Abstract

We suggest vortex phase elements to detect the polarization state of the focused incident beam. We analytically and numerically show that only the types of polarization (linear, circular, or cylindrical) can be distinguished in the low numerical aperture (NA) mode. Sharp focusing is necessary to identify the polarization state in more detail (direction or sign). We consider a high NA micro-objective and a diffractive axicon as focusing systems. We show that the diffractive axicon more precisely detects the polarization state than does the micro-objective with the same NA.

© 2015 Optical Society of America

1. Introduction

The mutual influence between optical phase vortices and polarization singularities–both their transformation into each other and the resulting angular momentum compensation or enhancement—is well studied [1–13]. Using the vortex phase to analyze laser field polarization properties has subsequently been proposed [14–16]. However, the inter-relation between scalar (phase) and vector (polarization) optical vortices can only be visualized using a high numerical aperture mode (e.g., sharp focusing) [15–19].

Besides scalar phase singularities (vortex phase, phase jumps) [3, 9], various polarization singularities of vector fields exist [20–22]. The inter-relations between two types of singularities in the angular moment of photons have been known for a long time [1, 2] and are successfully used in optical manipulations by micro- and nano-particles [7, 11–13, 23]. There are a variety of applications of light beams with phase and polarization singularities, including microscopy, lithography, and nonlinear optics [10, 12, 24–29]. Also, the inter-relation of polarization and phase singularities has been shown in anisotropic materials [30–33] and with sharp focusing [15, 16, 18, 34, 35].

As a rule, generation and analysis of light beams with phase or polarization singularities are carried out by means of the same device. For example, beams with vortex phases are formed by diffractive optics: spiral phase plates, fork-shaped gratings, and more complex multi-order diffractive optical elements [20, 36, 37]. To transform linearly polarized radiation generated by the majority of modern lasers into circularly polarized radiation, half-wave plates are used. For more complex polarization transformations, including the generation of cylindrical vector beams with radial and azimuthal polarization, segmented polarizing plates, subwavelength gratings, interference schemes, light modulators, and other devices are used [38–41].

The analysis of polarization singularities by interferometry requires complex devices and schemes with anisotropic elements [14, 42]. Using the inter-relation of polarization and phase singularities allows for more simple analytical schemes [16, 17, 43], but can only appreciably visually distinguish between homogeneously polarized (linear and circular polarization) and cylindrical (radial and azimuthal polarization) beams. More detailed detection of, for instance, directions of circular polarization and distinction between radial and azimuthal polarizations require sharp focusing [43].

We use vortex phase elements to detect the focused incident beam polarization state. Their complex-valued functions can be written as superpositions of optical vortices. The elements can be implemented using diffractive optics [44] and then added to the focusing system [15–19]. Further, we can insert a singularity into a focusing element structure [45].

A micro-objective [46], a parabolic mirror [47, 48], a diffractive lens [48–50], or an axicon [51–55] can provide sharp focusing. A parabolic mirror or diffractive lens was proposed to achieve [48] sharper focusing than in the case of a micro-objective. This proposal was confirmed experimentally for parabolic mirrors [47] and numerically for diffractive lenses [49, 50]. In addition, aplanatic lens focusing properties were enhanced by insertion of axicon structures [50, 56].

We analyze two cases of sharp focusing. In the first case, sharp focusing is achieved by a micro-objective. In this study, we perform simulations using the Debye approximation [57] and the plane wave expansion method [58]. In the second case, sharp focusing is achieved by a diffractive axicon. Since we use the finite-difference time-domain (FDTD) method, as part of a free-software package, Meep, the calculations here are more accurate [59]. The results obtained by the two methods are very similar.

2. Focusing by a micro-objective

We use the Debye approximation to simulate an aplanatic focusing optical system [57]:

E(ρ,φ,z)=ifλ0α02πB(θ,ϕ)T(θ)P(θ,ϕ)exp[ik(ρsinθcos(ϕφ)+zcosθ)]sinθdθdϕ,
Where (ρ,φ,z) are the cylindrical coordinates of the focal area, (θ,ϕ) are the spherical angular coordinates of the focusing system output pupil, B(θ,ϕ) is the transmission function, T(θ) is the pupil apodization function (T(θ)=cosθ for an aplanatic micro-objective), P(θ,ϕ) is the polarization vector, sinα=NA/n, n is the medium refractive index, k=2π/λ is the wavenumber, λ is the wavelength, and f is the focal length.

The polarization vector P(θ,ϕ) of the focusing system can be written in the following form [60]:

P(θ,ϕ)=[1+cos2ϕ(cosθ1)sinϕcosϕ(cosθ1)cosϕsinθsinϕcosϕ(cosθ1)1+sin2ϕ(cosθ1)sinϕsinθsinθcosϕsinθsinϕcosθ][a(ϕ,θ)b(ϕ,θ)c(ϕ,θ)],
Where a(θ,ϕ), b(θ,ϕ) , and c(θ,ϕ) are the polarization functions for the x-, y- and z-components of the incident beam, respectively. For types of polarization that are used most frequently, these functions do not depend on θ and can be rewritten in simpler forms.

If the transmission function is given by

B(θ,ϕ)=R(θ)Ω(ϕ),
where
Ω(ϕ)=m=M1M2dmexp(imϕ),
Equation (1) can be simplified for most polarization states [18, 19, 50]:
E(ρ,φ,z)=ikfm=M1M2dmimeimφ0αQm(ρ,φ,θ)R(θ)T(θ)sinθexp(ikzcosθ)dθ,
where vector Qm(ρ,φ,θ) depends on the input field polarization (t=kρsinθ) as follows:

– for linear x-polarization

Qm(ρ,φ,θ)=[Jm(t)+14[2Jm(t)ei2φJm+2(t)ei2φJm2(t)](cosθ1)i4[ei2φJm+2(t)ei2φJm2(t)](cosθ1)i2[eiφJm+1(t)eiφJm1(t)]sinθ],
– for linear y-polarization
Qm(ρ,φ,θ)=[i4[ei2φJm+2(t)ei2φJm2(t)](cosθ1)Jm(t)+14[2Jm(t)+ei2φJm+2(t)+ei2φJm2(t)](cosθ1)12[eiφJm+1(t)+eiφJm1(t)]sinθ],
– for left (“+”) or right (“−”) circular polarization
Qm(ρ,φ,θ)=12[Jm(t)+12[Jm(t)e±i2φJm±2(t)](cosθ1)±i{Jm(t)+12(Jm(t)+e±i2φJm±2(t))(cosθ1)}ie±iφJm±1(t)sinθ],
– for radial polarization
Qm(ρ,φ,θ)=12[i[eiφJm+1(t)eiφJm1(t)]cosθ[eiφJm+1(t)+eiφJm1(t)]cosθ2Jm(t)sinθ],
– for azimuthal polarization

Qm(ρ,φ,θ)=12[[eiφJm+1(t)+eiφJm1(t)]i[eiφJm+1(t)eiφJm1(t)]0].

On-axis (ρ = 0) intensity values vary with different polarization states (Eqs. (6)-(10)) and superposition of optical vortices (Eq. (4)). In practice, special combinations from Eq. (4) may be useful. For instance, we can describe binary phase element behavior as exp(imφ)±exp(imφ) [50]. |m|2 is the only case for which the on-axis intensity has a non-zero value. Table 1 shows how the on-axis intensity varies with the focused electric field components.

Tables Icon

Table 1. Dependence of on-axis intensity components due to superposition in Eq. (4)

Table 1 summarizes the dependencies of focal intensity distributions on the polarization and vortex phases and on the focusing system NA. When the NA value is low, we can assume sinθ0 and cosθ1. Table 2 shows the simulation results for low NA = 0.25. Two different types of polarization obviously differ: cylindrical (radial and azimuthal) and homogenous (linear and circular). When the NA value is high, we can assume sinθ1 and cosθε0. Table 3 shows that simulations using high NA give much more information about the polarization states.

Tables Icon

Table 2. Focal intensity distribution for Gaussian beam, NA = 0.25 (red color for x-component, green for y-component, picture size is 8λ × 8λ)

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Table 3. Focal intensity distribution for Gaussian beam, NA = 0.95 (red color for x-component, green for y-component, and blue for z-component; picture size is 2λ × 2λ)

Sharp focusing makes azimuthal polarization easily recognizable; it is the only type of polarization for which the intensity value at the central focal point is zero. Linear polarization can be detected by focal spot elongation in the direction of the polarization axis.

An additional first-order vortex phase can distinguish the circular polarization direction. A binary phase containing optical vortices of opposite signs can detect the orthogonal linear polarization.

3. Multi-order diffractive optical element

The previous results indicate that, for unequivocal detection of polarization type, multiple test vortex phases are required, even with sharp focusing [43]. Thus, simultaneously monitoring the effects of several optical vortices and their superposition is desirable. Combinations of optical vortices that can be generated by simple binary phase elements are practically convenient to this end [19, 26].

Multi-order binary optical elements can be used to subject a field of interest to several vortices, simultaneously [37, 61]. Figure 1 shows the binary phase transmission function, which receives the response of an analyzed beam to various combinations of phase vortices simultaneously in several diffractive orders in the focal plane.

 figure: Fig. 1

Fig. 1 Transmission function of the focusing system. (a) Binary phase of multi-order diffractive optical element. (b) Accordance of diffractive orders to combinations of optical vortices.

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Figure 2 shows the recognition of orthogonal linear polarization states in the absence or presence of vortex phases in an analyzed beam. The amplitude of the beam is a Gaussian function multiplied by radius. The phase of the beam is constant or a vortex of the first order.

 figure: Fig. 2

Fig. 2 Detection of orthogonal linear polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.

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The parameters of the calculation were: wavelength of incident radiation λ = 1 μm, focal length f = 101 μm, numerical aperture of micro-objective, NA = 0.99, radius of Gaussian beam is 50 μm, and focal area of interest is 15 μm × 15 μm.

From the modeling results, the presence or absence of a vortex phase of the first order is easily identified by the presence of a correlation peak in the corresponding diffractive order in the focal plane; the absence of an optical vortex corresponds to high intensity in the center of the focal plane.

Less obvious characteristics allow for the recognition of orthogonal linear polarizations. In particular, if the analyzed beam has no vortex phase, then nonzero intensity values of vertical diffractive orders (corresponding to cos(φ)) indicate x-polarization, while the zero intensity values indicate y-polarization (see the graphics of Fig. 2). The presence of a vortex phase in the analyzed beam overwhelms these more subtle responses, and polarization detection becomes uncertain.

Detection of orthogonal circular polarizations is shown in Fig. 3, in the absence and presence of a vortex phase in the analyzed beam. Detection of the vortex phase is similar to the previous case.

 figure: Fig. 3

Fig. 3 Detection of orthogonal circular polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.

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Detecting orthogonal circular polarizations is much easier than detecting linear polarizations, orthogonal circular polarization singularities are closely connected with phase singularities. We can show this by presenting circular polarization in polar components:

ex±iey=(ercosφeφsinφ)±i(ersinφ+eφcosφ)=exp(±iφ)[er±ieφ]

As follows from (11), circular polarization corresponds to the vortex phase of the first order with the same sign as the direction of polarization.

The inter-relation of the polarization singularity with the vortex phase can be seen by the nonzero intensity corresponding to vortex diffractive orders (Fig. 3), even though there is no vortex phase in the analyzed beams.

When the vortex phase is present in the analyzed beam, it is still easy to distinguish the type of circular polarization: if the directions of circular polarization and phase vortex are identical, the central diffractive order will have zero intensity. If the directions are opposite, then the central diffractive order will have nonzero intensity (see the graphics of Fig. 3).

Figure 4 shows detection of orthogonal cylindrical polarizations in the absence and presence of a vortex phase in an analyzed beam.

 figure: Fig. 4

Fig. 4 Detection of orthogonal cylindrical polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.

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Orthogonal cylindrical polarizations are visually obvious because this type of polarization is singular and connected with the vortex phase. In particular, for radial and azimuthal polarization, respectively:

er=excosφ+eysinφ=12exp(iφ)[exiey]+12exp(iφ)[ex+iey],
eφ=exsinφ+eycosφ=12exp(iφ)[ey+iex]+12exp(iφ)[ey+iex].

From expressions (12) and (13), a cylindrical polarization contains vortex phases of the first order of both signs. Such inter-relation between cylindrical polarizations and vortex phases is evident in the nonzero intensities of corresponding vortex diffractive orders (Fig. 4), even when there is no vortex phase in the analyzed beam.

The vortex phase in cylindrically polarized beam is detected by a correlation peak in the central diffractive order with zero intensity in any vortex (horizontal) diffractive order.

With a vortex phase in the analyzed beam, the type of polarization is still easy to distinguish: for radial polarization, vertical diffractive orders have nonzero value, and for azimuthal polarization, they have zero intensity (see the graphics of Fig. 4).

This research shows that, in conditions of sharp focusing, the multi-order diffractive optical element combined with various superpositions of vortex phases unequivocally distinguishes singular polarization states—circular, radial and azimuthal. The inter-relation of polarization and phase singularities provides unambiguous recognition.

Such inter-relation is absent for linear polarization, so detailed recognition of various types of linear polarizations by means of micro-objective focusing is complicated. Previous work [43] has shown that diffractive axicon focusing allows for better detection of linear polarization states. The NA of a micro-objective can be set to various values depending on radius and reaches its maximal value at the edge of an optical element. Therefore, light refracted at various angles (paraxial and non-paraxial) is combined at the focus. An axicon has identical NA at any radius - both in the center and at the edge; therefore, all focused light contains information corresponding to the same high value numerical aperture.

4. Focusing by the diffractive axicon

Plane wave expansion (PWE) can be used to explain focusing by an axicon [58]:

E(ρ,φ,z)=1λ2σ1σ202πM(σ,ϕ)(Fx(σ,ϕ)Fy(σ,ϕ))exp[ikz1σ2]exp[ikσρcos(φϕ)]σdσdϕ,
M(σ,ϕ)=[1cos2ϕ(1γ)cosϕsinϕ(1γ)σcosϕcosϕsinϕ(1γ)1sin2ϕ(1γ)σsinϕ],γ=1σ2,
(Fx(σ,ϕ)Fy(σ,ϕ))=0r002π(E0x(r,τ)E0y(r,τ))exp[ikrσcos(τϕ)]rdrdτ.
If the input field components are given in the form of vortex beams, as in Eq. (4):
E0(r,τ)=E0(r)exp(imτ),
we can simplify relations (14)-(16) in the following manner (t=kσρ) [54, 55]:

E(ρ,φ,z)=k2i2mexp(imφ)σ1σ2Sm(ρ,φ,σ)(Px(σ)Py(σ))exp(ikz1σ2)σdσ,
Sm(ρ,φ,σ)=[14{2Jm(t)(1+γ)+[ei2φJm+2(t)+ei2φJm2(t)](1γ)}i4[ei2φJm+2(t)ei2φJm2(t)](1γ)i2σ[eiφJm+1(t)eiφJm1(t)]i4[ei2φJm+2(t)ei2φJm2(t)](1γ)14{2Jm(t)(1+γ)[ei2φJm+2(t)+ei2φJm2(t)](1γ)}12σ[eiφJm+1(t)+eiφJm1(t)]],
(Px(σ)Py(σ))=0R(E0x(r)E0y(r))Jm(krσ)rdr.

We will use the diffractive phase axicon as a focusing element. Its complex-valued transmission function is given by

E0(r)=exp(ikα0r),
where α0 is a parameter defining axicon NA.

In this case, integral (20) can be approximated in the following way:

P(σ)δ(σα0),
where δ() is the delta-function.

Then the distribution in the focal domain is estimated as follows:

E(ρ,φ,z)i2mα0exp(imφ)exp(ikz1α02)Sm(ρ,φ,α0)(cxcy),
where (cx,cy) is the polarization vector for a homogenously polarized beam.

Thus, the distribution of Eq. (23) depends on the incident beam polarization state, the vortex phase order, and the focusing system NA (γ0=1α02,t=kα0ρ) as:

E(ρ,φ,z)α02(cxJm(t)(1+γ0)+12{[ei2φJm+2(t)(cxicy)+ei2φJm2(t)(cx+icy)]}(1γ0)cyJm(t)(1+γ0)12{[ei2φJm+2(t)(cyicx)+ei2φJm2(t)(cy+icx)]}(1γ0)α0[eiφJm+1(t)(cy+icx)+eiφJm1(t)(cyicx)]).

When NA values are low (α0ε0 and γ01),

|E(ρ,φ,z)|2ε2(|cx|2+|cy|2)Jm2(kερ),
and there is no difference between linear and circular polarizations.

When NA values are high (α01 and γ0ε0), it is possible to recognize the polarization state, even on the optical axis (ρ = 0), for specific values of m:

- for m = 0:

|E(0,0,z)|2|cx|2+|cy|24
- for m = ± 1:
|E(0,0,z)|2|cyicx|24
- for m = ± 2:

|E(0,0,z)|2|cx±icy|2+|cy±icx|216

Equation (27) allows us to easily recognize the circular polarization direction: in the case of m = 1, zero value in the central focal point indicates left-circular polarization (cy=icx), and a non-zero value indicates right-circular polarization (cy=icx); in the case of m = −1, the situation is reversed.

In the case of cylindrical beams, it is more convenient to rewrite the matrix Eq. (19) with cylindrical components (cr,cφ):

Smc(ρ,φ,σ)=12[iγ[eiφJm+1(t)eiφJm1(t)][eiφJm+1(t)+eiφJm1(t)]γ[eiφJm+1(t)+eiφJm1(t)]i[eiφJm+1(t)eiφJm1(t)]2σJm(t)0].

Then we have the following instead of (23):

Smc(ρ,φ,α0)=12[iγ0[eiφJm+1(t)eiφJm1(t)][eiφJm+1(t)+eiφJm1(t)]γ0[eiφJm+1(t)+eiφJm1(t)]i[eiφJm+1(t)eiφJm1(t)]2α0Jm(t)0].

When NA values are low (α0ε0 and γ01), the longitudinal component is very small, and, therefore, the total intensity distributions for radial (cφ=0) and azimuthal (cr=0) polarizations are similar. When NA values are high (α01 and γ0ε0), the intensity distributions clearly depended on the polarization state, even along the optical axis (ρ = 0):

- for m = 0:

|E(0,0,z)|2|cr|2
- for m = ± 1:

|E(0,0,z)|2|cφ|2

Equations (31) and (32) allow us to easily distinguish between radial and azimuthal polarizations: in the case of m = 0, the zero value in the central focal point indicates azimuthal polarization, and a non-zero value indicates radial polarization; in the case of m = ± 1, the situation is reversed.

Homogenous (linear or circular) polarizations give non-zero values for |m|2, whereas cylindrical (radial or azimuthal) polarizations give non-zero values in the central focal point only for |m|1.

Table 4 shows the intensity distributions in the focal plane (the plane with the highest possible on-axis intensity) calculated according to Eqs. (18)-(21), for α0 = 0.95.

Tables Icon

Table 4. Intensity distributions of Gaussian beam focused by the axicon in Eq. (21) with α0 = 0.95 (red color for x-component, green for y-component, and blue for z-component, picture size 4λ × 4λ)

Comparing Tables 3 and 4 reveals that the diffractive axicon provides sharper focusing than an aplanatic lens with the same NA. This axicon characteristic is very useful when more confident detection of an incident beam polarization state is required. For instance, the focal spot elongation for linearly polarized beam in the direction of the polarization axis becomes more obvious. In addition, strengthening the longitudinal component allows us to positively recognize orthogonal cylindrical and circular polarization states. When a second-order optical vortex is used, distinctions between the focal images become essential.

In order to confirm our results, we performed more accurate calculations using the finite-difference time-domain (FDTD) method, provided as part of a free-software package, Meep [59].

Simulation parameters were as follows: wavelength λ = 0.532 μm; the computational domain size is x, y∈[–6.5λ; 6.5λ], z∈[–6λ; 6λ]; the absorbing layer PML thickness is 2λ; the space discretization is λ/30; the time discretization is λ/(60c), where c is the speed of light; and the substrate thickness of the axicon is 8λ. The diffractive axicon numerical aperture is NA = 0.95 and the refractive index is n = 1.46; thus, the microrelief height is 1.087λ.

Table 8 shows the transverse intensity images of the Gaussian beam, the Laguerre-Gaussian mode (0,1) and the Hermite-Gaussian mode (0,1) focused by the axicon (Eq. (21)) with NA = 0.95.

Tables Icon

Table 5. Intensity distributions in the transverse planes of the Gaussian beam, Laguerre-Gaussian mode (0,1) and Hermite-Gaussian mode (0,1) focused by the axicon with NA = 0.95 (blue color for z-component, yellow for all-components, picture size 6λ × 6λ)

From the simulation results, we can see that there is a good agreement between both methods. However, there are also quantitative differences that result from real optical characteristics, which have been considered in FDTD-calculations (e.g., the substrate element thickness and the distance between the source and the element [54, 55, 62]).

5. Conclusions

It has been shown, both analytically and numerically, that it is possible to distinguish only types of polarization (linear, circular or cylindrical) using a low-NA mode. In order to perform a more detailed analysis of a polarization state, sharp focusing should be used.

Sharp focusing easily distinguishes azimuthal polarization; it is the only type of polarization for which there is zero intensity value in the central focal point. Linear polarization can be detected by focal spot elongation in the direction of the polarization axis. An additional first-order vortex phase can be used to recognize circular polarization direction. The orthogonal linear polarization can be detected with the use of a binary phase that contains optical vortices of opposite signs.

We have shown that, with sharp focusing, the multi-order diffractive optical element combined with various superpositions of vortex phases can unequivocally distinguish singular polarization states (circular, radial and azimuthal). Unambiguous recognition is provided by the inter-relation of polarization and phase singularities.

We show that the diffractive axicon provides sharper focusing than a micro-objective with the same NA. The greater the focusing, the more confident the detection of an incident beam polarization state. For instance, in the case of a linear polarization, the focal spot elongation in the direction of the polarization axis becomes more obvious. In addition, strengthening the longitudinal component allows us to positively recognize the orthogonal cylindrical and circular polarization states. The second-order optical vortex provides essential distinctions between the focal images.

Numerical simulations are performed for a high-NA diffractive axicon by means of the plane wave expansion method and FDTD. There is good agreement between the methods, with small quantitative differences.

Thus, singular phase elements inserted in a high-NA focusing system can be used to detect different incident beam polarization states.

Acknowledgments

The work was financially supported by the Russian Foundation for Basic Research (grants 13-07-00266, 14-07-31079 mol_a) and by the Ministry of Education and Science of Russian Federation.

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30. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef]   [PubMed]  

31. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]   [PubMed]  

32. A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18(10), 10848–10863 (2010). [CrossRef]   [PubMed]  

33. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010). [CrossRef]   [PubMed]  

34. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009). [CrossRef]  

35. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef]   [PubMed]  

36. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992). [CrossRef]  

37. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998). [CrossRef]  

38. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef]   [PubMed]  

39. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002). [CrossRef]   [PubMed]  

40. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).

41. S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012). [CrossRef]   [PubMed]  

42. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002). [CrossRef]   [PubMed]  

43. S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

44. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley & Sons Inc., 2002).

45. S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

46. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

47. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef]   [PubMed]  

48. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef]   [PubMed]  

49. N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001). [CrossRef]  

50. S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010). [CrossRef]   [PubMed]  

51. V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007). [CrossRef]   [PubMed]  

52. C. J. Zapata-Rodríguez and A. Sánchez-Losa, “Three-dimensional field distribution in the focal region of low-Fresnel-number axicons,” J. Opt. Soc. Am. A 23(12), 3016–3026 (2006). [CrossRef]   [PubMed]  

53. V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

54. S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012). [CrossRef]  

55. S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013). [CrossRef]  

56. S. N. Khonina, N. L. Kazanskiĭ, A. V. Ustinov, and S. G. Volotovskiĭ, “The lensacon: nonparaxial effects,” J. Opt. Technol. 78(11), 724–729 (2011). [CrossRef]  

57. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]  

58. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, “Theory of Waves,” Nauka, Moscow, 384 p (1979).

59. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010). [CrossRef]  

60. S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004). [CrossRef]  

61. K. Singh, Perspectives in Engineering Optics (Anita Publications, Delhi, 2003).

62. S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013). [CrossRef]  

References

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  13. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
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  15. L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004).
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  16. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
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  17. I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010).
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  18. S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
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  19. S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013).
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  20. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).
  21. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
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  22. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
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  23. V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).
  24. P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004).
    [Crossref] [PubMed]
  25. M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).
  26. Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).
  27. D. L. Andrews, Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier Inc., 2008).
  28. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–204 (2011).
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  29. S. N. Khonina and I. Golub, “How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy,” J. Opt. Soc. Am. A 29(10), 2242–2246 (2012).
    [Crossref] [PubMed]
  30. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003).
    [Crossref] [PubMed]
  31. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref] [PubMed]
  32. A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18(10), 10848–10863 (2010).
    [Crossref] [PubMed]
  33. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010).
    [Crossref] [PubMed]
  34. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
    [Crossref]
  35. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
    [Crossref] [PubMed]
  36. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
    [Crossref]
  37. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
    [Crossref]
  38. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990).
    [Crossref] [PubMed]
  39. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002).
    [Crossref] [PubMed]
  40. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).
  41. S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
    [Crossref] [PubMed]
  42. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
    [Crossref] [PubMed]
  43. S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).
  44. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley & Sons Inc., 2002).
  45. S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).
  46. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  47. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
    [Crossref] [PubMed]
  48. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
    [Crossref] [PubMed]
  49. N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
    [Crossref]
  50. S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010).
    [Crossref] [PubMed]
  51. V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
    [Crossref] [PubMed]
  52. C. J. Zapata-Rodríguez and A. Sánchez-Losa, “Three-dimensional field distribution in the focal region of low-Fresnel-number axicons,” J. Opt. Soc. Am. A 23(12), 3016–3026 (2006).
    [Crossref] [PubMed]
  53. V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).
  54. S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
    [Crossref]
  55. S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
    [Crossref]
  56. S. N. Khonina, N. L. Kazanskiĭ, A. V. Ustinov, and S. G. Volotovskiĭ, “The lensacon: nonparaxial effects,” J. Opt. Technol. 78(11), 724–729 (2011).
    [Crossref]
  57. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  58. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, “Theory of Waves,” Nauka, Moscow, 384 p (1979).
  59. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
    [Crossref]
  60. S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
    [Crossref]
  61. K. Singh, Perspectives in Engineering Optics (Anita Publications, Delhi, 2003).
  62. S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
    [Crossref]

2013 (4)

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013).
[Crossref]

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
[Crossref]

2012 (5)

2011 (3)

2010 (5)

2009 (3)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).

2008 (3)

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

2007 (3)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[Crossref] [PubMed]

2006 (2)

C. J. Zapata-Rodríguez and A. Sánchez-Losa, “Three-dimensional field distribution in the focal region of low-Fresnel-number axicons,” J. Opt. Soc. Am. A 23(12), 3016–3026 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2005 (2)

A. S. Desyatnikov, L. Torner, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Prog. Opt. 47, 219–319 (2005).

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

2004 (7)

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004).
[Crossref] [PubMed]

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004).
[Crossref]

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[Crossref] [PubMed]

S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[Crossref]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003).
[Crossref] [PubMed]

2002 (3)

2001 (3)

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

M. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219 (2001).

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

1998 (1)

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

1997 (1)

1994 (1)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

1990 (1)

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

1936 (2)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137(3453), 31 (1936).
[Crossref]

Alferov, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
[Crossref] [PubMed]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Angelsky, O. V.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

Biener, G.

Bokor, N.

Bomzon, Z.

Brasselet, E.

Chen, Z.

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

Ciattoni, A.

Cincotti, G.

Cottrell, D. M.

Courtial, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Dai, G.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Davidson, N.

Davis, J. A.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

Desyatnikov, A.

Desyatnikov, A. S.

A. S. Desyatnikov, L. Torner, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Prog. Opt. 47, 219–319 (2005).

Dhayalan, V.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

Dholakia, K.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Ebihara, T.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

Fadeyeva, T. A.

Ford, D. H.

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Freund, I.

Golub, I.

Hasman, E.

Hayashi, N.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Helseth, L. E.

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004).
[Crossref]

Holbourn, A. H. S.

A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137(3453), 31 (1936).
[Crossref]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Izdebskaya, Y. V.

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Kalosha, V. P.

Karpeev, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
[Crossref] [PubMed]

Kazanskii, N. L.

Kazanskiy, N. L.

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

Khonina, S. N.

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013).
[Crossref]

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
[Crossref]

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
[Crossref]

S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
[Crossref] [PubMed]

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

S. N. Khonina and I. Golub, “How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy,” J. Opt. Soc. Am. A 29(10), 2242–2246 (2012).
[Crossref] [PubMed]

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

S. N. Khonina, N. L. Kazanskiĭ, A. V. Ustinov, and S. G. Volotovskiĭ, “The lensacon: nonparaxial effects,” J. Opt. Technol. 78(11), 724–729 (2011).
[Crossref]

S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010).
[Crossref] [PubMed]

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Kimura, W. D.

Kivshar, Y. S.

Kleiner, V.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

Krolikowski, W.

Laukkanen, J.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Levenson, M. D.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

Meixner, A. J.

Mokhun, A. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Mokhun, I. I.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Moreno, I.

Morikawa, Y.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Morozov, A. A.

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

Munro, P.

Neshev, D. N.

Nesterenko, D. V.

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–204 (2011).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

Palma, C.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Pereira, S. F.

S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[Crossref]

Pu, J.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Pustovoi, I. A.

S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Ruiz, I.

Sánchez-Losa, A.

Savel’ev, D. A.

S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
[Crossref]

Savelyev, D. A.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
[Crossref]

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

Serafimovich, P. G.

S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
[Crossref]

Sergienko, N.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Shvedov, V. G.

Simpson, N. B.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Skidanov, R. V.

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

Soifer, V. A.

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Soskin, M.

M. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219 (2001).

Soskin, M. S.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Stadler, J.

Stafeev, S. S.

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

Stamnes, J. J.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

Stanciu, C.

Stupperich, C.

Tan, S. M.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Tidwell, S. C.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

A. S. Desyatnikov, L. Torner, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Prog. Opt. 47, 219–319 (2005).

Török, P.

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Turunen, J.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Unno, Y.

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

Ustinov, A. V.

van de Nes, A. S.

S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[Crossref]

Vasnetsov, M. V.

M. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219 (2001).

Volotovskii, S. G.

Volotovsky, S. G.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010).
[Crossref] [PubMed]

Volyar, A. V.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Yao, A. M.

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Zapata-Rodríguez, C. J.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).

Zhao, D.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

Adv. Opt. Photon. (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–204 (2011).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).

Appl. Opt. (1)

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

J. Exp. Theor. Phys. (1)

S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
[Crossref]

J. Microlithogr., Microfabr., Microsyst. (2)

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

J. Mod. Opt. (4)

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58(9), 748–760 (2011).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

J. Opt. (1)

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Opt. Technol (1)

S. N. Khonina, D. A. Savel’ev, I. A. Pustovoĭ, and P. G. Serafimovich, “Diffraction at binary microaxicons in the near field,” J. Opt. Technol 79(10), 22–29 (2012).
[Crossref]

J. Opt. Technol. (1)

Laser Photon. Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Nature (1)

A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137(3453), 31 (1936).
[Crossref]

Opt. Commun. (4)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004).
[Crossref]

S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[Crossref]

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

Opt. Eng. (1)

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Opt. Lett. (8)

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[Crossref] [PubMed]

V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[Crossref] [PubMed]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002).
[Crossref] [PubMed]

S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
[Crossref] [PubMed]

Opt. Mem. Neur. Netw. (1)

S. N. Khonina, D. V. Nesterenko, A. A. Morozov, R. V. Skidanov, and V. A. Soifer, “Narrowing of a light spot at diffraction of linearly-polarized beam on binary asymmetric axicons,” Opt. Mem. Neur. Netw. 21(1), 17–26 (2012).

Phys. Part. Nucl. (1)

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Proc. SPIE (1)

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

Prog. Electromagnetics Res. (1)

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

Prog. Opt. (4)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

M. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219 (2001).

A. S. Desyatnikov, L. Torner, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Prog. Opt. 47, 219–319 (2005).

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

Other (5)

D. L. Andrews, Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier Inc., 2008).

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (John Wiley & Sons Inc., 2002).

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K. Singh, Perspectives in Engineering Optics (Anita Publications, Delhi, 2003).

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Figures (4)

Fig. 1
Fig. 1 Transmission function of the focusing system. (a) Binary phase of multi-order diffractive optical element. (b) Accordance of diffractive orders to combinations of optical vortices.
Fig. 2
Fig. 2 Detection of orthogonal linear polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.
Fig. 3
Fig. 3 Detection of orthogonal circular polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.
Fig. 4
Fig. 4 Detection of orthogonal cylindrical polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.

Tables (5)

Tables Icon

Table 1 Dependence of on-axis intensity components due to superposition in Eq. (4)

Tables Icon

Table 2 Focal intensity distribution for Gaussian beam, NA = 0.25 (red color for x-component, green for y-component, picture size is 8λ × 8λ)

Tables Icon

Table 3 Focal intensity distribution for Gaussian beam, NA = 0.95 (red color for x-component, green for y-component, and blue for z-component; picture size is 2λ × 2λ)

Tables Icon

Table 4 Intensity distributions of Gaussian beam focused by the axicon in Eq. (21) with α0 = 0.95 (red color for x-component, green for y-component, and blue for z-component, picture size 4λ × 4λ)

Tables Icon

Table 5 Intensity distributions in the transverse planes of the Gaussian beam, Laguerre-Gaussian mode (0,1) and Hermite-Gaussian mode (0,1) focused by the axicon with NA = 0.95 (blue color for z-component, yellow for all-components, picture size 6λ × 6λ)

Equations (32)

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E ( ρ , φ , z ) = i f λ 0 α 0 2 π B ( θ , ϕ ) T ( θ ) P ( θ , ϕ ) exp [ i k ( ρ sin θ cos ( ϕ φ ) + z cos θ ) ] sin θ d θ d ϕ ,
P ( θ , ϕ ) = [ 1 + cos 2 ϕ ( cos θ 1 ) sin ϕ cos ϕ ( cos θ 1 ) cos ϕ sin θ sin ϕ cos ϕ ( cos θ 1 ) 1 + sin 2 ϕ ( cos θ 1 ) sin ϕ sin θ sin θ cos ϕ sin θ sin ϕ cos θ ] [ a ( ϕ , θ ) b ( ϕ , θ ) c ( ϕ , θ ) ] ,
B ( θ , ϕ ) = R ( θ ) Ω ( ϕ ) ,
Ω ( ϕ ) = m = M 1 M 2 d m exp ( i m ϕ ) ,
E ( ρ , φ , z ) = i k f m = M 1 M 2 d m i m e i m φ 0 α Q m ( ρ , φ , θ ) R ( θ ) T ( θ ) sin θ exp ( i k z cos θ ) d θ ,
Q m ( ρ , φ , θ ) = [ J m ( t ) + 1 4 [ 2 J m ( t ) e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) i 2 [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] sin θ ] ,
Q m ( ρ , φ , θ ) = [ i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) J m ( t ) + 1 4 [ 2 J m ( t ) + e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) 1 2 [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ J m ( t ) + 1 2 [ J m ( t ) e ± i 2 φ J m ± 2 ( t ) ] ( cos θ 1 ) ± i { J m ( t ) + 1 2 ( J m ( t ) + e ± i 2 φ J m ± 2 ( t ) ) ( cos θ 1 ) } i e ± i φ J m ± 1 ( t ) sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] cos θ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] cos θ 2 J m ( t ) sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 0 ] .
e x ± i e y = ( e r cos φ e φ sin φ ) ± i ( e r sin φ + e φ cos φ ) = exp ( ± i φ ) [ e r ± i e φ ]
e r = e x cos φ + e y sin φ = 1 2 exp ( i φ ) [ e x i e y ] + 1 2 exp ( i φ ) [ e x + i e y ] ,
e φ = e x sin φ + e y cos φ = 1 2 exp ( i φ ) [ e y + i e x ] + 1 2 exp ( i φ ) [ e y + i e x ] .
E ( ρ , φ , z ) = 1 λ 2 σ 1 σ 2 0 2 π M ( σ , ϕ ) ( F x ( σ , ϕ ) F y ( σ , ϕ ) ) exp [ i k z 1 σ 2 ] exp [ i k σ ρ cos ( φ ϕ ) ] σ d σ d ϕ ,
M ( σ , ϕ ) = [ 1 cos 2 ϕ ( 1 γ ) cos ϕ sin ϕ ( 1 γ ) σ cos ϕ cos ϕ sin ϕ ( 1 γ ) 1 sin 2 ϕ ( 1 γ ) σ sin ϕ ] , γ = 1 σ 2 ,
( F x ( σ , ϕ ) F y ( σ , ϕ ) ) = 0 r 0 0 2 π ( E 0 x ( r , τ ) E 0 y ( r , τ ) ) exp [ i k r σ cos ( τ ϕ ) ] r d r d τ .
E 0 ( r , τ ) = E 0 ( r ) exp ( i m τ ) ,
E ( ρ , φ , z ) = k 2 i 2 m exp ( i m φ ) σ 1 σ 2 S m ( ρ , φ , σ ) ( P x ( σ ) P y ( σ ) ) exp ( i k z 1 σ 2 ) σ d σ ,
S m ( ρ , φ , σ ) = [ 1 4 { 2 J m ( t ) ( 1 + γ ) + [ e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( 1 γ ) } i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( 1 γ ) i 2 σ [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( 1 γ ) 1 4 { 2 J m ( t ) ( 1 + γ ) [ e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( 1 γ ) } 1 2 σ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] ] ,
( P x ( σ ) P y ( σ ) ) = 0 R ( E 0 x ( r ) E 0 y ( r ) ) J m ( k r σ ) r d r .
E 0 ( r ) = exp ( i k α 0 r ) ,
P ( σ ) δ ( σ α 0 ) ,
E ( ρ , φ , z ) i 2 m α 0 exp ( i m φ ) exp ( i k z 1 α 0 2 ) S m ( ρ , φ , α 0 ) ( c x c y ) ,
E ( ρ , φ , z ) α 0 2 ( c x J m ( t ) ( 1 + γ 0 ) + 1 2 { [ e i 2 φ J m + 2 ( t ) ( c x i c y ) + e i 2 φ J m 2 ( t ) ( c x + i c y ) ] } ( 1 γ 0 ) c y J m ( t ) ( 1 + γ 0 ) 1 2 { [ e i 2 φ J m + 2 ( t ) ( c y i c x ) + e i 2 φ J m 2 ( t ) ( c y + i c x ) ] } ( 1 γ 0 ) α 0 [ e i φ J m + 1 ( t ) ( c y + i c x ) + e i φ J m 1 ( t ) ( c y i c x ) ] ) .
| E ( ρ , φ , z ) | 2 ε 2 ( | c x | 2 + | c y | 2 ) J m 2 ( k ε ρ ) ,
| E ( 0 , 0 , z ) | 2 | c x | 2 + | c y | 2 4
| E ( 0 , 0 , z ) | 2 | c y i c x | 2 4
| E ( 0 , 0 , z ) | 2 | c x ± i c y | 2 + | c y ± i c x | 2 16
S m c ( ρ , φ , σ ) = 1 2 [ i γ [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] γ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 2 σ J m ( t ) 0 ] .
S m c ( ρ , φ , α 0 ) = 1 2 [ i γ 0 [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] γ 0 [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 2 α 0 J m ( t ) 0 ] .
| E ( 0 , 0 , z ) | 2 | c r | 2
| E ( 0 , 0 , z ) | 2 | c φ | 2

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