13]). Among them, several types of representations based on the plane-wave angular spectrum have been reported in the last years [5,14,15]. Such kind of decomposition has revealed to be useful because it allows one to separate the contribution of the propagating and evanescent waves. In particular, the propagating electric-field solution has been written as the sum of two terms [16]: one of them is transverse to the propagation direction; another one exhibits a nonzero longitudinal component, and its associated magnetic field is also transverse. Such analytical description differs from alternative proposals appeared in the literature, also based on the plane-wave spectrum (see [5] and references therein).
In addition, such formalism enables us to analytically incorporate the presence of evanescent waves [1,5,17,18] when their contribution is significant. These waves are receiving increasing attention due, for instance, to their possibilities for subwavelength resolution. As it will be shown in the present paper, the evanescent field can also be written as the sum of transverse and longitudinal parts. We will study and compare the relative weight of all these terms on the basis of the plane-wave angular spectrum of the field. Attention will be focused on beams with radial polarization distributions [2,4,15,[19][20][21][22][23][24][25], a class of fields with important potential applications. More specifically, in the present work we will consider radially polarized beams that retain this character upon free propagation [15]. We refer to them as RPM fields (radially polarized maintained fields). Doughnut-like beams with radial polarization represent il-lustrative examples of these fields (used, for example, as light sources in super-resolution processes). This paper is organized as follows. In the next section, the formalism and the key definitions are introduced. The field is decomposed as the sum of propagating and evanescent waves. As occurs for the propagation term, the evanescent part split, in turn, into transverse and longitudinal components. Section 3 considers RPM beams and analyzes the relative contribution of the transverse and longitudinal terms of both the propagating and the evanescent parts. The width of the plane-wave angular spectrum of the field is shown to play a central role in the comparative significance of the longitudinal component of the propagating wave. The above results are applied to several illustrative examples in Sect. 4. Finally, the main conclusions are summarized in Sect. 5.

Formalism and key 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 [16]: where k is the wavenumber, and, for simplicity, the timedependent factor exp(−iωt) is been omitted. In these equations,Ẽ andH denote the spatial Fourier transform of E and H , respectively. For simplicity, from now on, we will choose z as the direction of propagation of the beam. Instead of using Cartesian coordinates, for the sake of convenience, we will further use cylindrical coordinates R, θ and z, i.e., along with polar coordinates, ρ and φ, related to the transverse Cartesian Fourier-transform variables u, v by the expressions In terms of these variables, a general solution of the Maxwell equations in the Fourier space can formally be written as follows: with 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 > 0.
Since the electric field should obey the Maxwell equation ∇ · E = 0, the Cartesian components ofẼ 0 should satisfy the conditioñ The vectorH is then obtained fromẼ 0 in the form where We see from (7b) that the third component of vector σ 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 (4), the electric field solution of the Maxwell equations at any transverse plane z reads whereẼ 0 should fulfil (5a), (5b). To our purposes, let us now split this field E(R, θ, z) as the sum of two terms: where The first term, E pr , involves a superposition of plane waves and represents the contribution of the propagating waves (some authors [26] refer to this term as the homogeneous part of the angular spectrum). The second term of (9), E ev , should be understood as a superposition of inhomogeneous waves that decay at different rates along the propagation axis. As is well known, close to the initial plane, the evanescent part of the exact solution would be significant enough in the nonparaxial case. Otherwise, E ev is negligible.
Let us now choose a reference system formed by the orthogonal unitary vectors e 1 and e 2 , namely [16], The propagating field can then be written in the form where the functions a(ρ, φ) and b(ρ, φ) give the projection ofẼ 0 onto the vectors e 1 and e 2 , respectively, i.e., the dot symbolizing the inner product. It is interesting to remark that, for the propagating field E pr , conditions (5a), (5b) reduces tõ where is a unitary vector associated to each plan-wave. Thus, the triad s, e 1 , and e 2 constitute a mutually orthogonal system of unit vectors (see Fig. 1). In fact, the choice of these three vectors characterizes the formalism used in this paper. Let us finally consider the field solution associated to the evanescent term (cf. (9) and (10b)). In a similar way to that used for the propagating term, we can formally write the evanescent part of the field in the form [18] where the unit vector e 1 was defined before, and are vectors with complex components (again, the double sign + and − refers to z > 0 and z < 0, respectively), and Note that e ev · s ev = 0 for any z. In addition, the field E ev fulfils condition (6) when ρ ∈ [1, ∞]. It should also 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). This can be used, for example, to find the ranges of both, the propagation distance and the beam size for which the evanescent waves are significant.
This evanescent part can be though as a superposition of inhomogeneous waves whose constant phase surfaces are planes orthogonal to the (nonunitary) transverse vector s 0 , defined in the form and whose constant-amplitude planes are perpendicular to the propagation axis z. The global field E(R, θ, z) would finally read (see (9)) E(R, θ, z) It should be noted that the exact solution of the Maxwell equations given by (20) is valid for both paraxial and nonparaxial regimes. The angular spectrumẼ 0 at some initial plane would characterize each particular beam.

Radially polarized maintained fields
Let us now consider a general RPM beam. These fields exhibit four main properties, namely, (a) The function a(ρ, φ) (see (13a)) equals zero [15].
(b) The plane-wave angular spectrum is independent of φ. This means that b(ρ, φ) = b(ρ) (see (13b)). (c) It can also be shown from (12) In particular, the general propagation law of E pr reduces to the simple form [27] where u R = (cos θ, sin θ, 0) and u z = (0, 0, 1) are unitary vectors in the radial and longitudinal directions, respectively, and J 0 and J 1 being the Bessel functions of the first kind of order 0 and 1, respectively. Since u R is orthogonal to the z-axis, the first term of the left-hand side of (22) would also be both radially polarized and transverse to the propagation axis at any plane z. The polarization state of E pr can be characterized by means of the so-called degree of linear polarization, Λ, introduced by Lekner [3]. Thus, in terms of the functions C T (R, z) and C L (R, z) given by (23a) and (23b), we have and so that We see that Λ is rotationally symmetric around the z-axis (as expected), and the field is no longer linearly polarized after propagation (except along the z-axis, where Λ(0, z) = 1 for any z).
The relative contributions of the transverse and longitudinal terms of the propagating field can then be compared each other through the ratio (I pr ) L /(I pr ) T , where (I pr ) L and (I pr ) T denote the squared modulus of the longitudinal and transverse parts of E pr , integrated over the whole transverse plane. Since the (longitudinal and transverse) components of E pr are written in terms of their plane-wave spectrum, it follows by applying the Parseval theorem that (I pr ) L and (I pr ) T do no change upon free propagation. Consequently, it would suffice to calculate the ratio (I pr ) L /(I pr ) T at the initial plane. By taking into account the value of E pr (R, θ, 0) given by (21a), we obtain so that It is clear from (23a), (23b) and (27) that C T , C L , and the ratio (I pr ) L /(I pr ) T are determined by the function b(ρ) (cf. (13b)).
To get deeper insight into the physical meaning of function b(ρ), let us consider the ratio (at any z) where (E pr ) z denotes the longitudinal component of the propagating wave. In terms of b(ρ), η z becomes But (29) can be understood as the (normalized) secondorder moment, ρ 2 b , of the function b(ρ). In other words, η z = ρ 2 b characterizes the angular size of the region where |b(ρ)| 2 is important. Consequently, for those RPM fields whose angular spectra exhibit small values of ρ 2 b , the weight of the longitudinal component of the propagating wave would be small, compared with |E pr | 2 . In the limit ρ 2 b 1, we approach the paraxial regime. On the contrary, high values of ρ 2 b (close to 1) mean that we are considering the nonparaxial case.
With regard to the evanescent wave associated to RPM fields, it can be shown that this wave splits into transverse and longitudinal components, in a similar way to that occurs for the propagating field. Furthermore, the relative weight of such components can be compared by using the ratio (I ev ) L /(I ev ) T , where (I ev ) L and (I ev ) T represent the squared modulus (integrated over the whole plane) of the longitudinal and transverse components associated to the evanescent part of the field. By using (21b), at the plane z = 0 we obtain and In these equations, (I ev ) L and (I ev ) L refer to the plane z = 0. In addition, the ratio (I ev ) L (I ev ) T can also be written in the form Accordingly, at any plane z, we finally have In other words, for the evanescent term of RPM beams, the contribution of its longitudinal component is always greater than the contribution of the transverse one at any plane transverse to the propagation direction.

Examples
We will next analyze two kinds of RPM beams characterized by their respective plane-wave spectrum. Although these fields have been investigated in the literature in recent years, however, to our knowledge, this is the first time in which the behavior and significance of the transverse and longitudinal components of their associated evanescent waves are studied and compared.
To begin with, let us consider a vectorial distribution f 1 (x, y) at plane z = 0 whose angular spectrumf 1 takes the form [27] where f (ρ) is a scalar function independent of φ. We see from (34) that the angular spectrum of f 1 only exhibits a longitudinal component. Unfortunately, this implies that f 1 does not fulfil the Maxwell equations, and therefore it cannot represent a realistic light field. However, on the basis of the formalism reported in Sect. 2 (see (20)), it can be shown that the following field, generated byf 1 , obeys the Maxwell equations: The second example is, in a sense, complementary to the first case, because now the starting point is a vector function f 2 whose plane-wave angular spectrumf 2 (ρ, φ) exhibits a radial (transverse) distributioñ with vanishing longitudinal component. Although f 2 cannot be accepted as a realistic light field, (35) is still valid, and the field E 2 (obtained after substitutingf 1 byf 2 in this expression) represents another solution of the Maxwell equations. In this case we get (cf. (20)) It remains to select a function f (ρ) that characterizes the RPM beams chosen in the examples. We will consider an angular spectrum with a doughnut-shape profile of the form with D = 1 kw 0 , where w 0 denotes a parameter closely connected with the transverse size of the field at the beam waist. Figure 2 plots the squared modulus of the components of the electric fields E 1   The same as in Fig. 2, but now the curves are plotted at the plane z = λ Fig. 5 The same as in Fig. 3, but now the curves are plotted at the plane z = λ -Close to the z-axis, the magnitudes of the components of the propagating part associated to E 1 are smaller than the respective components of E 2 . -The opposite occurs for the evanescent waves. In other words, near the propagation axis, the evanescent part of E 1 predominates with respect to the evanescent term associated to E 2 .
In terms of parameter w 0 , Fig. 3 compares the contributions (i.e., integrated squared modulus) of the transverse and longitudinal components of both the propagating and the evanescent waves at plane z = 0. From this figure it is apparent that -The ratio (I ev ) L /(I ev ) T increases almost linearly with w 0 and takes quite similar values for both fields E 1 and E 2 . We see that (I ev ) L reaches much higher values than (I ev ) T for wider beams. The above characteristics keep essentially valid after propagating a short distance from the initial plane. Figures 4  and 5 illustrate the behavior at the plane z = λ. The main qualitative differences with regard to the plane z = 0 can finally be summarized as follows: -The presence of ripples (circles around the z-axis) in both the transverse and the longitudinal components of the evanescent wave; -Both fields, E 1 and E 2 , exhibit quite similar spatial distributions of the components of their respective evanescent terms. This does not happen for the propagating wave in the vicinity of the z-axis. -The contribution of the evanescent wave has significantly reduced, as expected.

Conclusions
In terms of the plane-wave angular spectrum of a light beam, the propagating and the evanescent parts of the nonparaxial field have been split into transverse and longitudinal components. For radially polarized maintained beams (i.e., ra-dially polarized fields that retain this character upon propagation), analytical expressions have been provided that allow one to compute both the spatial shape of such components and the ratio between their respective contributions, integrated over the whole transverse plane. In addition, this formalism enables one to establish the range for which a nonparaxial regime should be considered.
In the examples, a comparison has been made between the behavior of two types of doughnut-like beams with radial polarization, along with their evolution under short-distance propagation.