Abstract

In terms of the angular spectrum representation, general expressions are given to describe the free-space propagation of electromagnetic fields with radial or azimuthal polarization structure at a transverse plane. The transverse distributions of the radial, azimuthal and longitudinal components of these fields are also analysed. In particular, the on-axis behavior upon free propagation is studied. Furthermore, the special but important case of those fields that retain their polarization character (radial or azimuthal) under propagation is also considered. The analytical results are illustrated by application to some examples.

©2008 Optical Society of America

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

E(x,y,z)=E˜(u,v,z)exp[ik(xu+yv)]dudv,

and a similar expression for H. In Eq. (1) (x, y, z) are the Cartesian variables, k is the wavenumber of the light, and 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]

E˜(ρ,ϕ,z)=E0˜(ρ,ϕ)exp(ikz1ρ2),

with

H˜(ρ,ϕ,z)=s(ρ,ϕ)×E˜(ρ,ϕ,z),

where is the spatial Fourier transform of H, 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(ρ,ϕ)=(ρcosϕ,ρsinϕ,1ρ2) . 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:

E(R,θ,z)=0102π[a(ρ,ϕ)e1(ϕ)+b(ρ,ϕ)e2(ρ,ϕ)]×
exp[ikRρcos(ϕθ)]expikz1ρ2ρdρdϕ

where a(ρ,ϕ)= 0·e 1 and b(ρ,ϕ)= 0·e 2 are integrable functions that define the spectrum of the field, e 1=(sinϕ, -cosϕ, 0) and e2=(1ρ2cosϕ,1ρ2sinϕ,ρ) 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, ER and E θ, namely,

ER(R,θ,z)=cosθEx(R,θ,z)+sinθEy(R,θ,z),
Eθ(R,θ,z)=sinθEx(R,θ,z)+cosθEy(R,θ,z),

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:

ER(R,θ,z)=0102π[a(ρ,ϕ)sin(θϕ)+b˜(ρ,ϕ)cos(θϕ)]×
    exp[ikρRcos(θϕ)]exp(ikz1ρ2)ρdρdϕ,
Eθ(R,θ,z)=0102π[a(ρ,ϕ)cos(θϕ)+b˜(ρ,ϕ)sin(θϕ)]×
exp[ikρRcos(θϕ)]exp(ikz1ρ2)ρdρdϕ,

where b˜(ρ,ϕ)b(ρ,ϕ)1ρ2 . 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))

0102π[a(ρ,ϕ)cos(θϕ)+b˜(ρ,ϕ)sin(θϕ)]exp[ikρRcos(θϕ)]ρdρdϕ=0.

This equation can also be written in the form

02πdϕ01a(ρ,ϕ)ρ{exp[ikRρcos(θϕ)]}ρdρ+
01dρ02πb˜(ρ,ϕ)ϕ{exp[ikRρcos(θϕ)]}dϕ=0.

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

I1=02πdϕ{[ρa(ρ,ϕ)exp[ikRρcos(θϕ)]]}ρ=0ρ=1
02π01[ρaρ]exp[ikRρcos(θϕ)]dρdϕ

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

I1=02π01[ρaρ]exp[ikRρcos(θϕ)]dρdϕ.

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

I2=-02π01[b~ϕ]exp[ikRρcos(θϕ)]dρdϕ,

and Eq. (8) takes the form

02π01[ρaρ+b~ϕ]exp[ikRρcos(θϕ)]dρdϕ=0.

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

ρaρ+b~ϕ=0.

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

a(ρ,ϕ)=1ρF(ρ,ϕ)ϕ,
b~(ρ,ϕ)=F(ρ,ϕ)ρ,

where F can be any mathematically well-behaved function in the domain we are considering. Recall that a(1,ϕ)=0, and (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:

Erad(R,θ,z)=0102π[1ρF(ρ,ϕ)ϕe1(ϕ)+11ρ2F(ρ,ϕ)ρe2(ρ,θ)]×
exp[ikRρcos(θϕ)]exp(ikz1ρ2)ρdρdϕ.

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

F(ρ,ϕ)=mexp(imϕ)fm(ρ),m=0,±1,±2,

where

fm(ρ)=12π02πF(ρ,ϕ)exp(imϕ)dϕ,

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

Erad(R,θ,z)=[miexp(imθ)Am(R,z)]uR(θ)+[mexp(imθ)Bm(R,z)]uθ(θ)+
+[mexp(imθ)Cm(R,z)]uz

where

Am(R,z)=π(i)m01exp(ikz1ρ2)×
{mfm(ρ)[Jm1(kRρ)+Jm+1(kRρ)]+ρfm(ρ)[Jm+1(kRρ)-Jm1(kRρ)]}dρ,
Bm(R,z)=π(i)m01exp(ikz1ρ2)×
{mfm(ρ)[Jm1(kRρ)Jm+1(kRρ)]+ρfm(ρ)[Jm+1(kRρ)+Jm1(KRρ)]}dρ,
Cm(R,z)=2π(i)m01ρ2fm(ρ)1ρ2Jm(kRρ)exp(ikz1ρ2)dρ,

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.,

uR=(cosθ,sinθ,0),
uθ=(sinθ,cosθ,0),
uz=(0,0,1).

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 Bm(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

(Erad)x(0,z)=iA1(0,z)+iA1(0,z),
(Erad)y(0,z)=A1(0,z)+A1(0,z),
(Erad)z(0,z)=2π01fm=0(ρ)1ρ2ρ2J0(kRρ)exp(ikz1ρ2)dρ,

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

  1. Those fields with f m=0=0 in the expansion (16.a) do not exhibit longitudinal component all along the z-axis.
  2. For those fields with f m=1=f m=-1=0, the transverse field vanishes at axial points.
  3. When f m=1=0 and f m=-1≠0 (or vice versa), the transverse field is circularly polarized on the propagation axis.

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 fm(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 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

0102π[a(ρ,ϕ)sin(θϕ)+b~(ρ,ϕ)cos(θϕ)]exp[ikρRcos(θϕ)]ρdρdϕ=0.

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))

aϕρb~ρ=0.

The solution of this equation can be written in the form

a(ρ,ϕ)=Gρ,
b~(ρ,ϕ)=1ρGϕ,

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

Eazim(R,θ,z)=0102π[Gρe1(ϕ)+1ρ11ρ2Gϕe2(ρ,θ)]×
exp[ikRρcos(θϕ)]exp(ikz1ρ2)ρdρdϕ.

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

G(ρ,ϕ)=mexp(imϕ)gm(ρ),

and Eq. (25) becomes

Eazim(R,θ,z)=[mexp(imθ)Um(R,z)]uR(θ)+[miexp(imθ)Vm(R,z)]uθ(θ)+
+[mexp(imθ)Wm(R,z)]uz,

where

Um(R,z)=π(i)m01exp(ikz1ρ2)×
{ρgm(ρ)[Jm1(kRρ)+Jm+1(kRρ)]+mgm(ρ)[Jm1(kRρ)Jm+1(kRρ)]}dρ,
Vm(R,z)=π(i)m01exp(ikz1ρ2)×
{ρgm(ρ)[Jm+1(kRρ)Jm1(kRρ)]mgm(ρ)[Jm1(kRρ)+Jm+1(kRρ)]}dρ,
Wm(R,z)=2mπ(i)m101ρgm(ρ)1ρ2Jm(kRρ)exp(ikz1ρ2)dρ,

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 Um(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

(Eazim)x(0,z)=U1(0,z)+U1(0,z),
(Eazim)y(0,z)=iU1(0,z)iU1(0,z),
(Eazim)z(0,z)=0.

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

Eazim(R,θ,z)=H(R,z)uθ(θ),

where

H(R,z)=2πi01dG(ρ)dρJ1(kRρ)exp(ikz1ρ2)ρdρ.

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

F(ρ,ϕ)=δ(ρa)exp(ipϕ),a(0,1),

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:

ER[R˜Jp(R˜)+2iz˜dJpdR˜]exp(ikz1a2)exp(ipθ),
Eθ2pz˜Jp(R˜)R˜exp(ikz1a2)exp(ipθ),

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

R˜=kaR,
z˜=ka21a2z,

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, |ER|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 Jp() 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 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.

Download Full Size | PPT Slide | PDF

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

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

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

q=Eθ2ER2+Eθ2,

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 in the limit z→∞. The curves correspond to the cases p=1 (continuous line); p=2 (dotted line); and p=3 (dashed line).

Download Full Size | PPT Slide | PDF

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 (=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 (=0) the transverse field components vanish, and the field only exhibits a longitudinal component.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

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.

Acknowledgments

The research work leading to this paper has been supported by the Ministerio de Educación y Ciencia (project FIS2007-63396) and by CM-UCM (Research group 910335). We also thank Alejandro Manjavacas and Gemma Piquero their valuable help.

References and links

1. A. A. Tovar, “Production and propagation of cylindrical polarized Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998). [CrossRef]  

2. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000). [CrossRef]  

3. R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000). [CrossRef]  

4. J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002). [CrossRef]  

5. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002). [PubMed]  

6. D. J. Armstrong, M. C. Philips, and A. V. Smith, “Generation of radially polarized beams with an image rotating resonator,” Appl. Opt. 42, 3550–3554 (2003). [CrossRef]   [PubMed]  

7. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A , 20, 1974–1980 (2003). [CrossRef]  

8. N. Pasilly, R. de S. Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J-F Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005). [CrossRef]  

9. M. S. Roth, E. W. Wyss, H. Glur, and H. P. Weber, “Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens,” Opt. Lett. 30, 1665–1667 (2005). [CrossRef]   [PubMed]  

10. D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006). [CrossRef]  

11. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24, 636–643 (2007). [CrossRef]  

12. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713 (2007). [CrossRef]   [PubMed]  

13. P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996). [CrossRef]   [PubMed]  

14. P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998). [CrossRef]  

15. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999). [CrossRef]  

16. C. J. R. Sheppard, “Polarization of almost-planes waves,” J. Opt. Soc. Am. A 17, 335–341 (2000). [CrossRef]  

17. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001). [CrossRef]  

18. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002). [CrossRef]  

19. G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005). [CrossRef]  

20. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006). [CrossRef]  

21. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006). [CrossRef]  

22. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16, 2845–2858 (2008). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. A. A. Tovar, “Production and propagation of cylindrical polarized Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
    [Crossref]
  2. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
    [Crossref]
  3. R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
    [Crossref]
  4. J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
    [Crossref]
  5. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
    [PubMed]
  6. D. J. Armstrong, M. C. Philips, and A. V. Smith, “Generation of radially polarized beams with an image rotating resonator,” Appl. Opt. 42, 3550–3554 (2003).
    [Crossref] [PubMed]
  7. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A,  20, 1974–1980 (2003).
    [Crossref]
  8. N. Pasilly, R. de S. Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J-F Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
    [Crossref]
  9. M. S. Roth, E. W. Wyss, H. Glur, and H. P. Weber, “Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens,” Opt. Lett. 30, 1665–1667 (2005).
    [Crossref] [PubMed]
  10. D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006).
    [Crossref]
  11. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24, 636–643 (2007).
    [Crossref]
  12. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713 (2007).
    [Crossref] [PubMed]
  13. P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
    [Crossref] [PubMed]
  14. P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [Crossref]
  15. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [Crossref]
  16. C. J. R. Sheppard, “Polarization of almost-planes waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
    [Crossref]
  17. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
    [Crossref]
  18. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
    [Crossref]
  19. G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005).
    [Crossref]
  20. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
    [Crossref]
  21. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006).
    [Crossref]
  22. R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16, 2845–2858 (2008).
    [Crossref] [PubMed]

2008 (1)

2007 (2)

2006 (3)

D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
[Crossref]

P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006).
[Crossref]

2005 (3)

2003 (2)

2002 (3)

J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[PubMed]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

2001 (1)

2000 (3)

C. J. R. Sheppard, “Polarization of almost-planes waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
[Crossref]

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

1999 (1)

1998 (2)

P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

A. A. Tovar, “Production and propagation of cylindrical polarized Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[Crossref]

1996 (1)

Aït-Ameur, K.

Armstrong, D. J.

Blit, S.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Bosch, S.

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
[Crossref]

Carnicer, A.

Chaumet, P. C.

Chu, X.

G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005).
[Crossref]

Davidson, N.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Deng, D.

Deng, D. M.

Denis, R. de S.

Fiesem, A. A.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Glur, H.

Gori, F.

Guo, Q.

Hierle, R.

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16, 2845–2858 (2008).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
[Crossref]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
[Crossref]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express 16, 2845–2858 (2008).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
[Crossref]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678–1680 (2001).
[Crossref]

Movilla, J. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Oron, R.

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Pääkkönen, P.

Pasilly, N.

Philips, M. C.

Piquero, G.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

Roch, J-F

Roth, M. S.

Saghafi, S.

Sheppard, C. J. R.

Smith, A. V.

Tervo, J.

Török, P.

P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[Crossref] [PubMed]

Tovar, A. A.

Treussart, F.

Turunen, J.

J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[PubMed]

Vahimaa, P.

P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express 10, 949–959 (2002).
[PubMed]

J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

Varga, P.

P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[Crossref] [PubMed]

Weber, H. P.

Wu, L.

Wyss, E. W.

Yang, X.

Zhao, L.

G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005).
[Crossref]

Zhou, G.

G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Oron, S. Blit, N. Davidson, and A. A. Fiesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

J. Mod. Opt. (1)

J. Tervo, P. Vahimaa, and J. Turunen, “On the propagation-invariant and self-imaging intensity distributions of electromagnetic fields,” J. Mod. Opt. 49, 1537–1543 (2002).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams,” J. Opt. A: Pure Appl. Opt. 8, 524–530 (2006).
[Crossref]

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (2)

J. Phys. D (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D 33, 1817–1822 (2000).
[Crossref]

Opt. Commun. (1)

P. Varga and P. Török, “The Gaussian wave solution of Maxvell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

G. Zhou, X. Chu, and L. Zhao, “Propagation characteristics of TM Gaussian beam,” Opt. Laser Technol. 37, 470–474 (2005).
[Crossref]

Opt. Lett. (3)

Prog. Quantum Electron. (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarizad laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Squared modulus of the fields defined by Eqs. (32) and (33) in terms of 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.
Fig. 2.
Fig. 2. Squared modulus of the radial field component, |ER |2 (dotted line), and azimuthal component, |Eθ |2 (continuous line) in terms of , at the transverse plane z=λ. The curves are plotted for p=1. Remember that, on the propagation axis, the field is circularly polarized.
Fig. 3.
Fig. 3. The same as in Fig. 2, but now calculated at the transverse plane z=20 λ.
Fig. 4.
Fig. 4. Ordinates give the ratio q (defined by Eq. (35)) in terms of in the limit z→∞. The curves correspond to the cases p=1 (continuous line); p=2 (dotted line); and p=3 (dashed line).
Fig. 5.
Fig. 5. The same as in Fig. 2 but now with p=2. On the axis (=0) the transverse field components vanish, and the field only exhibits a longitudinal component.
Fig. 6.
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.

Equations (64)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y , z ) = E ˜ ( u , v , z ) exp [ i k ( xu + yv ) ] dudv ,
E ˜ ( ρ , ϕ , z ) = E 0 ˜ ( ρ , ϕ ) exp ( ikz 1 ρ 2 ) ,
H ˜ ( ρ , ϕ , z ) = s ( ρ , ϕ ) × E ˜ ( ρ , ϕ , z ) ,
E ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) e 1 ( ϕ ) + b ( ρ , ϕ ) e 2 ( ρ , ϕ ) ] ×
exp [ ik R ρ cos ( ϕ θ ) ] exp ikz 1 ρ 2 ρ d ρ d ϕ
E R ( R , θ , z ) = cos θ E x ( R , θ , z ) + sin θ E y ( R , θ , z ) ,
E θ ( R , θ , z ) = sin θ E x ( R , θ , z ) + cos θ E y ( R , θ , z ) ,
E R ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) sin ( θ ϕ ) + b ˜ ( ρ , ϕ ) cos ( θ ϕ ) ] ×
        exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz 1 ρ 2 ) ρ d ρ d ϕ ,
E θ ( R , θ , z ) = 0 1 0 2 π [ a ( ρ , ϕ ) cos ( θ ϕ ) + b ˜ ( ρ , ϕ ) sin ( θ ϕ ) ] ×
exp [ i k ρ R cos ( θ ϕ ) ] exp ( i kz 1 ρ 2 ) ρ d ρ d ϕ ,
0 1 0 2 π [ a ( ρ , ϕ ) cos ( θ ϕ ) + b ˜ ( ρ , ϕ ) sin ( θ ϕ ) ] exp [ i k ρ R cos ( θ ϕ ) ] ρ d ρ d ϕ = 0 .
0 2 π d ϕ 0 1 a ( ρ , ϕ ) ρ { exp [ i k R ρ cos ( θ ϕ ) ] } ρ d ρ +
0 1 d ρ 0 2 π b ˜ ( ρ , ϕ ) ϕ { exp [ i k R ρ cos ( θ ϕ ) ] } d ϕ = 0 .
I 1 = 0 2 π d ϕ { [ ρ a ( ρ , ϕ ) exp [ i k R ρ cos ( θ ϕ ) ] ] } ρ = 0 ρ = 1
0 2 π 0 1 [ ρ a ρ ] exp [ i k R ρ cos ( θ ϕ ) ] d ρ d ϕ
I 1 = 0 2 π 0 1 [ ρ a ρ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ .
I 2 = - 0 2 π 0 1 [ b ~ ϕ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ ,
0 2 π 0 1 [ ρa ρ + b ~ ϕ ] exp [ ikRρ cos ( θ ϕ ) ] d ρd ϕ = 0 .
ρ a ρ + b ~ ϕ = 0 .
a ( ρ , ϕ ) = 1 ρ F ( ρ , ϕ ) ϕ ,
b ~ ( ρ , ϕ ) = F ( ρ , ϕ ) ρ ,
E rad ( R , θ , z ) = 0 1 0 2 π [ 1 ρ F ( ρ , ϕ ) ϕ e 1 ( ϕ ) + 1 1 ρ 2 F ( ρ , ϕ ) ρ e 2 ( ρ , θ ) ] ×
exp [ ikRρ cos ( θ ϕ ) ] exp ( ikz 1 ρ 2 ) ρd ρd ϕ .
F ( ρ , ϕ ) = m exp ( i m ϕ ) f m ( ρ ) , m = 0 , ± 1 , ± 2 ,
f m ( ρ ) = 1 2 π 0 2 π F ( ρ , ϕ ) exp ( i m ϕ ) d ϕ ,
E r a d ( R , θ , z ) = [ m i exp ( i m θ ) A m ( R , z ) ] u R ( θ ) + [ m exp ( i m θ ) B m ( R , z ) ] u θ ( θ ) +
+ [ m exp ( i m θ ) C m ( R , z ) ] u z
A m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ m f m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] + ρ f m ( ρ ) [ J m + 1 ( k R ρ ) - J m 1 ( k R ρ ) ] } d ρ ,
B m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ m f m ( ρ ) [ J m 1 ( k R ρ ) J m + 1 ( k R ρ ) ] + ρ f m ( ρ ) [ J m + 1 ( k R ρ ) + J m 1 ( K R ρ ) ] } d ρ ,
C m ( R , z ) = 2 π ( i ) m 0 1 ρ 2 f m ( ρ ) 1 ρ 2 J m ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
u R = ( cos θ , sin θ , 0 ) ,
u θ = ( sin θ , cos θ , 0 ) ,
u z = ( 0 , 0 , 1 ) .
( E r a d ) x ( 0 , z ) = i A 1 ( 0 , z ) + i A 1 ( 0 , z ) ,
( E r a d ) y ( 0 , z ) = A 1 ( 0 , z ) + A 1 ( 0 , z ) ,
( E r a d ) z ( 0 , z ) = 2 π 0 1 f m = 0 ( ρ ) 1 ρ 2 ρ 2 J 0 ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
0 1 0 2 π [ a ( ρ , ϕ ) sin ( θ ϕ ) + b ~ ( ρ , ϕ ) cos ( θ ϕ ) ] exp [ i k ρ R cos ( θ ϕ ) ] ρ d ρ d ϕ = 0 .
a ϕ ρ b ~ ρ = 0 .
a ( ρ , ϕ ) = G ρ ,
b ~ ( ρ , ϕ ) = 1 ρ G ϕ ,
E azim ( R , θ , z ) = 0 1 0 2 π [ G ρ e 1 ( ϕ ) + 1 ρ 1 1 ρ 2 G ϕ e 2 ( ρ , θ ) ] ×
exp [ i k R ρ cos ( θ ϕ ) ] exp ( i k z 1 ρ 2 ) ρ d ρ d ϕ .
G ( ρ , ϕ ) = m exp ( i m ϕ ) g m ( ρ ) ,
E azim ( R , θ , z ) = [ m exp ( i m θ ) U m ( R , z ) ] u R ( θ ) + [ m i exp ( i m θ ) V m ( R , z ) ] u θ ( θ ) +
+ [ m exp ( i m θ ) W m ( R , z ) ] u z ,
U m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ ρ g m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] + m g m ( ρ ) [ J m 1 ( k R ρ ) J m + 1 ( k R ρ ) ] } d ρ ,
V m ( R , z ) = π ( i ) m 0 1 exp ( i k z 1 ρ 2 ) ×
{ ρ g m ( ρ ) [ J m + 1 ( k R ρ ) J m 1 ( k R ρ ) ] m g m ( ρ ) [ J m 1 ( k R ρ ) + J m + 1 ( k R ρ ) ] } d ρ ,
W m ( R , z ) = 2 m π ( i ) m 1 0 1 ρ g m ( ρ ) 1 ρ 2 J m ( k R ρ ) exp ( i k z 1 ρ 2 ) d ρ ,
( E azim ) x ( 0 , z ) = U 1 ( 0 , z ) + U 1 ( 0 , z ) ,
( E azim ) y ( 0 , z ) = i U 1 ( 0 , z ) i U 1 ( 0 , z ) ,
( E azim ) z ( 0 , z ) = 0 .
E azim ( R , θ , z ) = H ( R , z ) u θ ( θ ) ,
H ( R , z ) = 2 π i 0 1 d G ( ρ ) d ρ J 1 ( kRρ ) exp ( ikz 1 ρ 2 ) ρ d ρ .
F ( ρ , ϕ ) = δ ( ρ a ) exp ( ip ϕ ) , a ( 0 , 1 ) ,
E R [ R ˜ J p ( R ˜ ) + 2 i z ˜ dJ p d R ˜ ] exp ( ikz 1 a 2 ) exp ( ip θ ) ,
E θ 2 p z ˜ J p ( R ˜ ) R ˜ exp ( ikz 1 a 2 ) exp ( ip θ ) ,
R ˜ = kaR ,
z ˜ = ka 2 1 a 2 z ,
q = E θ 2 E R 2 + E θ 2 ,

Metrics