## 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.

© 2012 OSA

## 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, 2]. 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], optical trapping and manipulating nanoparticles [4], 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], the vector angular spectrum method [6], the multiscale singular perturbation method [7], and Richards-Wolf theory [8]. 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], spirally polarized beams [9], controllable dark-hollow beams [10], hollow Gaussian beams [11], vectorial Laguerre-Bessel-Gaussian beams [12], Lorentz-Gaussian beams [13], Gaussian vortex beams [5], partially coherent dark hollow beams [14], and beams diffracted at a circular aperture [15–17] and a rectangular aperture [18].

Recently, interest in cylindrical vector laser beams has been prompted by their intriguing applications in single molecule detection [19], optical micro-fabrication [20], 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] explored the focal shift in vector beams under the weak focusing condition. Youngworth and Brown [22] reported the high numerical aperture focusing of cylindrical vector beams in the nonparaxial limit. Deng *et al*. [23] studied the nonparaxial and paraxial propagation of radially polarized elegant beams. Rashid *et al*. [24] analyzed the focal field of high order cylindrical vector beams in the limit of high numerical aperture. Deng *et al*. [25] and Baberjee *et al*. [26] investigated the nonparaxial and paraxial propagation of radially polarized Gaussian beams, respectively. Especially, Dorn *et al*. [27] 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*. [5,28], 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, 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, 29]

**ê**

*and*

_{r}**ê**

*are the unit vectors in the polar coordinate system (*

_{ϕ}*r*,

*ϕ*),

*A*(

*r*) represents the radial-dependent amplitude,

*m*is the azimuthal topological charge, and

*φ*

_{0}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

*φ*

_{0}= 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]

*γ*=

*kρ*/

*ξ*, $\xi =\sqrt{{z}^{2}+{\rho}^{2}}$,

*k*= 2

*π*/

*λ*, and

*λ*is the wavelength. Note that the weak nonparaxial approximation takes the form $\sqrt{{z}^{2}+{\rho}^{2}+{r}^{2}-2\rho r\text{cos}\left(\varphi -\theta \right)}\simeq \sqrt{{z}^{2}+{\rho}^{2}}+\frac{{r}^{2}}{2\sqrt{{z}^{2}+{\rho}^{2}}}-\frac{\rho r\text{cos}\left(\varphi -\theta \right)}{\sqrt{{z}^{2}+{\rho}^{2}}}$ [5,28], which holds true under the conditions of

*ω*<

*λ*and

*z*≥

*ω*

^{2}/(2

*λ*) [30], where

*ω*is the waist width. This approximation to describe the nonparaxial propagation of light beams is well known [12, 17, 23].

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

*J*(·) is the Bessel function of

_{m}*m*th-order.

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

*mθ*−

*θ*+

*φ*

_{0}.

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, 25]

where*ω*

_{0}is the waist radius of the Gaussian beam, and

*E*

_{0}is an amplitude constant. One takes $\alpha =1/{\omega}_{0}^{2}$ 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 $\alpha =1/{\omega}_{0}^{2}+ik/\left(2f\right)$.

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

*m*] > −1, Re[

*β*] > 0,

*γ*> 0,

*t*=

*γ*

^{2}/(8

*β*), and

*I*(·) is the modified Bessel function of

_{m}*m*th-order, we yield the analytical results

*β*=

*α*−

*ik*/(2

*ξ*) and

*t*=

*γ*

^{2}/(8

*β*). 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 *φ*_{0} = 0, we deduce the nonparaxial propagation of a radially polarized vector beam from Eq. (8) as follows

*q*=

*ξ*−

*ik*/(2

*α*). The obtained result is coincident with the one reported previously [5]. For an azimuthally polarized vector beam (

*m*= 1 and

*φ*

_{0}=

*π*/2), we yield

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 (*z*^{2} +*ρ*^{2})^{1/2} ≈ *z* +*ρ*^{2}/(2*z*) ≈ *z*. Accordingly, we obtain the propagation expressions for the azimuthal-variant vector beam under the paraxial approximation as

*γ*′ =

*kρ*/

*z*,

*β*′ =

*α*−

*ik*/(2

*z*), and

*t*′ =

*γ*′

^{2}/(8

*β*′). In particular, for a radially polarized vector beam (

*m*= 1 and

*φ*

_{0}= 0), one gets [5, 26]

*q*≃

*z*−

*ik*/(2

*α*). For the case of

*m*= 1 and

*φ*

_{0}=

*π*/2, we obtain the paraxial propagation of an azimuthally polarized vector beam as

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, 29]. To define the width of a vector beam, in general, one adopts an encircled-power criterion, i.e., the width *ρ*_{0} of a vector beam as that radius within 80% of the beam’s power is enclosed. Accordingly, the width *ρ*_{0} of a focused vector beam at the propagation distance *z* satisfies [21]

*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*by

_{G}*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,

**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, *E*_{0} = 1 (a.u.), *ω*_{0} = 1 *μ*m, and *f* = 4 *μ*m [5]. Detailed numerical simulations have been performed using the formulae derived in Section 2 and the results are shown in Figs. 1–3.

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 *φ*_{0} = *π*/4 at the lens’ geometrical focus (*x* − *y* plane). The intensity patterns are normalized by the maximum of the global intensity
${I}_{G}^{\mathit{Max}}\left(x,y,f\right)$. 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]. The intensity distributions of a vector beam with *m* = 1 and *φ*_{0} = *π*/4 are quite different from those of radial polarized vector beam (*m* = 1 and *φ*_{0} = 0) exhibiting the radial and longitudinal components only or of azimuthal polarized vector beam (*m* = 1 and *φ*_{0} = *π*/2) just having the azimuthal component [5, 22], suggesting that the focal field distribution could be manipulated by altering the initial phase of the vector beam *φ*_{0}. Consequently, the *φ*_{0}-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] pointed out, the difference of the intensity distributions under the nonparaxial and paraxial approximations could be large with the parameter *ω*_{0}/*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
${I}_{G}^{\mathit{Max}}\left(x,0,z\right)$. 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].

This focal shift is confirmed in Fig. 2. Using Eq. (14), we obtain the beam waist *ρ*_{0} of nonparaxial focused vector beam with *m* = 1 and *φ*_{0} = *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, *ω*_{0} = 1 *μ*m, and *f* = 4 *μ*m, we estimate the focal shift of the vector beam with *m* = 1 for *φ*_{0} = *π*/4 to be |(*z* − *f*)/*f*| ≃ 0.45, which is very close to that of a radially polarized beam [5]. It is noteworthy that the focal shift obtained in our case is nearly independent of *φ*_{0}.

Figures 3(a) and 3(b) display the nonparaxial energy flux distributions of a vector beam (*m* = 1 and *φ*_{0} = *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 *ω*_{0} ∼ 2*λ*. Nevertheless, the results obtained by the nonparaxial and paraxial theories are identical under the conditions of *ω*_{0} ≫ *λ* and *z* ≫ *λ*.

For the case of paraxial propagation, we take the parameters as *λ* = 532 nm, *E*_{0} = 1 (a.u.), *ω*_{0} = 2.5 mm, *f* = 8 mm [4], 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 *φ*_{0} = 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, 24, 29], although with different focusing condition and different pupil apodization function.

Figure 5 illustrates the beam waist *ρ*_{0} 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].

Using Eq. (15), we simulate the paraxial energy flux distributions of vector beams at focus (*x*–*y* plane) by taking *λ* = 532 nm, *E*_{0} = 1 (a.u.), *ω*_{0} = 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 *φ*_{0} = 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 *φ*_{0} 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 *φ*_{0} = 0 to *π*/2, indicating that the vector beam with the azimuthal polarization concentrates much more energy flux than that of the radial polarization.

## 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).

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