1 Introduction

Recently, more and more attention is being paid to Airy beam which is one kind of nondiffracting beam. Besides the properties of nondiffraction and self-healing which are also possessed by other nondiffracting beams, such as Bessel beam [1, 2], Bessel–Gaussian beam [3] and Mathieu beam [4, 5], Airy beam displays its unique properties such as self-acceleration and it can be realized in one-dimensional domain [6, 7], while other nondiffracting beams can only be realized in higher dimensions. Due to its unique properties, Airy beam has found wide applications, such as optical micromanipulation [8–11], plasma guidance [12, 13], vacuum electron acceleration [14], light bullets [15, 16], optical routing [17] and atmospheric optics [18, 19]. A variety of methods have been developed to generate Airy beam [7, 20–24]. Circular Airy beam has been introduced as one kind of abruptly autofocusing beam [25], whose maximum intensity remains almost constant during propagation in free space and suddenly increases by orders of magnitude right before its focal point. Circular Airy beam has been generated in experiment and used for trapping and guiding microparticles [26, 27]. All aforementioned literatures have been confined to scalar Airy beam.

In the past decades, cylindrical vector beams with spatially nonuniform state of polarization, such as radially polarized (RP) beam and azimuthally polarized (AP) beam, have been studied extensively due to their interesting tight-focusing properties and have found wide application in microscopy, lithography, free-space optical communications, electron acceleration, proton acceleration, particle trapping, material processing, optical data storage, high-resolution metrology, super-resolution imaging, plasmonic focusing and laser machining [28–44]. When a RP beam is focused by a high numerical aperture (NA) lens, a strong longitudinal electric field appears and the tightly focused beam spot of a RP beam is much smaller than that of a linearly polarized beam [28–31], which is useful for particle acceleration and fluorescent imaging. When an AP beam is focused by a high NA lens, a strong magnetic field on the optical axis is generated while the electric field is purely transverse and displays subwavelength dark channel [28–31], which is useful for atom guiding and trapping. Recently, cylindrical vector circular Airy beam was introduced, and a RP circular Airy beam was generated in experiment, and the abrupt polarization self-transitions associated with spin–orbit interaction of the vector circular Airy beam was demonstrated both theoretically and experimentally [45]. In this letter, our aim is to study the tight-focusing properties of an AP circular Airy beam and a RP circular Airy beam, and carry out experimental generation of an AP circular airy beam. Some interesting and useful results are found.

2 Tight-focusing properties of cylindrical vector circular Airy beams

In this section, we study the tight-focusing properties of cylindrical vector circular Airy beams. The scheme of tight focusing of cylindrical vector circular Airy beams by a high NA thin lens is shown in Fig. 1.

Fig. 1
figure 1

Scheme of tight focusing of cylindrical vector circular Airy beams by a high NA thin lens. f is the focal length of the thin lens, and Q(r,ϕ) is an observation point in the receiver plane

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First, we analyze the tight focusing of an AP circular Airy beam. In the cylindrical coordinates, the electric field of an AP circular Airy beam in source plane can be expressed as [45]

$$\vec{E}(\rho ,\varphi ) = \text{Ai}\left( {\frac{{r_{0} - \rho }}{\omega }} \right)\exp \left[ {a\left( {\frac{{r_{0} - \rho }}{\omega }} \right)} \right]{\vec{e}}_{{\varphi }} ,$$
(1)

where Ai(·) denotes the Airy function [6, 25], ρ and φ are the radial and azimuthal (angle) coordinates in the source plane, a is an exponential decay factor, r 0 is the radius of the primary ring and ω is a scaling factor. \(\vec{e}_{\varphi }\) is the unit vector along the azimuthal direction. The radius of the ring with maximum intensity is about r 0 − ω, and its full width at half maximum (FWHM) is about 2.28ω [26].

Let us consider the tight focusing of an AP circular Airy beam by a high NA thin lens in free space as shown in Fig. 1. Using the Richards–Wolf vectorial diffraction integral [46], the electric field of the focused AP circular Airy beam near the focal region can be expressed as [47].

$$\vec{E}(r,\phi ) = \left[ \begin{gathered} E_{r} (r,\phi ) \hfill \\ E_{\varphi } (r,\phi ) \hfill \\ E_{z} (r,\phi ) \hfill \\ \end{gathered} \right] = \frac{2\pi f}{\lambda }\int_{0}^{{\theta_{\hbox{max} } }} {\left[ {\begin{array}{*{20}c} 0 \\ {A_{\varphi } (\theta )} \\ 0 \\ \end{array} } \right]\sqrt {\cos \theta } \sin \theta J_{1} (kr\sin \theta )\exp (ikz\cos \theta )\text{d}\theta } ,$$
(2)

where r and ϕ are the radial and azimuthal (angle) coordinates in the receiver plane, k = 2π/λ is the wave number with λ being the wavelength of incident beam, f is the focal length of the high NA thin lens, θ is an NA angle, and θ max is the maximal NA angle, which is related to the NA by the formula \(\theta_{\hbox{max} } = \arcsin (\text{NA})\), J 1(·) denotes the Bessel function of the first kind of order one, A φ (θ) is the pupil apodization function at the objective aperture and is derived by setting ρ = f sin θ in Eq. (1). It is clear from Eq. (2) that the field near the focal region remains AP.

Applying Eqs. (1) and (2), we can study the tight-focusing properties of an AP circular Airy beam numerically. For the sake of simplicity, we assume f = 3 mm, NA = 0.95 throughout this paper. We calculate in Fig. 2a, b the normalized intensity distribution of a tightly focused AP circular Airy beam in the x–y plane at z = 0 and the normalized intensity of such beam in the x–z plane with y = 0, respectively. The initial beam parameters of the incident beam are set as r0 = 1.8 mm, ω = 0.03 mm and a = 0.25. One finds from Fig. 2a that the focused beam in the focal plane turns to be an AP Bessel-like beam whose central intensity is zero due to its polarization singularity. The FWHM of the ring is about 0.49λ. From Fig. 2b, one finds that the depth of focus (DOF) of the dark channel is extremely long and is about 52λ. To show this properties more clearly, we calculate in Fig. 2c the one-dimensional normalized intensity distribution (y = 0) versus x at several propagation distances. We observe from Fig. 2c that the normalized intensity distribution remains almost invariant over the distance from z = −26λ to z = 26λ, which means that the focused AP circular Airy beam near the focal region displays the properties of nondiffraction, and this phenomenon may be explained by the fact that the focused beam becomes a AP Bessel-like beam near the focal region, which is a diffractive-free beam. In previous literatures, a dark channel with FWHM about 0.5λ and DOF about 26λ is formed in Ref. [48] through tight focusing a double-ring-shaped AP beam by a high NA thin lens with an annular aperture (AA) and a dark channel with FWHM about 0.5λ and DOF about 48λ is formed in Ref. [49] through tight focusing a double-ring-shaped AP beam by a high NA lens axicon. Compared to the methods mentioned in those two papers, we do not employ any AA or axicon in our focusing system, while we can obtain a dark channel with smaller FWHM and longer DOF. Our results will be useful for atom guiding and trapping, where a dark channel with long DOF is required [50, 51].

Fig. 2
figure 2

Normalized intensity distribution of a tightly focused AP circular Airy beam with r0 = 1.8 mm, ω = 0.03 mm and a = 0.25 mm. a Normalized intensity distribution in the x–y plane at z = 0. b Normalized intensity distribution in the x–z plane with y = 0. c One-dimensional normalized intensity distribution (y = 0) versus x at several propagation distances

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To learn about the influences of the beam parameters r 0 and ω on the FWHM and the DOF of the dark channel, we calculate in Fig. 3a–c the normalized intensity distribution of a tightly focused AP circular Airy beam in the x–z plane with y = 0 for different values of r 0 and in Fig. 3d the one-dimensional normalized intensity distribution (y = 0) versus x at z = 0 for different values of r 0 with ω = 0.03 mm and a = 0.25. One finds from Fig. 3a–c that the FWHM and the DOF of the dark channel are closely determined by the parameter r 0. The values of the FWHM and the DOF decrease as r 0 increases. Our numerical results show that for r 0 = 1.0 mm, 1.5 mm and 2.5 mm, the values of the FWHM are about 0.87λ, 0.59λ, 0.35λ, and the values of the DOC are about 115λ, 68λ, 26.5λ. It is clear from Fig. 3d that the transverse beam profile at z = 0 remains almost unchanged while the beam size varies as r 0 varies.

Fig. 3
figure 3

Normalized intensity distribution of a tightly focused AP circular Airy beam in the x–z plane with y = 0 for a r0 = 1.0 mm b r0 = 1.5 mm c r0 = 2.5 mm d One-dimensional normalized intensity distribution (y = 0) versus x at z = 0

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We calculate in Fig. 4 the DOF and the FWHM of the dark channel versus the beam parameter ω with r 0 = 1.8 mm and a = 0.25 mm. One finds that the DOF of dark channel increases with the decrease of ω, while the FWHM remains almost invariant as ω varies. Our numerical results (not shown here to save space) also show that the influence of the beam parameter a on the DOF and the FWHM of the dark channel is quite small. From above discussions, we come to the conclusion that we can modulate the values of the DOF and FWHM by varying the beam parameters r 0 and ω of the incident AP circular Airy beam.

Fig. 4
figure 4

DOF and FWHM of the dark channel versus the beam parameter ω

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Now, we analyze the tight focusing of a RP circular Airy beam. The electric field of a RP circular Airy beam in the source plane in the cylindrical coordinates can be expressed as [45].

$$\vec{E}(\rho ,\varphi ) = \text{Ai}\left( {\frac{{r_{0} - \rho }}{\omega }} \right)\exp \left[ {a\left( {\frac{{r_{0} - \rho }}{\omega }} \right)} \right]\vec{e}_{r} ,$$
(3)

where \(\vec{e}_{r}\) is the unit vector along the radial direction. Comparing Eqs. (1) and (3), one finds that the intensity distribution of a RP circular Airy beam is the same with that of an AP circular Airy beam in the source plane, while their tight-focusing properties are totally different as shown later.

According to the Richards–Wolf vectorial diffraction integral [46], the electric field of the focused RP circular Airy beam near the focal region can be expressed as [47].

$$\vec{E}(r,\phi ) = \left[ \begin{gathered} E_{r} (r,\phi ) \hfill \\ E_{\varphi } (r,\phi ) \hfill \\ E_{z} (r,\phi ) \hfill \\ \end{gathered} \right] = \frac{\pi f}{\lambda }\int_{0}^{{\theta_{\hbox{max} } }} {\left[ {\begin{array}{*{20}c} {l_{r} (\theta )\sin 2\theta J_{1} (kr\sin \theta )} \\ 0 \\ {2il_{r} (\theta )\sin^{2} \theta J_{0} (kr\sin \theta )} \\ \end{array} } \right]\sqrt {\cos \theta } \exp (ikz\cos \theta )\text{d}\theta } ,$$
(4)

where J 0(·) denotes the first kind of Bessel function of order zero. l r (θ) is the pupil apodization function at the objective aperture and is derived by setting ρ = f sin θ in Eq. (3). Different from the case of tight focusing of an AP circular Airy beam, one finds from Eq. (4) that the longitudinally polarized field E z near the focal region appears besides the RP field E r . Thus, both E r and E z should be taken into consideration together in the focal region for the case of tight focusing of a RP circular Airy beam.

The RP intensity, the longitudinally polarized intensity and the total intensity of a tightly focused RP circular Airy beam near the focal region are given as I r  = |E r |2, I z  = |E z |2 and I = |E r |2 + |E z |2, respectively. Applying Eqs. (3) and (4), we calculate in Fig. 5a–c the normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–y plane at z = 0. Figure 5d shows the one-dimensional normalized intensity distributions (y = 0) versus x at z = 0. The parameters of the RP circular Airy beam are chosen as r 0 = 2.5 mm ω = 0.03 mm and a  = 0.25 mm. One finds from Fig. 5 that the RP intensity distribution of the focused RP circular Airy beam in the focal plane (z = 0) is similar to that of a RP Bessel beam whose central intensity is zero due to its polarization singularity at the beam center (see Fig. 5a), and the longitudinally polarized intensity distribution at the focal plane is similar to that of Bessel beam of order zero whose intensity distribution displays a bright spot at the beam center (see Fig. 5b). The ratio of the peak intensity of the longitudinally polarized field to that of the radial polarized field (i.e., I zmax/I rmax) is about 6.7. Thus, the contribution of the longitudinally polarized intensity to the total intensity plays a dominant role at the focal plane, and the total intensity distribution looks like the intensity distribution of the longitudinally polarized field (see Fig. 5c, d). The values of FWHM for I r , I z and I at z = 0 are about 0.35, 0.41 and 0.5λ, respectively.

Fig. 5
figure 5

a–c Normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–y plane at z = 0. d One-dimensional normalized intensity distributions (y = 0) versus x at z = 0

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Figure 6a–c show the normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–z plane with y = 0, r 0 = 2.5 mm, ω = 0.03 mm and a = 0.25 mm, and Fig. 6d shows one-dimensional normalized longitudinally polarized intensity distribution versus z with x = 0 and y = 0. From Fig. 6a, b, one finds that both the RP intensity distribution and the longitudinally polarized intensity distribution maintain their beam shapes invariant for extremely long distance. The DOF inferred from the longitudinally polarized intensity distribution is approximately 27.5λ (see Fig. 6c, d). In other words, the focused RP circular Airy beam near the focal plane also exhibits the property of nondiffraction. This phenomenon can be explained by the fact that the RP field and longitudinally polarized field of the tightly focused RP circular Airy beam become RP Bessel-like beam and Bessel-like beam of order 0, respectively, and both beams display the properties of nondiffraction. The super long needle about 27.5λ of the strong longitudinally polarized field obtained in this paper will be useful for particle acceleration and fluorescent imaging. In Ref. [35], a long needle about 4λ is obtained by tight focusing a RP beam with the help of a binary-phase optical element. In our case, the super long needle is obtained without applying any phase element.

Fig. 6
figure 6

a–c Normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–z plane with y = 0 and r0 = 2.5 mm. d One-dimensional normalized longitudinally polarized intensity distribution versus z with x = 0 and y = 0

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To learn about the influence of the beam parameter r 0 on the tight-focusing properties of a RP circular Airy beam, we calculate in Figs. 7 and 8 the normalized intensity distributions of a tightly focusing RP circular Airy beam near the focal plane for two different values of r 0. One finds from Figs. 7b, d and 8b, d that the DOF of the longitudinally polarized field increases as the value of r 0 increases, and reaches approximately 54λ and 98λ for r 0 = 1.8 mm and r 0 = 1.2 mm, respectively. The values of FWHM of the longitudinally polarized field at z = 0 are about 0.59λ and 0.89λ for r 0 = 1.8 mm and r 0 = 1.2 mm, respectively. However, the ratio of I zmax/I rmax decreases as the value of r 0 increases (see Figs. 7a, b, 8a, b), which means that the energy carried by the RP field increases as the value of r 0 increases. As the contribution of the RP intensity to the total intensity increases, the total intensity distribution is significantly influenced by the RP field. As shown in Fig. 8c, for the case of r 0 = 1.2 mm, the ratio of I zmax/I rmax reaches to 0.56 which means the energy of the longitudinally polarized field is quite small, and the total intensity distribution in x–z plane displays a “semi” dark channel.

Fig. 7
figure 7

a–c Normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–z plane with y = 0 and r0 = 1.8 mm. d One-dimensional normalized longitudinally polarized intensity distribution versus z with x = 0 and y = 0

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Fig. 8
figure 8

a–c Normalized intensity distributions I r /I max, I z /I max and I/I max of a tightly focused RP circular Airy beam in the x–z plane with y = 0 and r0 = 1.2 mm. d One-dimensional normalized longitudinally polarized intensity distribution versus z with x = 0 and y = 0

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Figure 9 shows the ratio of I zmax/I rmax (solid line) and DOF (dashed line) of the longitudinally polarized field of a tightly focused RP circular Airy beam in the focal plane (z = 0) versus the beam parameter r 0 with ω = 0.03 mm and a = 0.25 mm. One finds from Fig. 9 that the ratio of I zmax/I rmax decreases as the value of r 0 decreases, while the DOF of the longitudinally polarized field increases. Our numerical results show that the total intensity distribution in the x–z plane displays needle-like distribution only when I zmax/I rmax > 1. Thus, one should choose suitable value of r 0 in order to obtain a super long needle of strong longitudinally polarized field.

Fig. 9
figure 9

Ratio of I zmax/I rmax (solid line) and DOF (dashed line) of the longitudinally polarized field of a tightly focused RP circular Airy beam in the focal plane (z = 0) versus the beam parameter r 0

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3 Experimental generation of an azimuthally polarized circular Airy beam

In Ref. [45], experimental generation of a RP circular Airy beam was reported. In this section, we carry out experimental generation of an AP circular Airy beam.

The experimental setup for generating an AP circular Airy beam is shown in Fig. 10. A linearly polarized laser beam (λ = 532 nm) generated by a diode-pumped solid-state laser passes through a beam expander (BE) and then goes toward a reflective phase spatial light modulator (SLM, Holoeye Pluto-vis), which acts as a phase filter programed by a computer software. The phase filter loaded on the reflective phase SLM (shown as inset of Fig. 10) is generated by the method proposed by G. Papazoglou et al. [26]. First, the Fourier transform (FT) of a circular Airy source \(F\left\{ {E(\sqrt {x^{2} + y^{2} } )} \right\} \equiv A(k_{x} ,k_{y} )\exp [i\varPhi (k_{x} ,k_{y} )]\) is calculated numerically. Second, we encode both amplitude and phase values to a phase-only function Ψ(k x , k y ) ≡ exp [iA ’(k x , k y )Φ(k x , k y )] [26, 52]. The beam from the reflective phase SLM is then Fourier transformed by a thin lens with focal length f = 400 mm. In the FT plane, a linearly polarized circular Airy beam is generated. An AA located in the FT plane is used to block the undesired light spot in the beam center. After passing through an azimuthal polarization converter (APC) which is produced by the company Arcoptix and is located just behind the AA, the linearly polarized circular Airy beam turns to be an AP circular Airy beam. The APC does not alter the beam profile of the Airy beam. The charge-coupled device is used to measure the intensity distribution of the generated AP circular Airy beam.

Fig. 10
figure 10

Experimental setup for generating an AP circular Airy beam. DPSSL diode-pumped solid-state laser, BE beam expander, SLM spatial light modulator, AA annular aperture, APC azimuthal polarization converter, CCD charge-coupled device, PC personal computer

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Figure 11a, b shows our experimental results of the intensity distribution of the generated AP circular Airy beam with different values of the beam parameters r 0 and ω. The beam parameters r 0 and ω are controlled through adjusting the parameters of the phase filter loaded on the reflective phase SLM. Figure 6c, d shows our experimental results of the intensity distribution of the generated AP circular Airy beam after passing through a linear polarizer for two different values (π/2 and 0) of the transmission angle. One finds from Fig. 6 that the AP circular Airy beam is indeed generated in our experiment. Our experiment results (not shown here to save space) also show that the evolution properties of the intensity distribution of an AP circular Airy beam on propagation in free space are similar to those of a RP circular Airy beam as shown in [45].

Fig. 11
figure 11

a, b Experimental results of the intensity distribution of the generated AP circular Airy beam with different values of the beam parameters r 0 and ω. c, d Experimental results of the intensity distribution of the generated AP circular Airy beam after passing through a linear polarizer for two different values (π/2 and 0) of the transmission angle

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4 Conclusion

As a summary, we have studied the tight-focusing properties of cylindrical vector circular Airy beams numerically, and we have found that we can generate a super long subwavelength dark channel with FWHM about 0.49λ and DOF about 52λ near the focal region for the case of tight focusing of an AP circular Airy beam, and we can generate a super long needle with DOF about 27.5λ of strong longitudinally polarized field for the case of tight focusing of a RP circular Airy beam. Furthermore, we have reported experimental generation of an AP circular Airy beam. Our results will be useful for atom guiding and trapping, particle acceleration and fluorescent imaging.