Nonparaxial and paraxial focusing of azimuthal-variant vector beams

doi:10.1364/OE.20.017684

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Nonparaxial and paraxial focusing of azimuthal-variant vector beams

Bing Gu and Yiping Cui »View Author Affiliations

Optics Express, Vol. 20, Issue 16, pp. 17684-17694 (2012)
http://dx.doi.org/10.1364/OE.20.017684

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▸ Article Outline
– Abstract
1. Introduction
2. Theory
3. Numerical results and discussions
4. Conclusion
– References and links

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Abstract

Based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation, we investigate the propagation behavior of a lowest-order Laguerre-Gaussian beam with azimuthal-variant states of polarization. We present the analytical expressions for the radial, azimuthal, and longitudinal components of the electric field with an arbitrary integer topological charge m focused by a nonaperturing thin lens. We illustrate the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations.

1. Introduction

During the past decade, the beam vectorial characteristic and propagation behavior have received extensive attention due to the academic interest and technological applications [1

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

, 2

J. S. Ahn, H. W. Kihm, J. E. Kihm, D. S. Kim, and K. G. Lee, “3-dimensional local field polarization vector mapping of a focused radially polarized beam using gold nanoparticle functionalized tips,” Opt. Express 17, 2280–2286 (2009). [CrossRef] [PubMed]

]. Owing to the advent of new optical structures, such as micro-cavities and photonic crystals, the beam width is comparable or even small than the wavelength. Besides, high numerical aperture focusing has been adopted in near-field optical microscope [3

A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006). [CrossRef] [PubMed]

], optical trapping and manipulating nanoparticles [4

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010). [CrossRef]

], and etc. Accordingly, the beam’s divergence angle becomes large. Under the above-mentioned conditions, the conventional paraxial theory fails at describing the beam propagation. Fortunately, many propagation approaches beyond the paraxial approximation have been developed to describe the nonparaxial vectorial beam propagation, including the Rayleigh-Sommerfeld integrals [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

], the vector angular spectrum method [6

X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. 36, 295–297 (2011). [CrossRef] [PubMed]

], the multiscale singular perturbation method [7

D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagation in free space,” J. Opt. Soc. Am. A 24, 3317–3325 (2007). [CrossRef]

], and Richards-Wolf theory [8

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

]. By application of these approaches, research efforts have been focused on the nonparaxial propagation of a variety of laser beams, such as the cylindrically polarized Laguerre-Gaussian beams [1

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

], spirally polarized beams [9

R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]

], controllable dark-hollow beams [10

Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25, 537–542 (2008). [CrossRef]

], hollow Gaussian beams [11

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049 (2009). [CrossRef]

], vectorial Laguerre-Bessel-Gaussian beams [12

Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942–11951 (2007). [CrossRef] [PubMed]

], Lorentz-Gaussian beams [13

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26, 141–147 (2009). [CrossRef]

], Gaussian vortex beams [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

], partially coherent dark hollow beams [14

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011). [CrossRef]

], and beams diffracted at a circular aperture [15

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003). [CrossRef] [PubMed]

–17

X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express 18, 7064–7075 (2010). [CrossRef] [PubMed]

] and a rectangular aperture [18

K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004). [CrossRef]

Recently, interest in cylindrical vector laser beams has been prompted by their intriguing applications in single molecule detection [19

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]

], optical micro-fabrication [20

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011). [CrossRef] [PubMed]

], and so on. To gain an insight on the underlying physical mechanisms for the light-matter interaction, it is desirable to precisely determine both the focal field and the propagation behavior of the focused cylindrical vector beam. Greene et al. [21

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]

] explored the focal shift in vector beams under the weak focusing condition. Youngworth and Brown [22

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

] reported the high numerical aperture focusing of cylindrical vector beams in the nonparaxial limit. Deng et al. [23

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

] studied the nonparaxial and paraxial propagation of radially polarized elegant beams. Rashid et al. [24

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009). [CrossRef]

] analyzed the focal field of high order cylindrical vector beams in the limit of high numerical aperture. Deng et al. [25

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

] and Baberjee et al. [26

P. P. Banerjee, G. Cook, and D. R. Evans, “A q-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A 26, 1366–1374 (2009). [CrossRef]

] investigated the nonparaxial and paraxial propagation of radially polarized Gaussian beams, respectively. Especially, Dorn et al. [27

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

] first experimentally demonstrated the strong focusing of a radially polarized field distribution with annular aperture.

In this work, we address the nonparaxial and paraxial propagations of an azimuthal-variant vector beam with arbitrary integer topological charge m. The vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation developed by Kotlyar et al. [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

,28

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13, 075703 (2011). [CrossRef]

], which has been validated by the finite-difference time domain, is adopted to study the nonparaxial propagation of azimuthal-variant vector beams. We present the analytical expressions for the radial, azimuthal, and longitudinal components of the lowest-order Laguerre-Gaussian beam with an arbitrary integer topological charge m focused by a nonaperturing thin lens. By numerical illustration, we investigate the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations. For the special case of m = 1, the results are in agreement with the ones reported previously [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

, 26

2. Theory

In the polar coordinate system, the transverse electric field distribution of an azimuthal-variant vector beam at the plane z = 0 can be expressed by [1

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

, 29

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20, 149–157 (2012). [CrossRef] [PubMed]

]

E (r, ϕ, 0) = E_{r} (r, ϕ, 0) {\hat{e}}_{r} + E_{ϕ} (r, ϕ, 0) {\hat{e}}_{ϕ},

(1)

where

E_{r} (r, ϕ, 0) = A (r) cos (m ϕ - ϕ + φ_{0}),

(2a)

E_{ϕ} (r, ϕ, 0) = A (r) sin (m ϕ - ϕ + φ_{0}) .

(2b)

Here ê_r and ê_ϕ are the unit vectors in the polar coordinate system (r,ϕ), A(r) represents the radial-dependent amplitude, m is the azimuthal topological charge, and φ₀ is the initial phase of the vector beam. Two extreme cases of vector beams are the radially and azimuthally polarized vector beams for m = 1 with φ₀ = 0 and π/2, respectively. Note that the azimuthal-variation vector beam belongs to a kind of local linearly polarized vector field. And that the spatial distribution of states of polarization is dependent on the azimuthal angle ϕ only.

Based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation, the three-dimensional electric field for the propagating beam in free space along the +z direction can be given by [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

]

\begin{array}{l} E_{r} (ρ, θ, z) & = & - \frac{i k z}{2 π ξ^{2}} exp (i k ξ) \\ \times \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, ϕ + θ, 0) cos ϕ - E_{ϕ} (r, ϕ + θ, 0) sin ϕ] \\ \times exp (\frac{i k r^{2}}{2 ξ}) exp (- i γ r cos ϕ) r d r d ϕ, \end{array}

(3a)

\begin{array}{l} E_{ϕ} (ρ, θ, z) & = & - \frac{i k z}{2 π ξ^{2}} exp (i k ξ) \\ \times \int_{0}^{\infty} \int_{0}^{2 π} [E_{r} (r, ϕ + θ, 0) sin ϕ + E_{ϕ} (r, ϕ + θ, 0) cos ϕ] \\ \times exp (\frac{i k r^{2}}{2 ξ}) exp (- i γ r cos ϕ) r d r d ϕ, \end{array}

(3b)

\begin{array}{l} E_{z} (ρ, θ, z) & = & \frac{i k}{2 π ξ^{2}} exp (i k ξ) \\ \times \int_{0}^{\infty} \int_{0}^{2 π} {E_{r} (r, ϕ + θ, 0) [r - ρ cos ϕ] + E_{ϕ} (r, ϕ + θ, 0) ρ sin ϕ} \\ \times exp (\frac{i k r^{2}}{2 ξ}) exp (- i γ r cos ϕ) r d r d ϕ, \end{array}

(3c)

where γ = kρ/ξ,

ξ = \sqrt{z^{2} + ρ^{2}}

, k = 2π/λ, and λ is the wavelength. Note that the weak nonparaxial approximation takes the form

\sqrt{z^{2} + ρ^{2} + r^{2} - 2 ρ r cos (ϕ - θ)} ≃ \sqrt{z^{2} + ρ^{2}} + \frac{r^{2}}{2 \sqrt{z^{2} + ρ^{2}}} - \frac{ρ r cos (ϕ - θ)}{\sqrt{z^{2} + ρ^{2}}}

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

,28

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13, 075703 (2011). [CrossRef]

], which holds true under the conditions of ω < λ and z ≥ ω²/(2λ) [30

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002). [CrossRef]

], where ω is the waist width. This approximation to describe the nonparaxial propagation of light beams is well known [12

Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942–11951 (2007). [CrossRef] [PubMed]

, 17

X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express 18, 7064–7075 (2010). [CrossRef] [PubMed]

, 23

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

The integrations over ϕ for an integer m ≥ 0 can be accomplished using the identities

\int_{0}^{2 π} cos (m ϕ + φ_{0}) exp [- i x cos (ϕ - θ)] d ϕ \equiv 2 π {(- i)}^{m} J_{m} (x) cos (m θ + φ_{0}),

(4a)

\int_{0}^{2 π} sin (m ϕ + φ_{0}) exp [- i x cos (ϕ - θ)] d ϕ \equiv 2 π {(- i)}^{m} J_{m} (x) sin (m θ + φ_{0}),

(4b)

where J_m(·) is the Bessel function of mth-order.

Substituting Eq. (2) into Eq. (3) and using Eq. (4), we obtain

E_{r} (ρ, θ, z) = \frac{{(- i)}^{m + 1} k z}{ξ^{2}} exp (i k ξ) cos Ψ \int_{0}^{\infty} A (r) exp (\frac{i k r^{2}}{2 ξ}) J_{m} (γ r) r d r,

(5a)

E_{ϕ} (ρ, θ, z) = \frac{{(- i)}^{m + 1} k z}{ξ^{2}} exp (i k ξ) sin Ψ \int_{0}^{\infty} A (r) exp (\frac{i k r^{2}}{2 ξ}) J_{m} (γ r) r d r,

(5b)

E_{z} (ρ, θ, z) = \frac{{(- i)}^{m + 1} k}{ξ^{2}} exp (i k ξ) cos Ψ \int_{0}^{\infty} A (r) [ρ J_{m} (γ r) - i r J_{m - 1} (γ r)] exp (\frac{i k r^{2}}{2 ξ}) r d r,

(5c)

where Ψ = mθ − θ + φ₀.

For the sake of simplicity, we only concern with the propagation of a lowest-order Laguerre-Gaussian beam (i.e., radially polarized elegant Gaussian beam) with the electric field distribution in the initial plane given by [23

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

, 25

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

]

A (r) = E_{0} (\frac{\sqrt{2}}{ω_{0}}) r exp (- α r^{2}),

(6)

where ω₀ is the waist radius of the Gaussian beam, and E₀ is an amplitude constant. One takes

α = 1 / ω_{0}^{2}

for the beam under propagational diffraction in free space. For the beam focused by a nonaperturing thin lens with a geometric focal length of f, we have

α = 1 / ω_{0}^{2} + i k / (2 f)

Inserting Eq. (6) into Eq. (5) and making use of the integral theorems [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

]

\int_{0}^{\infty} r^{2} e^{- β r^{2}} J_{m} (γ r) d r = \frac{\sqrt{π}}{4 β^{3 / 2}} e^{- t} [2 t I_{(m - 2) / 2} (t) - (m - 1 + 2 t) I_{m / 2} (t)],

(7a)

\int_{0}^{\infty} r^{3} e^{- β r^{2}} J_{m - 1} (γ r) d r = \frac{\sqrt{π} γ}{16 β^{5 / 2}} e^{- t} [(m + 1 - 4 t) I_{(m - 2) / 2} (t) + (m - 3 + 4 t) I_{m / 2} (t)],

(7b)

where Re[m] > −1, Re[β] > 0, γ > 0, t = γ²/(8β), and I_m(·) is the modified Bessel function of mth-order, we yield the analytical results

\begin{array}{l} E_{r} (ρ, θ, z) & = & \frac{{(- i)}^{m + 1} \sqrt{π} E_{0} k z}{2 \sqrt{2} β^{3 / 2} ω_{0} ξ^{2}} exp (i k ξ - t) cos Ψ \\ \times [2 t I_{(m - 2) / 2} (t) - (m - 1 + 2 t) I_{m / 2} (t)], \end{array}

(8a)

\begin{array}{l} E_{ϕ} (ρ, θ, z) & = & \frac{{(- i)}^{m + 1} \sqrt{π} E_{0} k z}{2 \sqrt{2} β^{3 / 2} ω_{0} ξ^{2}} exp (i k ξ - t) sin Ψ \\ \times [2 t I_{(m - 2) / 2} (t) - (m - 1 + 2 t) I_{m / 2} (t)], \end{array}

(8b)

\begin{array}{l} E_{z} (ρ, θ, z) & = & \frac{- {(- i)}^{m + 1} \sqrt{π} E_{0} k}{2 \sqrt{2} β^{3 / 2} ω_{0} ξ^{2}} exp (i k ξ - t) cos Ψ \\ \times {[\frac{i γ}{4 β} (m + 1 - 4 t) - 2 ρ t] I_{(m - 2) / 2} (t) \\ + [\frac{i γ}{4 β} (m - 3 + 4 t) + ρ (m - 1 + 2 t)] I_{m / 2} (t)}, \end{array}

(8c)

where β = α − ik/(2ξ) and t = γ²/(8β). Equation (8), which is the basic result of the present work, gives a general three-dimensional electric field of the nonparaxial propagation of azimuthal-variant vector beams.

For the case of m = 1 and φ₀ = 0, we deduce the nonparaxial propagation of a radially polarized vector beam from Eq. (8) as follows

E_{r} (ρ, θ, z) = \frac{- E_{0} k^{2} ρ z}{2 \sqrt{2} ω_{0} α^{2} q^{2} ξ} exp (i k ξ - \frac{k^{2} ρ^{2}}{4 α q ξ}),

(9a)

E_{ϕ} (ρ, θ, z) = 0,

(9b)

E_{z} (ρ, θ, z) = \frac{i E_{0} k}{\sqrt{2} ω_{0} α^{2} q^{2}} (1 + \frac{i k ρ^{2}}{2 q}) exp (i k ξ - \frac{k^{2} ρ^{2}}{4 α q ξ}),

(9c)

where q = ξ − ik/(2α). The obtained result is coincident with the one reported previously [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

]. For an azimuthally polarized vector beam (m = 1 and φ₀ = π/2), we yield

E_{r} (ρ, θ, z) = 0,

(10a)

E_{ϕ} (ρ, θ, z) = \frac{- E_{0} k^{2} ρ z}{2 \sqrt{2} ω_{0} α^{2} q^{2} ξ} exp (i k ξ - \frac{k^{2} ρ^{2}}{4 α q ξ}),

(10b)

E_{z} (ρ, θ, z) = 0 .

(10c)

The paraxial propagation result can be regarded as a special case of the nonparaxial result described by Eq. (8). Under the paraxial approximation, one gets (z² +ρ²)^1/2 ≈ z +ρ²/(2z) ≈ z. Accordingly, we obtain the propagation expressions for the azimuthal-variant vector beam under the paraxial approximation as

\begin{array}{l} E_{r p} (ρ, θ, z) & = & \frac{{(- i)}^{m + 1} \sqrt{π} E_{0} k}{2 \sqrt{2} {β^{'}}^{3 / 2} ω_{0} z} exp (i k z - t^{'}) cos Ψ \\ \times [2 t^{'} I_{(m - 2) / 2} (t^{'}) - (m - 1 + 2 t^{'}) I_{m / 2} (t^{'})], \end{array}

(11a)

\begin{array}{l} E_{ϕ p} (ρ, θ, z) & = & \frac{{(- i)}^{m + 1} \sqrt{π} E_{0} k}{2 \sqrt{2} {β^{'}}^{3 / 2} ω_{0} z} exp (i k z - t^{'}) sin Ψ \\ \times [2 t^{'} I_{(m - 2) / 2} (t^{'}) - (m - 1 + 2 t^{'}) I_{(m / 2)} (t^{'})], \end{array}

(11b)

\begin{array}{l} E_{z p} (ρ, θ, z) & = & \frac{- {(- i)}^{m + 1} \sqrt{π} E_{0} k}{2 \sqrt{2} {β^{'}}^{3 / 2} ω_{0} z^{2}} exp (i k z - t^{'}) cos Ψ \\ \times {[\frac{i γ^{'}}{4 β^{'}} (m + 1 - 4 t^{'}) - 2 ρ t^{'}] I_{(m - 2) / 2} (t^{'}) \\ + [\frac{i γ^{'}}{4 β^{'}} (m - 3 + 4 t^{'}) + ρ (m - 1 + 2 t^{'})] I_{m / 2} (t^{'})}, \end{array}

(11c)

where γ′ = kρ/z, β′ = α − ik/(2z), and t′ = γ′²/(8β′). In particular, for a radially polarized vector beam (m = 1 and φ₀ = 0), one gets [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

, 26

]

E_{r p} (ρ, θ, z) = \frac{- E_{0} k^{2} ρ}{2 \sqrt{2} ω_{0} α^{2} q^{2}} exp (i k z + \frac{i k ρ^{2}}{2 q}),

(12a)

E_{ϕ p} (ρ, θ, z) = 0,

(12b)

E_{z p} (ρ, θ, z) = \frac{i E_{0} k}{\sqrt{2} ω_{0} α^{2} q^{2}} (1 + \frac{i k ρ^{2}}{2 q}) exp (i k z + \frac{i k ρ^{2}}{2 q}),

(12c)

where q ≃ z−ik/(2α). For the case of m = 1 and φ₀ = π/2, we obtain the paraxial propagation of an azimuthally polarized vector beam as

E_{r p} (ρ, θ, z) = 0,

(13a)

E_{ϕ p} (ρ, θ, z) = \frac{- E_{0} k^{2} ρ}{2 \sqrt{2} ω_{0} α^{2} q^{2}} exp (i k z + \frac{i k ρ^{2}}{2 q}),

(13b)

E_{z p} (ρ, θ, z) = 0 .

(13c)

For a cylindrical vector beam, its focused field is the so-called doughnut light field with a central dark spot and an outer bright ring [24

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009). [CrossRef]

, 29

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20, 149–157 (2012). [CrossRef] [PubMed]

]. To define the width of a vector beam, in general, one adopts an encircled-power criterion, i.e., the width ρ₀ of a vector beam as that radius within 80% of the beam’s power is enclosed. Accordingly, the width ρ₀ of a focused vector beam at the propagation distance z satisfies [21

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]

]

\frac{\int_{0}^{2 π} \int_{0}^{ρ_{0}} I_{G} (ρ, θ, z) ρ d ρ d θ}{\int_{0}^{2 π} \int_{0}^{\infty} I_{G} (ρ, θ, z) ρ d ρ d θ} = 0.8,

(14)

where I_G = |E_r|² + |E_ϕ|² + |E_z|² is the nonparaxial intensity of the vector beam. For the case of the paraxial propagation, one should replace I_G by I_GP = |E_rp|² + |E_ϕp|² + |E_zp|² in Eq. (14).

The energy flux distribution at the z plane can be given by the time-average of the z component of the Poynting vector,

S_{z} = \frac{1}{2} Re {[E (ρ, θ, z) \times H^{*} (ρ, θ, z)]}_{z} .

(15)

Here Re[·] is the real part and the asterisk denotes the complex conjugate. The magnetic fields H(ρ, θ, z) under the nonparaxial and paraxial approximations are easily obtained by taking the curl of Eqs. (8) and (11), respectively.

3. Numerical results and discussions

To investigate the nonparaxial propagation characteristics of the focused azimuthal-variant vector beams, we take the typical parameters as λ = 633 nm, E₀ = 1 (a.u.), ω₀ = 1 μm, and f = 4 μm [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

]. Detailed numerical simulations have been performed using the formulae derived in Section 2 and the results are shown in Figs. 1–3.

Fig. 1 Nonparaxial intensity patterns (top row) and the corresponding cross-section intensity profiles when y = 0 (middle row) of a vector beam with m = 1 for φ₀ = π/4 at the plane of the lens’ geometrical focus by taking λ = 633 nm, E₀ = 1 (a.u.), ω₀ = 1 μm, and f = 4 μm. The intensity patterns are normalized by

I_{G}^{Max} (x, y, f)

. Circles in (e)–(h) give the corresponding paraxial intensity profiles. The bottom row gives the nonparaxial intensity patterns through the focus, normalized by

I_{G}^{Max} (x, 0, z)

. Dotted and dashed lines in (l) are positions of the true focus and the lens’ geometrical focus, respectively.

Fig. 2 Beam waists ρ₀ of nonparaxial (paraxial) focused vector beam with m = 1 and φ₀ = π/4 through focus for λ = 633 nm, E₀ = 1 (a.u.), ω₀ = 1 μm, and f = 4 μm.

Fig. 3 Nonparaxial energy flux patterns of a vector beam with m = 1 at the planes of (a) the lens’ geometrical focus and (b) the true focus. Solid lines (circles) in (c) and (d) denote the corresponding nonparaxial (paraxial) energy flux profiles for y = 0 at z = f and z = 0.55 f, respectively. Numerical parameters: φ₀ = π/4, λ = 633 nm, E₀ = 1 (a.u.), ω₀ = 1 μm, and f = 4 μm.

First we explore the nonparaxial intensity distributions of a vector beam near the focal region of a nonaperturing thin lens. As an example, Figs. 1(a)–1(d) show the radial, azimuthal, longitudinal, and global intensity patterns of a vector beam with m = 1 for φ₀ = π/4 at the lens’ geometrical focus (x − y plane). The intensity patterns are normalized by the maximum of the global intensity

I_{G}^{Max} (x, y, f)

. As displayed in Figs. 1(a)–1(d), the radial and azimuthal intensity patterns have the cylindrical symmetry and the dark center, whereas the longitudinal intensity pattern has center light spot. Besides, the ratio of the maximum intensities of the longitudinal and transverse fields increases with decreasing the lens’ focal length, which has a trend, consistent with that of a radially polarized vector beam [22

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

]. The intensity distributions of a vector beam with m = 1 and φ₀ = π/4 are quite different from those of radial polarized vector beam (m = 1 and φ₀ = 0) exhibiting the radial and longitudinal components only or of azimuthal polarized vector beam (m = 1 and φ₀ = π/2) just having the azimuthal component [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

, 22

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

], suggesting that the focal field distribution could be manipulated by altering the initial phase of the vector beam φ₀. Consequently, the φ₀-dependent energy flux distribution is predictable. To more clearly show the nonparaxial intensity patterns, the middle row in Fig. 1 displays the corresponding intensity profiles alone the x-axis for y = 0. For the sake of comparison, the paraxial intensity profiles obtained by Eq. (11) are also illustrated by circles in Figs. 1(e)–1(h). It should be noted that there is a little difference between the nonparaxial and paraxial optical intensity distributions. As Jia et al. [17

X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express 18, 7064–7075 (2010). [CrossRef] [PubMed]

] pointed out, the difference of the intensity distributions under the nonparaxial and paraxial approximations could be large with the parameter ω₀/λ decreasing. Figures 1(i)–1(l) display the nonparaxial intensity patterns through the focus (x − z plane and y = 0), which are normalized by the maximum of the global intensity

I_{G}^{Max} (x, 0, z)

. As can be seen from Fig. 1(l), the optical power is somewhat more concentrated before z = f, and therefore the beam is likely to be narrower there. Apparently, the true focus occurs not at the lens’ geometric focus but rather closer to the lens. This is a well-known focal shift that has been discussed previously [21

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]

This focal shift is confirmed in Fig. 2. Using Eq. (14), we obtain the beam waist ρ₀ of nonparaxial focused vector beam with m = 1 and φ₀ = π/4 against the propagation distance z, with the same parameters as used in Fig. 1. As shown in Fig. 2, the focused beam converges gradually, subsequently reaches the true focus with a minimum beam waist at z ≃ 0.55 f, and then diverges strongly. The asymmetry in the beam waist on either side of the true focus is obvious, completely different from that of the paraxial result (circles in Fig. 2). This difference between the nonparaxial and paraxial results is anticipated because the paraxial approximation is inapplicable when the beam waist is comparable to the wavelength. In the case of λ = 633 nm, ω₀ = 1 μm, and f = 4 μm, we estimate the focal shift of the vector beam with m = 1 for φ₀ = π/4 to be |(z − f)/f| ≃ 0.45, which is very close to that of a radially polarized beam [5

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]

]. It is noteworthy that the focal shift obtained in our case is nearly independent of φ₀.

Figures 3(a) and 3(b) display the nonparaxial energy flux distributions of a vector beam (m = 1 and φ₀ = π/4) at the planes of the lens’ geometrical focus and the true focus, respectively, with the same parameters as used in Fig. 1. Solid lines (circles) in Figs. 3(c) and 3(d) corresponding to Figs. 3(a) and 3(b) are the nonparaxial (paraxial) cross-section energy flux profiles at y = 0 for z = f and z = 0.55 f, respectively. As shown in Fig. 3, the energy flux distributions of the vector beam have the so-called doughnut patterns with the on-axis energy null and annular energy distribution. Besides, the energy flux is most concentrated just at the true focus than any other place. The difference between the nonparaxial and paraxial energy flux is observable, as shown in Figs. 3(c) and 3(d). This is because that the paraxial propagation approximately describes the beam propagation in our case of ω₀ ∼ 2λ. Nevertheless, the results obtained by the nonparaxial and paraxial theories are identical under the conditions of ω₀ ≫ λ and z ≫ λ.

For the case of paraxial propagation, we take the parameters as λ = 532 nm, E₀ = 1 (a.u.), ω₀ = 2.5 mm, f = 8 mm [4

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010). [CrossRef]

], and simulate the three-dimensional intensities of the vector beams with different topological charges near the region of focus using Eq. (11). As examples, Fig. 4 illustrates the paraxial intensity patterns of vector beams with φ₀ = 0 for m = 1, 3, and 5 at focus (top row) and though focus when y = 0 (lower row). All intensity patterns are normalized by the maximum of I_G(x, 0, f). Interestingly, the focused field of the vector beam has a doughnut-shaped intensity profile with the characteristic of axially symmetric profile, as shown in Fig. 4. Furthermore, the radius of the doughnut field increases with increasing the topological charge m of the vector beam. These results are comparable with the well-known results of a high order cylindrical vector beams [21

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]

, 24

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009). [CrossRef]

, 29

B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20, 149–157 (2012). [CrossRef] [PubMed]

], although with different focusing condition and different pupil apodization function.

Fig. 4 Normalized paraxial intensity patterns of vector beams with different topological charges m at focus (top row) and through focus (lower row), by taking φ₀ = 0, λ = 532 nm, E₀ = 1 (a.u.), ω₀ = 2.5 mm, and f = 8 mm.

Figure 5 illustrates the beam waist ρ₀ of paraxial focused vector beam with different values of m through the lens’ geometrical focus, with the same parameters as used in Fig. 4. The distinct symmetry in the beam waist of the beam with different m on either side of the geometric focus can be seen, indicating that the vector beam reaches a minimum waist just at the lens’ geometric focus. As displayed in Fig. 5, the change of the beam waist near focus weakens gradually as the value of m increases. When the beam goes far away from focus, the beam waists of vector beams with different topological charges are nearly identical. It is noteworthy that the focal shift is negligible because the Fresnel number of our system is very large. Actually, the focal shift would be significant for the system with a narrow beam, long wavelength, and long lens focal length [21

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]

Fig. 5 Beam waists ρ₀ of the paraxial focused vector beams with different topological charges through focus, by taking φ₀ = 0, λ = 532 nm, E₀ = 1 (a.u.), ω₀ = 2.5 mm, and f = 8 mm.

Using Eq. (15), we simulate the paraxial energy flux distributions of vector beams at focus (x–y plane) by taking λ = 532 nm, E₀ = 1 (a.u.), ω₀ = 2.5 mm, f = 8 mm, and z = f. It is found that the paraxial energy flux distributions have the cylindrical symmetry and dark center, which are similar to the intensity distributions (see Fig. 4). Figure 6(a) shows the normalized cross-section energy flux profiles of the vector beams with φ₀ = 0 for m = 1 (solid line), 3 (dashed line), and 5 (dotted line). As illustrated in Fig. 6(a), the ring of the energy flux distributions at the focal plane becomes larger and thicker when the topological charge of the vector beam increases. Moreover, the size of dark center without energy flux is larger with increasing the topological charge. The effect of the parameter φ₀ on the paraxial energy flux distribution is analyzed, as illustrated in Fig. 6(b). From Fig. 6(b), one finds that the thickness of the energy flux ring becomes narrower when the initial phase of the vector beam changes from φ₀ = 0 to π/2, indicating that the vector beam with the azimuthal polarization concentrates much more energy flux than that of the radial polarization.

Fig. 6 Paraxial cross-section energy flux profiles of the vector beams with (a) different m for φ₀ = 0 and (b) m = 3 for different φ₀ at the plane of z = f, by taking λ = 532 nm, E₀ = 1 (a.u.), ω₀ = 2.5 mm, f = 8 mm, and y = 0.

4. Conclusion

In summary, we have investigated the nonparaxial diffraction of a lowest-order Laguerre-Gaussian beam with azimuthal-variant states of polarization based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation. We have presented the analytical expressions for the radial, azimuthal, and longitudinal components of the electric field with an arbitrary integer topological charge focused by a nonaperturing thin lens. By numerical simulations, we have illustrated the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations. In a word, we have investigated the focal field and the propagation behavior of the focused cylindrical vector beam with an arbitrary integer topological charge. With the help of the analytical three-dimensional focal field, it is easily to gain an insight on the novel effects for the interaction of vector field with the matter.

Acknowledgments

This work was supported by the National Science Foundation of China (Grant: 11174160) and the Program for New Century Excellent Talents in University (Grant: NCET-10-0503).

References and links

1.	A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998). [CrossRef]
2.	J. S. Ahn, H. W. Kihm, J. E. Kihm, D. S. Kim, and K. G. Lee, “3-dimensional local field polarization vector mapping of a focused radially polarized beam using gold nanoparticle functionalized tips,” Opt. Express 17, 2280–2286 (2009). [CrossRef] [PubMed]
3.	A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006). [CrossRef] [PubMed]
4.	X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010). [CrossRef]
5.	V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010). [CrossRef]
6.	X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. 36, 295–297 (2011). [CrossRef] [PubMed]
7.	D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagation in free space,” J. Opt. Soc. Am. A 24, 3317–3325 (2007). [CrossRef]
8.	B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959). [CrossRef]
9.	R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]
10.	Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25, 537–542 (2008). [CrossRef]
11.	D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049 (2009). [CrossRef]
12.	Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15, 11942–11951 (2007). [CrossRef] [PubMed]
13.	G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26, 141–147 (2009). [CrossRef]
14.	X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102, 205–213 (2011). [CrossRef]
15.	B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003). [CrossRef] [PubMed]
16.	S. Guha and G. D. Gillen, “Description of light propagation through a circular aperture using nonparaxial vector diffraction theory,” Opt. Express 13, 1424–1447 (2005). [CrossRef] [PubMed]
17.	X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express 18, 7064–7075 (2010). [CrossRef] [PubMed]
18.	K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004). [CrossRef]
19.	L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]
20.	C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011). [CrossRef] [PubMed]
21.	P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef] [PubMed]
22.	K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
23.	D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24, 636–643 (2007). [CrossRef]
24.	M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 11, 065204 (2009). [CrossRef]
25.	D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006). [CrossRef]
26.	P. P. Banerjee, G. Cook, and D. R. Evans, “A q-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A 26, 1366–1374 (2009). [CrossRef]
27.	R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
28.	V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. 13, 075703 (2011). [CrossRef]
29.	B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express 20, 149–157 (2012). [CrossRef] [PubMed]
30.	A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002). [CrossRef]

OCIS Codes
(260.1960) Physical optics : Diffraction theory
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: May 9, 2012
Revised Manuscript: July 1, 2012
Manuscript Accepted: July 1, 2012
Published: July 19, 2012

Citation
Bing Gu and Yiping Cui, "Nonparaxial and paraxial focusing of azimuthal-variant vector beams," Opt. Express 20, 17684-17694 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17684

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References

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A15, 2705–2711 (1998). [CrossRef]
J. S. Ahn, H. W. Kihm, J. E. Kihm, D. S. Kim, and K. G. Lee, “3-dimensional local field polarization vector mapping of a focused radially polarized beam using gold nanoparticle functionalized tips,” Opt. Express17, 2280–2286 (2009). [CrossRef] [PubMed]
A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96, 153901 (2006). [CrossRef] [PubMed]
X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010). [CrossRef]
V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A27, 372–380 (2010). [CrossRef]
X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett.36, 295–297 (2011). [CrossRef] [PubMed]
D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagation in free space,” J. Opt. Soc. Am. A24, 3317–3325 (2007). [CrossRef]
B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959). [CrossRef]
R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A21, 2029–2037 (2004). [CrossRef]
Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A25, 537–542 (2008). [CrossRef]
D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B26, 2044–2049 (2009). [CrossRef]
Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express15, 11942–11951 (2007). [CrossRef] [PubMed]
G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B26, 141–147 (2009). [CrossRef]
X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102, 205–213 (2011). [CrossRef]
B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.28, 2440–2442 (2003). [CrossRef] [PubMed]
S. Guha and G. D. Gillen, “Description of light propagation through a circular aperture using nonparaxial vector diffraction theory,” Opt. Express13, 1424–1447 (2005). [CrossRef] [PubMed]
X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express18, 7064–7075 (2010). [CrossRef] [PubMed]
K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A21, 1613–1620 (2004). [CrossRef]
L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett.86, 5251–5254 (2001). [CrossRef] [PubMed]
C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett.106, 123901 (2011). [CrossRef] [PubMed]
P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express4, 411–419 (1999). [CrossRef] [PubMed]
K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000). [CrossRef] [PubMed]
D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B24, 636–643 (2007). [CrossRef]
M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt.11, 065204 (2009). [CrossRef]
D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B23, 1228–1234 (2006). [CrossRef]
P. P. Banerjee, G. Cook, and D. R. Evans, “A q-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A26, 1366–1374 (2009). [CrossRef]
R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt.13, 075703 (2011). [CrossRef]
B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express20, 149–157 (2012). [CrossRef] [PubMed]
A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun.202, 17–20 (2002). [CrossRef]

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