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Abstract
We demonstrate that our theoretical scheme developed in the previous study on the caustics of the abruptly autofocusing vortex beams [Xiao et al., Opt. Express 29, 19975 (2021) [CrossRef] ] is universal for all the axially symmetric vortex beams. Further analyses based on this method show the complex compositions of the vortex caustics in real space. Fine features of the global caustics are well reproduced, including their deviations from the trajectories of the host beams. Besides, we also show the possibility of tailoring the vortex caustics in paraxial optics based on our theory. The excellent agreements of our theoretical results with both numerical and experimental results confirm the validity of this scheme.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams are structured light fields with doughnut-shaped spatial profiles and phase singularities. Their distinctive feature is the presence of the helical phase fronts described as exp (ilθ), where l is the so-called topological charge and θ is the azimuthal angle [1]. Among the vortex beams, the spiral vortex beams are shown to be strong in terms of the structural stability [2,3]. Due to these notable properties, vortex beams have attracted widespread interests and play critical roles in various applications, like particle micromanipulation [4,5], microscopy [6], material processing [7–9], and optical communication [10,11]. By superimposing the helical phase on the axially symmetric phases of different host beams like the Gaussian beam, the Bessel beam and the abruptly autofocusing beam, a series of axially symmetric vortex beams (ASVBs) are investigated [12–15]. In this way, vortex beams can be adapted with diverse intensity tubes in focusing, diverging and non-diffracting shapes.
Two major schemes are basically implemented to interpret and engineer the light fields. The integral-based scheme is the mostly-used method, involving with the diffraction integral using the stationary phase method [16–19]. In some cases, the use of this method induces “singular points” whose intensity tends to infinity [20]. Various asymptotic methods have also been discussed to analyze the field around the irremovable singular points [20–23]. As the approximation is indispensable to simplify the diffraction integral, error estimates as well as complex algebra are required. In parallel, the differentiation-based scheme is also developed in the realm of the geometric optics. In this scheme, vortex light fields can be decomposed into families of rays emerging from any cross-section along propagation [24,25]. And the envelope of the ensemble of rays constitutes the framework of the caustics [24,26]. In this way, a relatively simple and efficient method without further approximation is available to interpret the caustics of ASVBs and to predict the focusing properties in real space [27]. However, some clarifications of the compositions of caustics in real space have not been discussed. In addition, for applications like microscopy, laser ablation and fabrication of two-photon polymerization, tailoring the caustics have also been an increasing interest in recent years [28–32]. With a comprehensive understanding of the caustics, engineering the caustics of the ASVBs is also an issue of fundamental importance in applications.
In this paper, we show a developed approach to classify the caustics of different ASVBs. In Section 2, we firstly introduce the theoretical results as a set of concise expressions developed for reproducing the caustics of the ASVBs. Comparing with the existing results, these expressions are obtained without any mathematical approximation. Then, the third Section compares our theories with the numerical simulations and the experimental results for several exemplary beams, illustrating the geometries of the vortex caustics. In particular, finer features can be well outlined in more intuitive physical images as shown in the discussion of the Bessel vortex beam and the abruptly autofocusing vortex beam, which are absent in previous studies. Furthermore, we also developed a method to engineer vortex beams with tailored novel tubular caustics based on our results. Our theoretical results are in excellent agreement with both numerical simulations and experimental results.
2. Caustics of the ASVBs
2.1 Analytical results
From a given axially symmetric host beam with the phase profile ϕhost (r), the vortex version can be synthesized by combining the spiral phase ϕvortex (θ) =lθ with ϕhost (r) in any cross-section along propagation [27]:
Fig. 1. Schematics of the axially symmetric vortex beams (ASVBs) in geometrical optics: (a) a single hyperboloid formed by rays emerging from a specific ring in the transverse plane z = 0; (b) Two different hyperboloids formed by the corresponding families of rays from the plane z = 0 (the blue and red half tubes), with their intersections with y = 0 plane (blue and red solid curves) projected into the bottom plane. The intersections between other hyperboloids and y = 0 plane are also superimposed (gray solid curves). The whole beam propagates in the z direction.
2.2 Discussions on the topology of the caustics
In this work, we focus on the caustics in real space of an optical setup (z >0) since the beam is usually generated in such ways. In general, the constituent points of the global caustics of the ASVBs can be rather complex: either a single set of {z1} (or{z2}) or both sets of {z1} and {z2} can get involved in constituting the caustic in real space (as will be shown in Section 3.2). After some algebra, we find that the expression below can well distinguish the above two general cases in terms of zw:
Table 1. Constituent points in real space (z > 0)
3. Numerical and experimental demonstration of the caustics of different ASBVs
3.1 Setup
We use the same setup as in [27] to demonstrate our analytical results in section 2. As shown in Fig. 2, the Gaussian beam centered at 1065 nm from a home-made ultrafast fiber laser is firstly injected into a beam expander. The output collimated beam with an expanded width of ∼8 mm is then incident upon a phase-only spatial light modulator (Holoeye SLM-Pluto, 1920×1080 pixels). The exposed power is controlled by adjusting the half-wave plate before a polarizer. With the phases of the synthesized vortex beams in Table 2 encoded on the SLM together with an additional grating phase, ASVBs can be generated right after SLM in the first order. Then a telescope is used to shrink the ASVBs down to the micron-scale. The telescope is built with a lens (focal length = 1 m) and a microscope objective (50×, NA = 0.8). An iris is installed close to MO1 to block the undesired orders, retaining the first order of the diffraction lights as shown in the inset of Fig. 2. An imaging system made of another identical microscope objective (MO2) followed by a lens (focal length = 0.5 m) is used to capture the intensity profiles at each z position along beam propagation. The 2D side views of the intensity profiles are then extracted from the stacked data. Careful alignments only introduce weak perturbation in the setup. This guarantees that the vortex beams appear with negligible deviation from the ideal caustics.
Fig. 2. Schematic of the experimental setup. HWP: half-wave plate; P: polarizer; BE: beam expander; SLM: spatial light modulator; L: lenses; MO1 and MO2: microscope objectives.
Table 2. Phases of four vortex beams synthesized from specific host beams
3.2 Experimental demonstrations
In our previous study on vortex beams hosted by the abruptly autofocusing beams [27], it is proved that the caustics of their central tube can be well described by the set of constituent points determined by Eq. (3)-(5). To further demonstrate the universality of our formulae for the whole family of axially symmetric vortex beams (ASVBs), vortex beams with four additional host beams, such as Gaussian beams, Bessel beams, Bessel-like beams [33] and parabolic toroidal lens beams [34] are generated in this work. Table 2 lists their phases and the corresponding sin γ(r).
According to section 2, the caustics of the ASVBs with a given topological charge l are only related to the corresponding host phases. Without loss of generality, the topological charge l is chosen to be l = 5 for this section. Based on the angular spectrum method [35], we also numerically simulate different ASBVs with parameters corresponding to the setup in Fig. 2. The simulated and experimental results are shown in Fig. 3, where the measured 2D intensity profiles are extracted from the 3D data acquired with our setup. For Gaussian vortex beams, the phase of the host Gaussian beam as well as the corresponding total phase of the ASVB are shown in Fig. 3(a1) - (a2). The flat phase of the host Gaussian beam results in zw = 0 and B = 0 for any r, which gives z2 = – z1 with z1 > 0. According to Table 1, the caustic in real space is determined only by the single set of z1, shown by the blue dash-dotted line in Fig. 3(a3) - (a5). In the second example, the phases of the host Bessel beam with sin γ(r) ≡ 8.87×10–4 and the corresponding vortex beam are shown in Fig. 3(b1) - (b2). As Bessel vortex beam determines that R′(r) > 0 and zw < (B+Δ) /2A, the sign of z1 and z2 can be solved as z1 > 0 and z2 < 0. In this way, the caustic in real space is also determined by the single set of z1 as shown in Fig. 3(b3) - (b5) with the blue dash-dotted line. Besides, the set of (ρ(z1), z1) can be well approximated by the set of (R(r), zw(r)) for Bessel vortex beams. Comparing with the cylindrical caustic expected for Bessel vortex beams [23], our solution in Table 2 can well outline the finer features of the transitionally-expanding tube caustic as shown in [25]. The cylindrical caustic is the special case where the approximation l/(kr) ≪ sin γ is applied in the expression R(r) in Eq. (3). We stress here that the analytical results in this paper also address the puzzling correspondence between the ensemble of the hyperboloidal waists and the central tube caustics of Bessel vortex beams in our previous study [25].
Fig. 3. The results of different ASVBs with topological charge l = 5. (1)-(2): The phases of the host beams and the corresponding ASVBs. (3)-(5): The theoretical, numerical and experimental results for (a) Gaussian vortex beam, (b) Bessel vortex beam and (c) perfect optical vortex beam generated by parabolic vortex toroidal lens. Light blue dash-dot lines together with dark blue short lines represent the caustic defined by (ρ(z1), z1). Green dash-dot lines represent the caustic defined by (ρ(z2), z2). Orange dot lines represent part of the caustic formed by part of the hyperboloidal surface between (ρ(z1), z1) and (ρ(z2), z2) occurred in the perfect optical vortex.
A family of perfect optical vortices (POVs) are recently designed by integrating the spiral phase into the phase of the parabolic toroidal lens [34]. This POV with a large range of topological charge l has a quasi-static focal ring radius r0 at a focal length f. The set of equations determining the caustics of these beams are too complex in [34], resulting in analytical estimates in two special cases. Since our analytical results are deduced for any axially symmetric vortex beam without any further estimate, we also generate one of such beams to demonstrate the validity of our method. The corresponding beam parameters after the telescope in Table 2 are selected as f = 30µm and r0 = 7µm. After some algebra, we found the caustic of this specific POV can be decomposed into multiple sections. When a given r is small enough that zw(r) < [B+Δ⋅sgn(R′)] /2A and R′ > 0, the single set of z1(r) define the most front section of the caustic [the blue dash-dotted line in Fig. 3(c3)–(c5)]. As r grows until zw(r) > [B+Δ⋅sgn(R′)] /2A, both the solution sets in Eq. (5) are involved in forming the caustic: {(ρ(z1), z1(r))} define a tiny part of the caustic very close to the original focal plane of the host beam [dark blue lines in Fig. 3(c3)-(c5)]; {(ρ(z2), z2(r))} determine the long opening tube behind, as shown by the green dash-dotted line in Fig. 3(c3)-(c5). The intermediate caustic section [orange dots] in the range of [z1(rmax), z2(rmax)] between the short dark blue line and the green line is composed by part of the hyperboloid formed by the rays emerging from the edge of the effective aperture. Besides, when the vortex order is increased, the section {(ρ(z1), z1(r))} before the focus remains quasi-static and the section {(ρ(z2), z2(r))} after the focus is shorter. In short, the complex global caustics can be well analyzed with our method. Our analytical results are in excellent agreement with both numerical simulation and experimental results.
In addition, our results can well reproduce the deviation of the propagating behavior of the synthesized vortex beam from that of the corresponding host beam with specific parameters. The topological charge l is also chosen to be l = 5. Here, we firstly selected the Bessel-like beam in [33], i.e. a family of shape invariant beams, as the host beam. For specific Bessel-like beams with a linearly ramped central lobe along propagation, the parameters are selected as n = 2, m = 1 in Table 2. As for the Bessel-like beams featuring the linearly diverging central lobe, the parameters are chosen as a = –0.17 m–1, b = 0.0018. With R′ > 0 and zw < (B+Δ) /2A, z2 is negative. Therefore, the global caustic is also defined by the single set of z1, shown as the blue dash-dot lines in Fig. 4(a). For comparison, the central lobe profile of the Bessel-like beam without the spiral phase is superimposed as the purple line in Fig. 4(a3). It is evident that the spiral phase further accelerates the divergence along propagation. In the second Bessel-like beam [shown in Fig. 4(b)], we select the parameters as a = 0.056 m–1 and b = 0.0006 without changing n and m. Instead, this Bessel-like host beam has a linearly-tapered central lobe along propagation, presented by the purple line in Fig. 4(b3). Interestingly, the corresponding ASVB has an extremum on the ring caustic [shown by the single red circle in Fig. 4(b3)] determined by R′(rextrema) = 0. This behavior is totally different from the monotonous tapering of central lobe in the host beam along propagation. In fact, this deviation was also observed in our early work on abruptly autofocusing vortex (AAFV) beams hosted by the polynomial phase [27]. For such AAFV beams with specific parameters (for instance, n = 4, a = 0.3×10–4 m–3 and r0 = 1.08 mm as in [27]), the sets of z1 and z2 are both involved in constituting the global caustics. Besides the significant deviation of the vortex caustics from the host polynomial trajectories [shown by the purple line in Fig. 4(c3)], the introduction of the spiral phase into the host beam also brings the interesting feature: once the local maximum in the waist distribution R(r) exists, an extremum occurs accordingly on the global caustic despite of the original trajectory of the host beam as shown in Fig. 4. In short, the analyses based on our method can exactly reproduce the fine features of the caustic shape. This further demonstrates the validity of our analytical results, serving as a powerful tool for analyzing the vortex beams.
Fig. 4. Propagating behavior deviation of the ASVBs (l = 5) from their host beams. Bessel-like vortex beam with parameters: (a) a = –0.17 m–1, b = 0.0018 and (b) a = 0.056 m–1, b = 0.0006; (c) Abruptly autofocusing vortex beams with: n = 4, a = 0.3×10–4 m-3 and r0 = 1.08 mm. Purple lines in (a3) and (b3) show the central lobe size of the host Bessel-like beam without the spiral phase, and the purple line in (c3) represents the host polynomial trajectory. Blue dash-dot lines represent the caustic defined by (ρ(z1), z1) and green dash-dot lines represent the caustic defined by (ρ(z2), z2). Orange dot lines in (c1) - (c3) represent part of the caustic formed by part of the hyperboloidal surface between (ρ(z1), z1) and (ρ(z2), z2) like the perfect optical vortex. Insets of (b3) and (c3) show zoomed extrema points.
4. Engineering the caustics of the ASVBs in the paraxial optics
Previous sections show how to calculate the caustics of the ASVBs from the their phases ϕ(r,θ). With the help of the caustic expressions, we also demonstrate that the propagation behavior of the ASVBs can deviate significantly from that of the corresponding host beams due to the spiral phase. Therefore, a challenge arises here that the host beams cannot always well predict the tube shape of their vortex “brother” beams.
In fact, the inverse problem of tailoring the caustic by an engineered host phase is more interesting. The target profile ρ(z) = c(z) can be engineered by solving the host phase ϕhost(r) from zcaustic (r) =c–1(z) based on Eq. (4) and (5). However, this usually involves a mathematical challenge. Here, we clarify a fundamental limit and show the possibility of tailoring the tube shape based on our results with two preconditions. Since travelling waves are generated in numerous applications, this leads to a fundamental requirement that the topological charge l should not be very large within the radial contents of the beam, i.e. |l|≤ kr| cos γ(r)|. In practice, paraxial optics are adopted in most cases, requiring that tan γ(r)≃sin γ(r). If we further focus on the ASVBs with the azimuthal components of k vector much smaller than the corresponding radial components, we found r ≫ |–l /ϕ′(r,θ)| and sin2γ(r) + [l/(kr)]2 ≃ sin2γ(r). With these preconditions implemented in Eqs. (4) and (5), the tubular caustic can be well approximated as:
Fig. 5. Synthesized vortex beams of different vortex orders with: (a) quartic, (b) logarithmic, (c) parabolic and (d) exponential tubular profiles and tailored parameters listed in Table 3. The blue dash dotted lines represent the pre-engineered target caustics. The caustic profiles extracted from the simulation and experiments shown in the third column are all defined as in [27].
Table 3. Target profiles ρ(z) = c(z) of different shapes
In addition, two exemplary ASVBs with l = 6 are pre-engineered and shown in Fig. 6, presenting one beam carrying an opening central tube profile ρ(z) = a + bz with a = 2µm, b = 0.058 [Fig. 6(a)] and the other with a tapered central tube profile ρ(z) = a – bz with a = 5µm, b = 0.058 [Fig. 6(b)]. By inserting Eq. (7) into the above target tube profiles, the engineered phase can be readily obtained as ϕhost (r)= –(|l|/a)r ± r2bk/(2a). The blue solid lines represent the pre-determined caustics ρ(z). Figure 6(a3) and Fig. 6(b3) also present the simulated and measured tube profiles. The discrepancies from the target tapered profile in Fig. 6(b) mainly originate from the limited active aperture on the SLM (8.64mm×8.64 mm). This is proved by simulating the same beam with SLMs of different areas, where the larger active area (12mm×12 mm) allows a better agreement with the predefined geometry [Fig. 6(b5)]. Despite that such a large SLM is unavailable in our experiment, this distortion can be corrected in tailored caustics with a smaller initial ring and a less steep taper (a = 4µm, b = 0.038) as shown in Fig. 6(c).
Fig. 6. Numerical simulations and experimental results of several particular examples with the tunable hollow core radius. (a) ρ = a + bz with a = 2µm, b = 0.058; (b) ρ = a – bz with a = 5µm, b = 0.058 and (c) ρ = a – bz with a = 4µm, b = 0.038. SLMs with a smaller (7mm×7 mm) and a larger (12mm×12 mm) area are adopted in the simulation in (b4) and (b5), respectively. The blue dash dotted lines represent the pre-engineered target caustics. The caustic profiles extracted from the simulation and experiments shown in the third column are all defined as in [27].
5. Conclusion
In this paper, we demonstrate that the set of analytical equations developed in our previous study on the caustics of the abruptly autofocusing vortex beams [27] have a wider universality to well reproduce the caustics of the axially symmetric vortex beams. Based on a couple of vortex beams synthesized from different host beams, the universality is proved by the excellent agreements of our theory with the numerical and experimental results. Features of the vortex caustics can also be well addressed with our theoretical methods, including the components of the global caustics and the deviation of the vortex caustics from the host caustics. Besides, we have also shown that it is possible to pre-engineer the vortex caustics based on our theory in the paraxial regime where the polarization components can be decoupled. Interesting opportunities also arise for extending our work to the nonparaxial regime. We expect that these results will promote the development of numerous applications, such as material processing, microscopy, particle micromanipulation and synthesis of novel electro-magnetic wavepackets [36,37]. Also, the structural stability of the different solutions in the nonlinear regime can represent interesting future work.
Funding
National Natural Science Foundation of China (61605142, 61827821); Tianjin Research Program of Application Foundation and Advanced Technology of China (17JCJQJC43500); Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences (Open Fund of the State Key Laboratory of High Field Laser Physics); European Research Council (682032-PULSAR); Agence Nationale de la Recherche (ANR-15-IDEX-0003, ANR-17-EURE-0002).
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
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