In recent years, vortex beams (VBs) with orbital angular momentum (OAM) provide a new degree of freedom of light and have gradually broadened the scope of fields such as optical communications [1–3], optic micromanipulation [4, 5] and quantum information processing [6]. As a major branch, the Bessel-Gauss beam, proposed by Gori et al. [7] in 1987, attracted much attention due to its unique and interesting focusing characteristics. Radially polarized Bessel-Gauss beams have two intriguing features: Non-diffraction propagation and ultra-small spot focusing, showing unparalleled advantages in many fields [8–10]. Generally, a radially polarized beam (RPB) is an axially symmetric vector polarized beam with the axisymmetric distribution of polarization. A special modulated RPB is of interest because it gives a smaller focal spot size according to certain measures [11–13]. For example, in 2006, Kozawa and Sato [14] studied the focusing characteristics of a double-ring RPB through a high NA lens and found a bottle beam with practical applications. In 2020, Zeng et al. [15] deduced the statistical properties of radially polarized partially coherent FVBs, including average intensity, degree of polarization, and polarization state. With scientific research development, it has been considered a fact that RPB can produce smaller spots and stronger longitudinal fields than other beams, and now the research on designing the amplitude and phase of RPB is relatively mature. However, some studies have shown that other beams are also capable of producing spots of sub-wavelength size and have other excellent properties [16–18]. Our recent approach demonstrated that the traditional left-handed circularly polarized beam (CPB) can still produce light spots of the same size as the RPB, which has not been reported so far.

Without analyzing the complicated polarization distribution, CPB can be obtained in a simple way. It can be directly generated and converted based on a laser and a quarter-wave plate or simple hologram loaded onto a spatial light modulator (SLM). Among other applications, tight focused CPBs are useful in resolution improvement [19–21], plasmonics and nano-opticals applications [22, 23], light-matter interactions [24, 25], high-precision laser processing [26] and data storage [27]. In addition, the illustration of fractional phase is the key to ensure this special performance. It has been demonstrated that, by tightly focusing an incident CPB, a helical phase of the longitudinal component of the electric field can be created [28]. However, they were limited to only achieving longitudinal electric fields with topological charges depending on the handedness of the incoming circular polarization. In other words, they could not generate longitudinal electric fields with an arbitrary topological charge except for ±1 order, which is the real essential problem. Generally, whole vortex beams exp(ilφ) always possess an integer topological charge l, and methods for producing such beams are well developed. However, there are still many points worth studying in vortex beams with a fractional topological charge. Here, we verify the possibility of generating arbitrary fractional-order vortex modulated Bessel-Gauss beams with a strong longitudinal field, which is a novelty.

In this paper, we demonstrate for the first time that circularly polarized Bessel-Gauss beam can produce the same tight focusing ability as RPBs. Additionally, the possibility of generating Bessel-Gauss beams with arbitrary fractional-order vortex modulation with a strong longitudinal field is demonstrated. The tight focusing characteristics of the novel fractional-order vortex modulated Bessel-Gauss beams are studied in this paper, and the precise control of the number of lobes, radius, and layers is realized, which may promote broad development in optics and other related scientific fields, and is not only of academic interest but also of practical application value.

The focusing process of the optical system in Fig. 1 is as follows: After the light beam carrying the specified amplitude l₀(θ) is incident on the pupil filter P, an objective lens is used to generate the converging spherical wave at the focal sphere Ω. Under the sinusoidal condition, the spherical wave is propagated to point O on the axis controlled by the diffraction limit, and finally forms an electric field component along the x, y and z directions.

Figure 1. Schematic diagram of the focusing system. l0(θ) is the amplitude of the incident beam; P is the pupil filter; OL is the objective lens; Ω is the focus sphere; the center of which is O point, the radius f is the focal length of objective lens OL; and θ is the convergence angle.

The electric field component at O in Fig. 1 is represented by a rectangular coordinate system (x, y, z). To simplify the derivation, the model in this paper is established in a cylindrical coordinate system (ρ, φ, z) as the reference coordinate system. According to Debye diffraction theory [29], the electric fields of a circularly polarized Bessel-Gauss beam near the focus O can be expressed as [30–32]:

(1)$$E\left(\rho ,\varphi ,z\right)=E_{\rho }{e}_{\rho }+E_{\varphi }{e}_{\varphi }+{{\rm E}}_z{e}_z$$

where e_ρ, e_φ and e_z are unit vectors of radial, azimuth, and propagation direction, respectively. E_ρ, E_φ and E_z are amplitudes of three orthogonal components, which can be expressed as [33, 34]:

(2)$$E_{\rho }=\frac{-iA}{\pi }{\displaystyle {\int }_{\text{0}}^{\alpha }{\displaystyle {\int }_0^{2\pi }\sqrt{\cos \theta }\cdot \sin \theta \cdot V_{E\_\rho }(\theta ,\phi )\cdot T\cdot l_0(\theta )}}\cdot \exp \left\{ik\left[-\rho \sin \theta \cos (\varphi -\phi )+zcos\theta \right]\right\}d\phi d\theta ,$$

(3)$$E_{\varphi }=\frac{-iA}{\pi }{\displaystyle {\int }_{\text{0}}^{\alpha }{\displaystyle {\int }_0^{2\pi }\sqrt{\cos \theta }\cdot \sin \theta \cdot V_{E\_\varphi }(\theta ,\phi )\cdot T\cdot l_0(\theta )}}\cdot \exp \left\{ik\left[-\rho \sin \theta \cos (\varphi -\phi )+zcos\theta \right]\right\}d\phi d\theta$$

(4)$$E_z=\frac{-iA}{\pi }{\displaystyle {\int }_{\text{0}}^{\alpha }{\displaystyle {\int }_0^{2\pi }\sqrt{\cos \theta }\cdot \sin \theta \cdot V_{E\_z}(\theta ,\phi )\cdot T\cdot l_0(\theta )}}\cdot \exp \left\{ik\left[-\rho \sin \theta \cos (\varphi -\phi )+zcos\theta \right]\right\}d\phi d\theta$$

where A is the normalization constant, θ is the convergence angle, and φ is the azimuth angle concerning the x axis. ρ is the polar radius in polar coordinates. k = 2n₀π / λ, k is the wavenumber. α = arcsin(NA / n₀), where NA is the numerical aperture of the focusing objective lens, and its value in this paper is chosen as 0.95. n₀ is the refractive index in focusing space. l₀(θ) is the amplitude distribution of the incident beam, which can be expressed as [35]:

(5)$$l_0(\theta )=exp\left[-{\beta }_{\text{1}}{}^{\text{2}}{(\frac{\sin \theta }{\sin \alpha })}^2)\right]J_1(2{\beta }_{\text{2}}\frac{\sin \theta }{\sin \alpha })$$

where β1 is the ratio of the pupil radius to the incident beam waist, J₁(•) represents the first-order Bessel function of the first class, and β₂ represents the first-order Bessel function amplitude modulation parameter.

V_E is a propagation unit vector with three directions, namely V_{E_ρ}(θ, φ)、 V_{E_φ}(θ, φ) and V_{E_z}(θ, φ). If we study the characteristics of a circularly polarized Bessel-Gauss beam modulated by fractional-order vortex beam, V_E can be expressed as [36]:

(6)$$V_E=\left[\begin{array}{l}{V}_{E\_\rho }(\theta ,\phi )\\ V_{E\_\varphi }(\theta ,\phi )\\ V_{E\_z}(\theta ,\phi )\end{array}\right]=\left[\begin{array}{c}({\cos }^2\phi \cos \theta +{\sin }^2\phi )\pm icos\phi \sin \phi (\cos \theta -1)\\ \cos \phi \sin \phi (\cos \theta -1)\pm i(co{s}^2\phi +{\sin }^2\phi \cos \theta )\\ \sin \theta \cdot \exp (\pm i\phi )\end{array}\right]$$

where, when the symbol before i is “+,” it is right-handed circular polarization (RHCP); when the symbol before i is “−,” it is left-handed circular polarization (LHCP).

If we study the characteristics of a radially polarized Bessel-Gauss beam modulated by a fractional-order vortex beam, V_E can be expressed as [37]:

(7)$$V_E=\left[\begin{array}{l}{V}_{E\_\rho }(\theta ,\phi )\\ V_{E\_\varphi }(\theta ,\phi )\\ V_{E\_z}(\theta ,\phi )\end{array}\right]\text{=}\left[\begin{array}{c}\cos (\theta )\cdot \cos (\phi )\\ \cos (\theta )\cdot \sin (\phi )\\ \sin (\theta )\end{array}\right]$$

A special fractional vortex modulation phase is successively designed by term [38, 39]:

(8)$$T=\text{exp}\left\{i\cdot \left\{\text{phase}\left\{cos\left[(n-0.5)\phi +{\phi }_0\right]\right\}+(l+0.5)\phi \right\}\right\}$$

where the parameter n is the fractional-order phase modulation factor and l denotes topological charge l + 0.5, integer l = 0, ±1, ±2, ±3… and n = 0, ±1, ±2, ±3…, respectively. In fact, Eq. (8) corresponds to the lens apodization function in Richards-Wolf diffraction theory, which contains a modulation term {cos[(n − 0.5) φ + φ₀]} and fractional-order vortex phase (l + 0.5)φ with fractional topological charge. Based on the interaction between them, the focusing ability of the incident CPBs is consistent with that of RPBs. Here, the parameter l = 0 does not affect the modulation effect, so for the simplicity, we take l = 0. φ₀ represents the initial polarization of the illumination beam, and its value is 0 in this paper. So the Bessel-Gauss beams can be modulated accurately in focal plane.

It is worth noting that after the above phase modulation, LHCP and RHCP produce different focusing results. This is because the expressions of V_E of LHCP and RHCP are different, and they are modulated by the phase Eq. (8), which eventually leads to different results of E.

Eventually, the focal light intensity of the Bessel-Gauss beams modulated by the fractional-order vortex beams can be obtained using I = |E|².

In this paper, the tight focusing characteristics of Bessel-Gauss beams modulated by fractional-order vortex beams under vacuum conditions are simulated. It should be noted here that the unit in the picture is λ. The focusing characteristics of fractional-order vortex modulated Bessel-Gauss beams at a large numerical aperture (NA) are as follows:

Firstly, Fig. 2 shows the focusing characteristics of fractional-order vortex modulated Bessel-Gauss beams in the two cases of left-hand circular polarization (LHCP) and radial polarization. Under the conditions of NA = 0.95, the ratio of the pupil radius to the incident beam waist β₁ = 1 and the first-order Bessel function amplitude modulation parameter β₂ = 1, the effects of different fractional-order phase parameter n on the tight focusing characteristics of Bessel-Gauss beams are studied. In Fig. 2, it can be seen that both the LHCP and radial polarization generate strong longitudinal fields, and the change rules are basically similar.

Figure 2. When NA = 0.95, β1 = 1 and β2 = 1, the distribution of focal field intensity with different n values. (f)–(t) are LHCP; (u)–(ai) are radial polarization. Their corresponding vortex phases are shown in (a)–(e), respectively. The color bar (aj) shows the light intensity scale of (f)–(t) and (u)–(ai), respectively. LHCP, left-handed circular polarization.

Light spots appear in the longitudinal field of LHCP and radial polarization at different phase modulations, which vary with the azimuth angle. These light spots generally show a petal-like pattern, which will be temporarily called optical lobe in this paper.

From the perspective of the longitudinal field, the most intuitive phenomenon is that the number of optical lobes changes accordingly with the change of n. When n = 7, the number of lobes is 13; When n = 4, the number of lobes is 7; When n = 1, the number of lobes is 1; When n = −2, the number of lobes is 5; When n = −5, the number of lobes is 11. It was found that when n ≥ 1 (n is an integer), the number of optical lobes shows a change rule of (2n − 1). When n ≤ 0 (n is an integer), the number of lobes shows a change rule of (1 − 2n). That is to say, the quantitative relationship between the number of optical lobes and n is: |2n − 1|. In addition, it can be seen from Fig. 2 that the increase in the number of lobes will lead to a regular expansion of the entire spot size and the reef area, as shown in Figs. 2(k)–2(o) and Figs. 2(z)–2(ad). The results show that the n value can effectively adjust the number of optical lobes.

It is proved that a simple circular polarization can also generate a unique strong longitudinal field that is unique to the complex radial polarization, and the changes in spot size and energy intensity are also significant. Figure 3 numerically details the variation of the spot radius for the longitudinal field under LHCP and radial polarization. Here, the value marked in Fig. 3 represents the distance between the two peaks of the light spot, which is used to represent the size of the light spot in the longitudinal field under LHCP and radial polarization. We can clearly see that when n = 7, the spot radii of the longitudinal field under LHCP and radial polarization are 3.10λ and 3.12λ, respectively. When n = 4, the spot radii of the longitudinal field under LHCP and radial polarization are 1.84λ and 1.82λ, respectively. When n = −2, the spot radii of the longitudinal field under LHCP and radial polarization are 1.35λ and 1.32λ, respectively. When n = −5, the spot radii of the longitudinal field under LHCP and radial polarization are 2.67λ and 2.69λ, respectively. It can be seen that the adjustment of n to the longitudinal field spot radius is basically the same under LHCP and radial polarization. Figure 3 further confirms our view numerically.

Figure 3. Energy distribution in the vertical direction of the focal plane of the longitudinal field with different n values (n = 7, n = 4, n = 1, n = −2, n = −5). (a)–(e) are LHCP; (f)–(j) are radial polarization. LHCP, left-handed circular polarization.

Distributions of focal field intensity under different n under LHCP and RHCP are illustrated in Fig. 4. It can be clearly seen from the longitudinal field that the number of lobes in the longitudinal field under LHCP is clear, the border of each lobe is obvious, and a strong longitudinal field component is generated, as shown in Figs. 4(k)–4(o). The borders of the light lobes in the RHCP are no longer obvious, and the shape is not clear. The individual lobes start to connect together and gradually form a ring, as shown in Figs. 4(z)–4(ad).

Figure 4. When NA = 0.95, β1 = 1 and β2 = 1, the distribution of focal field intensity under different n. (f)–(t) are LHCP; (u)–(ai) are RHCP. Their corresponding vortex phases are shown in (a)–(e), respectively. The color bar (aj) shows the light intensity scale of (f)–(t) and (u)–(ai), respectively. LHCP, left-handed circular polarization; RHCP, right-handed circular polarization.

In addition, it was found that the energy of the longitudinal field under RHCP is obviously not as strong as that under LHCP, and it gradually dispersed. In order to show the change in energy distribution of the longitudinal field more intuitively, Fig. 5 plots the energy flow diagrams of the longitudinal field under LHCP and RHCP, respectively. In Figs. 5(a)–5(e) it can clearly be seen that the energy is evenly distributed at the position of each light lobe, and the energy is relatively concentrated. However, as shown in Figs. 5(f)–5(j), it was found that the energy flow is relatively scattered, the energy intensity is weak, and the outline of the light lobe is not clear.

Figure 5. Energy flow in the focal plane of the longitudinal field with different n values (n = 7, n = 4, n = 1, n = −2, n = −5); [(a)–(e) are LHCP; (f)–(j) are RHCP]. LHCP, left-handed circular polarization; RHCP, right-handed circular polarization.

Through the above analysis, we find that LHCP can produce the same strong longitudinal field effect as radial polarization, and compared with RHCP, only LHCP has the ability to create a small longitudinal field. Next, other focusing properties of fractional-order vortex modulated Bessel-Gauss beams under the condition of LHCP are studied.

Figure 6 illustrates the influence of the ratio of the pupil radius to the incident beam waist β₁ on the tight focusing characteristics of Bessel-Gauss beams under the conditions of NA = 0.95, the first-order Bessel function amplitude modulation parameter β₂ = 1, and the fractional-order phase modulation parameter n = 4. Figures 6(a)–6(l) show changes in the transverse field, longitudinal field, and total field of the optical lobes under LHCP, and also give the energy distribution curve in the vertical direction of the focal plane of the longitudinal field, as shown in Figs. 6(m)–6(p). The color bar 6(q) shows the light intensity scale of 6(a)–6(l), respectively. From the perspective of the longitudinal field, the number and shape of optical lobes remain unchanged in the process of β₁ gradually increasing, but the radius gradually increases. We numerically calculate the variation rule of the lobe radius as it increases from 1 to 2.5, that is, the reef area in the vertical direction of the longitudinal field is 1.84λ, 2.04λ, 2.34λ and 2.88λ, respectively. It can be seen that with the increase of β₁, the lobe radius increases significantly. The results show that the β₁ value can adjust the size of the lobe radius.

Figure 6. In LHCP, the focusing characteristics and energy distribution in the vertical direction of the focal plane of the longitudinal field under different β1 values (NA = 0.95, β2 = 1, n = 4). The color bar (q) shows the light intensity scale of (a)–(l), respectively. LHCP, left-handed circular polarization.

According to Fig. 6, we take β₁ as the ordinate, and the spot size is set to the abscissa (denoted by d here). As a result, the change curve of the ratio of β₁ to d is obtained, as shown in Fig. 7. When β₁ increased from 1 to 2.5, the ratio of β₁ to d was about 0.543, 0.735, 0.855, and 0.868, respectively. We can see that as β₁ increases, the ratio of β₁ to d increases more slowly, that is, the slope decreases. It was further verified that the increase of β₁ can lead to a nonlinear and slow growth trend of the spot size.

Figure 7. The variation curve of the ratio of the pupil radius to the incident beam waist β1 and the spot size d.

To obtain the effect of the first-order Bessel function amplitude modulation parameter β₂ on the focusing characteristics, we set NA = 0.95, the ratio of the pupil radius to the incident beam waist β₁ = 1, and the fractional-order phase modulation parameter n = 4. The focal spot patterns when the amplitude modulation parameters β₂ are 1.5, 2.5, 3.5, and 4.5 are obtained, as shown in Fig. 8. Figrues 8(a)–8(l) show changes in the transverse field, longitudinal field, and total field of a light spot under LHCP, and 8(m)–8(p) show changes in the contour energy distribution of a light beam in the longitudinal field. The color bar (q) shows the light intensity scale of 8(a)–8(l), respectively. From the perspective of the longitudinal field, the shape of the optical lobes begins to change when β₂ = 2.5. The seven originally independent lobes begin to connect together, and the area of the middle reef increases, as shown in Fig. 8(e). It can be seen that seven small light petals appear in the center of the reef area and begin to stratify like roses when β₂ = 3.5. And with the increase of β₂, the second light lobe gradually diffused outward, as shown in Fig. 8(h). The third layer of light lobes appears in the central reef area when β₂ = 4.5, and the number is still seven, as shown in Fig. 8(k). With the continuous increase of β₂, the third petal gradually diffuses around and the fourth petal begins to appear. In this way, the radius of the whole spot keeps increasing, the reef area keeps increasing, and the number of petal layers keeps increasing and diffusing from the periphery layer by layer. It plots the contour energy distribution of the focal plane of the longitudinal field, as shown in Figs. 8(m)–8(p). This more intuitively shows the change in the number of optical lobe layers with the increase of β₂. The results show that β₂ can accurately adjust the number of layers of the optical lobes.

Figure 8. In LHCP, the focal graph of light spot and contour energy distribution of the longitudinal field focal plane under different β2 values (NA = 0.95, β1 = 1, n = 4). The color bar (q) shows the light intensity scale of (a)–(l), respectively. LHCP, left-handed circular polarization.

In this paper, by comparing the tight focusing characteristics of fractional-order vortex modulated circularly polarized and radial polarized Bessel-Gauss beams, we found that simple and easy-to-control circular polarization can produce the same effect as the more complex radial polarization. At the same time, it also proves that only LHCP can create a small longitudinal field. This discovery gives researchers a new way of thinking, in which a strong longitudinal field can be achieved by simple circular polarization with fractional-order vortex modulation rather than complex radial polarization. Additionally, the possibility of generating arbitrary fractional-order vortex modulated Bessel-Gauss beams with a strong longitudinal field is demonstrated. A series of tight focusing characteristics of left-handed circular polarized Bessel-Gauss beams with arbitrary fractional-order vortex modulation are discussed. It is proved that it is possible to precisely control the number of lobes, layers and radii. The results show that the fractional-order phase modulation parameter n can effectively adjust the number of optical lobes. The ratio of the pupil radius to the incident beam waist β₁ can precisely adjust the radius of the optical lobes. The first-order Bessel function amplitude modulation parameter β₂ can adjust the number of layers of the optical lobes accurately. This new research presents the different effects of many optical parameters on the tight focusing properties of circular polarization with fractional-order vortex modulation, which has potential implications for more complex applications in optical micromanipulation, optical communications, and more.

At present, our research group is carrying out relevant experimental research, and it is expected to design and build a set of experimental platforms for measuring optical field detection under tight focusing conditions. Because the tight focus system is difficult to set up, it is necessary to be fully prepared. We believe that we will see some results in the near future.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

Parts of this work were supported by the National Key Research and Development Program of China (2018YFC 1313803).