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Abstract
We theoretically investigate the propagation dynamics of vectorial Mathieu and Weber tightly autofocusing beams, which are constructed based on nonparaxial Weber and Mathieu accelerating beams, respectively. They can automatically focus along the paraboloid and ellipsoid, and the focal fields represent the tightly focusing properties resembling that generated with a high NA lens. We demonstrate the influence of the beam parameters on the spot size and energy proportion of longitudinal component of the focal fields. It reveals that Mathieu tightly autofocusing beam supports a more superior focusing performance, of which the longitudinal field component with superoscillatory feature could be enhanced by decreasing the order and selecting the suitable interfocal separation of the beam. These results are expected to provide new insights for the autofocusing beams and the tight focusing of the vector beams.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Accelerating beams have received more attention in recent years due to their special propagation trajectories [1]. Airy beams, the famous classical type of self-accelerating beams, can be transmitted over long distances without diffraction while accelerating laterally [2–5]. By solving the analytical solutions of the wave equations, a series of novel accelerating beams are proposed, such as Bessel-like accelerating beam along a circular trajectory [6,7], Mathieu accelerating beam (NMAB) along on an elliptical trajectory [8,9], Weber accelerating beam (NWAB) along a parabolic trajectory [10], and fully vectorial accelerating diffraction-free Helmholtz beam [11]. Particularly, Airy beam is an approximate solution of the Weber beam [8]. Based on the accelerating beams, autofocusing beams are proposed and open up a new research field. Autofocusing beams can be automatically focused in free space without passing through a lens, of which the intensity increases abruptly at the focal point. The representative examples are abruptly autofocusing beams [12–14], circular Weber and Mathieu beams [15], and circular Pearce beams [16]. On the other hand, the optical caustic theory has become an important tool of constructing the autofocusing beams. Several types of autofocusing beams have been proposed, including circular swallowtail beams based on the catastrophe theory [17], one-dimensional multi-focus autofocusing optical beams [18], nonparaxial autofocusing beams [19,20], et al. These autofocusing beams show many potential applications in particle manipulation [21], super-resolution imaging [22], light bullets [23], and free-space optical communication [24].
On another hand, vector beams have attracted extensive research interest in the past decades [25–28]. Based on the special polarization structure, vector beams show many remarkable properties when tightly focused by a high NA lens. Typically, the radially polarized beam generates a strong longitudinal field and a sub-diffraction limit focal spot, while the azimuthally polarized beam produces a doughnut focal field [29]. Vector beams have many applications in laser micromachining [30], super-resolution microscopy [31], optical storage [32], micro manipulation [33], etc. Accordingly, it is significant to study the property of autofocusing beam with nonuniform polarization. It is found that autofocusing Airy beams with cylindrical polarizations can exhibit the abrupt polarization transitions [34,35]. By encoding cosine-azimuthal variant phases, vector abruptly autofocusing beam realizes the spin-dependent splitting [36]. Structured vectorial abruptly autofocusing beams can produce focal spots of various shapes [20]. Furthermore, tightly autofocusing beams (TABs) with cylindrical polarizations provide a new way to achieve tight focusing and to enhance the longitudinal field [37,38]. Therefore, to explore the novel vectorial features of different autofocusing beams is still an issue that is worth studying.
In this paper, we investigate the focusing characteristics of Weber and Mathieu tightly autofocusing beams (WTABs, MTABs) with radial polarizations, which focus along the paraboloid and ellipsoid trajectories, respectively. We analyzed the focusing performances in detail, including focal spot size and energy proportion of longitudinal component. A comparison of the focusing characteristics between different TABs and tightly focusing systems is summarized. The results show that as the transmission distance of tightly autofocusing beam decreases, the energy proportion of longitudinal field increases, and the focal spot shrinks. Namely, the tight focusing effect can be availably enhanced by reducing the focal length.
2. Theoretical principle
We begin our discussion with the nonparaxial accelerating solutions of Helmholtz equation. Weber accelerating beam, which transmits along the parabolic trajectory, is a particular solution for the parabolic coordinates (τ, σ) meeting z = τσ, x=(τ2 − σ2)/2 [8,9,39]. In the vectorial situation, we can use Rayleigh-Sommerfeld diffraction theory to simulate the propagation process of Weber accelerating beam, and to analyze the polarization characteristics during propagation. By transforming to the Cartesian coordinates, the expression of an x-polarized Weber beam is written as
where Wp can be either an odd or an even solution of the parabolic cylinder functions, m is the dimensionless parabolic momentum, k = 2π/λ is the wave number in vacuum, and α is the decay parameter used to ensure the finite energy. Figure 1(a) shows the propagation process of Weber accelerating beam with m = 50, where the white arrowheads depict the polarization directions of the main lobe. Obviously, the polarization rotates around x-axis, and the rotating angle reaches 42.76° when the main lobe arrives at z-axis. Similar to Ref. [37], the polarization rotation can also be extended to the radially-symmetric case of the x-z plane, producing a WTAB. Figure 1(d) depicts the schematic diagram of WTAB, which focuses along the paraboloid trajectory. A part of the radial polarization (blue arrows) is projected to z-axis and produces the longitudinal polarization, while the azimuthal polarization (green arrow) remains unchanged. Since the azimuthal polarization does not contribute to the longitudinal focal field, we will only discuss the radical polarization below. By converting Eq. (1) to the cylindrical coordinates, radially polarized WTAB can be written as where r denotes the radial coordinate.
Fig. 1. (a)-(c) Propagation processes of x-polarized nonparaxial Weber and Mathieu accelerating beams with m = 50, where the white arrows depict the polarization direction on the trajectories (dash lines). (d)-(f) Schematic diagram of the Weber and Mathieu tightly autofocusing beams, where the blue and green arrowheads denote the radial and azimuthal polarizations (er and eφ), and the red arrowheads depict the propagation trajectories of the beams. The second and third columns correspond to the case a > b and a < b for Mathieu beam, respectively.
Likewise, for the Mathieu solution of Helmholtz equation in the elliptical coordinates, it transmits along the elliptical trajectory. The elliptical coordinates (ξ, η) meet x = h sinhξ sinη, and z = h coshξ cosη, where h=|a2-b2|1/2 is the interfocal separation, ξ and η are the elliptic coordinates, a and b are the two semi-axes [8]. Here, a denotes the semi-axis corresponding to input plane. Similar to the construction of WTAB, we firstly calculated the polarization rotating during the propagation process of Mathieu accelerating beam (m = 50, h = 5µm) for the cases a > b and a < b, as shown in Figs. 1(b) and 1(c), respectively. The corresponding rotation angle of polarization is 74.01° and 65.31°, respectively, indicating that more energy would be converted into the longitudinal component. It should be noted that the more intense the self-bending of beam, the more intense the diffraction effect, and the more inconsistent the propagation trajectory with the prediction. By transforming Mathieu beams into the radially symmetric forms, we can obtain two types of radially polarized MTABs, of which the complex amplitude vectors for a > b and a < b is denoted by M1(r) and M2(r), respectively. Figures. 1(e)-(f) show the schematic diagram of MTABs. These two types of MTABs have the incident planes on the long and short semi-axes of the ellipsoid, respectively, and correspondingly the focal lengths are short and long semi-axes. Their amplitude vectors are expressed as
3. Results and discussions
In this section, we analyze the propagation dynamics of radially polarized WTAB and MTAB by the Rayleigh-Sommerfeld diffraction theory, and study the tightly autofocusing properties in detail. In the calculation, we uniformly set the wavelength of the beam λ=633 nm and the decay parameter α=105m-1. The initial intensity of the beam is normalized.
3.1 Focusing characteristics of WTAB
Firstly, we study the focusing characteristics of WTAB with m = 50. Figure 2(a) represents the transverse intensity distribution of radially polarized WTAB, which has a circularly symmetric profile. The energy of the beam is mainly concentrated in the main lobe and attenuates radially for the side lobes. Figures 2(d)-(f) depict the side views of the propagation processes of the total field, radially and longitudinally polarized components of WTAB, which reveals that the focusing trajectory of the main lobe of the beam has a parabolic trajectory (white dashed line). The inserts correspondingly depict the transverse intensity distributions at the focal plane, representing the circularly symmetric distributions, where the longitudinal component is analogous to a zero-order Bessel beam with a solid spot, and the radial component is like a hollow one-order Bessel beam. To reveal the abruptly focusing property, the intensity contrast along the z-axis is calculated as shown in Fig. 2(b). The position of the focal point is at 10.66 µm, which is identified as the location of the peak intensity. Obviously, the intensity stays at a low level until reaching the focal point, and then abruptly increases by 53.11 times. Figure 2(c) depicts the normalized intensity profiles of different polarization components at the focal point. The longitudinal component (Iz, red solid curve) is much higher than the radial one (Ir, green solid curve), and the former is 3.37 times higher than the latter. As for the azimuthal component, the intensity is always zero. The focal spot size can be denoted by the full-width at half-maximum (FWHM), and is calculated as 0.688λ. This indicates that WTAB produces a smaller focal spot than that generated by an objective with NA = 0.9 [40].
Fig. 2. (a) Initial transverse intensity distributions of WTAB. (b) Intensity contrast along the z-axis. (c) Intensity profiles of the total field (I), radially (Ir), and longitudinally (Iz) polarized components. (d)-(f) Side view of propagation processes of I, Ir, Iz in the logarithmic scale, where the insets show the transverse focal spots within 4λ squared areas.
3.2 Focusing characteristics of MTAB
Next, we investigate the focusing characteristics of two types of MTABs. In the case a < b, we choose h = 5µm, here. Figure 3(a) shows the transverse intensity distribution of radially polarized MTAB with m = 50, which is distributed axisymmetrically. Figures 3(d)-3(f) give the propagation process of the beam, showing that the light energy perfectly focuses along the elliptical (dashed lines) of which the long axis corresponds the focal length. From Fig. 3(g) we can see that the longitudinal and radial focal field of MTAB have the similar profiles to that of WTAB, but the longitudinal component is more dominant. The results in Figs. 3(b) and (c) show the intensity contrast along the z-axis and the intensity distributions at the focal plane, respectively. Comparing with WTAB, the focal length is reduced to 6.797 µm, and the maximum intensity has a small decrease (50.76 times of the input beam). From Fig. 3(c), it is seen evidently that the longitudinal component is further strengthened, the peak intensity of the longitudinal component is 8.2 times of the radial one. The dominant longitudinal field induces the further reduction of the focal spot size, which is 0.176λ2, smaller than that generated by the objective lens of NA = 0.95 [28].
Fig. 3. (a) Initial transverse intensity distributions of MTAB when a < b. (b) Intensity contrast along the z axis. (c) Intensity profiles of the total field (I), radially (Ir), and longitudinally (Iz) polarized components. (d)-(f) Side view of propagation processes of I, Ir, Iz in the logarithmic scale. (g) Intensity distribution of the longitudinal (top) and radial (bottom) focal field.
While for the case a > b of MTAB, we choose h = 5µm and m = 50. Similar to Fig. 3, the calculation results of the focal fields are shown in Fig. 4. Figure 4(a) gives the initial intensity distribution of MTAB, showing a larger diameter of main lobe than that of a < b. Figures 4(d)-(f) depict the beam focuses along the elliptical (dashed lines) of which the short axis corresponds the focal length. From Fig. 4(b), it can be seen that the focal length is further shortened to 4.492µm. The intensity distributions of focal spot in Fig. 4(c) reveal that the longitudinal component is enhanced again comparing with the case of a < b, and the peak intensity is 24.28 times of the radial one. This is consistent with the expectation of polarization rotation in Fig. 1. The focal spot size is 0.124λ2, smaller than that generated by an objective lens of NA = 0.95 combining with a diffractive optical element [41]. In general, a TAB with shorter focal length can generate a stronger longitudinal field and smaller focal spot, same as our previous work [37,38].
The focusing properties of MTABs are influenced by the value of the interfocal separation h, which mainly determines the radius of input beam for a > b and the focal length for a < b. To analyze the focusing properties, we mainly focus on the proportion of longitudinal component and the spot size of the focal field. Here, the energy proportion γ of the longitudinal field is defined as τz/(τz + τr), where τr,z = 2π∫|Er,z|2rdr. Figures 5(a) and 5(b) show the energy proportion γ and the spot size versus h, respectively, where the insets give the side views of the focusing processes of MTABs with different h in logarithmic scale. It is evident that as the increase of h, the focusing properties represent the opposite trend for the case of a > b and a < b. For the case of a > b, the proportion of longitudinal component increases and the spot size decreases with h. It indicates that the focusing performance can be enhanced by increasing h, which increases the radius of input beam but hardly changes the focal length. While for the case of a < b, the increasing of h obviously lengthens the focal length, and the focusing performance is weakened. It is worth noting that when the interfocal separation is small enough (e.g., h = 1µm), the two types of MTABs are in proximity to each other, and represent the similar focusing processes with quasi-circular trajectories. Importantly, we have chosen the maximum h = 5µm, because if h continues to increase, the severe diffraction would disrupt the focusing of MTAB, and would reduce the intensity of the focal field. As a result, to enhance the focusing performance of MTAB, we can choose the case of a > b and select h as large as possible.
3.3 Comparison of the focusing characteristics
In the following, we discuss the effect of parameter m on the focusing characteristics of radially polarized MTABs (h = 5µm) and WTABs. The parameter m, denoting the beam order, mainly determines the propagation trajectory in a certain parabolic or elliptic coordinate. Of course, it would also affect the focusing characteristics. Figures 6(a) and 6(b) depict the variations of the spot size and energy proportion of the longitudinal field with m, respectively. It is observed that the focusing performances of WTAB are inferior to those of MTABs on the whole. As m increases, the spot size of WTABs increases much faster than those of MTABs, which are less variable. While the energy proportion γ increases significantly with m for these three types of beams. Especially for MTAB (a > b), it can reach a higher γ for a small m, indicating that a purer longitudinal field is produced. When m = 40, the proportion of longitudinal component reaches γ=90.43%, and the FWHM of the focal field is 0.384λ (spot size is 0.116λ2), which exceeds the diffraction limit and approaches the size of the superoscillation [42]. Importantly, if m decreases below 40, continues to decrease, the propagation of MTAB will collapse, and the focusing performance would be greatly reduced. Therefore, we need to choose the appropriate m to guarantee the autofocusing process along the presetting trajectory.
As a comparison, Table 1 summarizes the focusing characteristics of radially polarized beams for different tightly focusing systems, which have been reported in other papers. Typically, a focal field with a strong longitudinally polarized component can be generated with a radially vector beam tightly focused by an high-NA objective lens [40]. Subsequently, annular aperture [40], binary optics element (BOE) [28], and diffractive optical element (DOE) [41] have been employed to enhance the longitudinal field and generate smaller focal spots. By employing TAB with spherical trajectory [37], the focal field has been further enhanced. WTAB and MTABs proposed in this paper can obviously generate the focal field with better performances than those for the traditional methods. Especially for MTABs, they can produce smaller focal spot and stronger longitudinal field than those generated with NA = 0.95 lens and DOE. These four types of TABs provide new possibilities to achieve tight focusing with different methods superior to high-NA lens.
Table 1. Summary of tightly focusing characteristics of various methods
4. Conclusion
In conclusion, we propose the new types WTAB and MTABs, which have profiles of higher order Mathieu and Weber functions. We numerically demonstrate the focusing properties of these beams, which have better performances than that generated by traditional methods with high-NA lens. We show that the longitudinal field can be effectively enhanced by reducing the order of WTAB and MTABs. MTABs represent better focusing properties than WTAB due to the larger polarization rotation during focusing. Especially for the MTAB of a > b, selecting proper interfocal separation can greatly enhance the tight focusing. This work is expected to contribute to expanding the research scopes of autofocusing beams, and will be useful for optical manipulation and super-resolution imaging.
Funding
National Key Research and Development Program of China (2022YFA1404800); National Natural Science Foundation of China(NSFC), (12074312, 12174309, 12074313); Fundamental Research Funds for the Central Universities (3102019JC008); the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX2021115).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper can be obtained from the authors upon reasonable request.
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