Since their debut in 2005^[1], optical helico-conical beams (HCBs) have attracted considerable attention throughout the structured light community. The HCBs can be generated from a product of helical and conical phase functions, which leads to a spiral intensity distribution at the focal plane when experiencing a Fourier transform^[1,2]. Unlike the conventional vortex beams, such as Laguerre–Gaussian beams^[3] and Bessel–Gaussian beams^[4], the nonseparable radial and angular phase components would bring about chirality features and unique propagation dynamics, which make the HCBs potential candidates in the fields of particle manipulation^[5–8], nanostructure fabrication^[9–11], and optical metrology^[12–14].

Recently, the generation and fabrication of more flexible HCBs were reported^[7,15–18]. Cheng et al. proposed a new kind of opening-controllable HCBs by modifying the radial exponent^[17]. Hu et al. developed multi-twisted beams consisting of multiple sub-HCBs with different azimuthal shift factors and demonstrated microsphere guidance experiments^[7]. Segura et al. developed a new family of helico-conical vector beams^[15]. Moreover, studies pertaining to the unusual propagation characteristics associated with the helico-conical phase also emerged progressively^[2,19–21]. Alonzo first revealed the 3D intensity distribution of HCBs in 2007^[2]. Then their self-healing properties were reported from both theoretical and experimental perspectives^[22,23]. The focusing and propagation characteristics of Airy beams and Bessel–Gaussian beams with a mixed helicon-conical phase have also been investigated in detail in the past few years^[20,21]. Most of the existing studies, however, are based on the ordinary helico-conical phase. Taking a broader perspective, some alternative optical configurations are observed as well due to their flexibility in engineering, such as arbitrary curvilinear beams^[24] and cross-phase structures^[25,26]. In addition, as one of the current popular research areas, most of the optical arrays (OAs) are focusing on conventional optical vortices^[27–29]. The existing research about HCBs has not yet conducted further reconstruction despite their research concern about various arrays^[7,16]. Consequently, the above issue encourages us to explore more complex HCB patterns to enrich the principles and applications of structured light fields.

In this work, we have demonstrated one kind of complex interference-pattern HCBs from both theoretical and experimental perspectives. The HCBs with elaborate fringe structures are generated based on tuning of complex amplitude and optical interference. Furthermore, we modify the expression of azimuthal term in the helico-conical phase in order to manipulate the field distributions; several typical cases are demonstrated. Through further comprehensive combinations, more sophisticated HCB patterns could be achieved to meet the requirements of various applications. The work presented in this paper could open a new pathway for the flexible manipulation of optical fields and is anticipated to find potential applications in related areas.

An HCB can be generated with a nonseparable product of helical and conical phase functions, which takes the following form^[1,7]: Φ(r,θ)=Φa(θ)Φr(r)=lθ(K−r/r0),(1)where (r, θ) represents polar coordinates, Φ is the phase distribution of the light field in the initial plane Ψ(r,θ), and Φa(θ)=lθ and Φr(r) refer to azimuthal and radial phase terms, respectively. The parameter l is the topological charge (TC), and r0 is a normalization factor of the radial coordinate, corresponding to the width of the aperture or radius of an incident beam^[2], K is a constant taking either 0 or 1. The phase expression in Eq. (1) indicates that the arising HCBs at the focal plane will produce spiral intensity patterns when experiencing a Fourier transform. A complete spiral pattern can be acquired when K=1, while only a truncated pattern can be obtained when K=0^[2].

In previous work, however, the curve trajectories of HCBs seem to be always narrow and sharp, as shown in Fig. 1(a). It restrains us from further constructing fine structures based on the existing patterns. A potential solution is to broaden the spiral trajectories of the HCBs so that the space can be created to adopt more details. It leads us readily to the two-dimensional convolution theorem in Fourier optics^[30], F[Ψ(r,θ)·h(r,θ)]=E(ρ,φ)*H(ρ,φ),(2)where (ρ, φ) represents polar coordinates at the observation plane, E(ρ,φ)=F(Ψ(r,θ)) is the complex amplitude distribution of HCB, h(r,θ) is a truncation function, H(ρ,φ) is the corresponding Fourier transform, and the symbol [*] represents the convolution operation. Equation (2) inspires us to take the h(r,θ) as a Gaussian-like function^[31], which is given by hG(r,θ)=exp(−r2/w02),(3)where w0 is the half-width of the Gaussian truncation. Therefore, the Fourier transform of hG(r,θ) is still a Gaussian-like function, and the convolution operation leads to the effect that each point on the observation plane makes a Gaussian-type extension, making the curve trajectory look stronger, as shown in Fig. 1(b). Additionally, w0 exactly corresponds to the half-width of the Gaussian beam in an experimental setting. Thus, the width of the curve trajectories of HCBs could be controlled by adjusting the size of the Gaussian truncation in the initial plane, as well as the radius of the incident Gaussian spot.

One of the most common strategies for fabricating fine structures is to construct optical vortex arrays^[32–36], which implies that the optical singularities are arranged according to either the distribution of the optical field or other predefined trajectories. Similarly, we superimpose HCBs of different TCs to produce optical interference fringes on the curve trajectories of the light field, which can be expressed as $$\begin{gathered} Ψ(r,θ)∝exp{i[l1θ(K−r/r0,1)]} \\ +exp{i[l2θ(K−r/r0,2)]}, \end{gathered}$$(4)where (l1, r0,1) and (l2, r0,2) are the TCs and normalized radial factors for the first and second light-field components, respectively. Notably, due to the nonseparability of the phase definition, the parameters l and r0 simultaneously affect the scale of the HCBs. In order to obtain patterns with clearly visible interference fringes, the interference components are supposed to be stretched into the same size by keeping the ratio l/r0^[1]. In terms of optical phase distribution, the number of fringes should be determined by the difference in TCs, i.e., Δl=l1−l2. Due to the inherent chirality properties of the HCBs, there may exist potential differences between the interference of the beams with the same-sign TC and their counterparts with opposite-sign TCs. In summary, by adjusting the values of l and r0, the interference HCBs with a striped structure can be obtained, and the number and style of the fringes are tunable as well.

Then let us look backward to Eq. (1). It should be noted that the azimuthal phase Φa(θ) is linearly related to the azimuth angle. In fact, we can further change the configuration of the HCBs by rewriting the relationship between the azimuthal angle and phase in the initial plane. In order to explore more reconfigured patterns, here we study three cases carrying nonlinear phases, including exponential type, sine type, and chirped-oscillation type cases, as shown in Fig. 2. The phase distribution of the exponential type case in the initial plane is expressed as ΦE(r,θ)=2πl(θ2π)n(K−r/r0),(5)where n is the exponential factor. Figure 2(a) plots the relationship between azimuthal phase Φa and θ when n equals several specific values. The original linear relationship is also plotted as a reference and l is set to 1 for simplicity in the successive section related to Fig. 2. As n decreases from 1 to 0, the opening of the HCB contracts until it closes, and it finally degenerates into a ring-like pattern. This phenomenon has been reported in previous work^[18]. The phase distribution for sine type case in the initial plane is expressed as ΦS(r,θ)=lsin(θ2·t)(K−r/r0),(6)where t is the scaling factor of the sine function. Similarly, Fig. 2(b) shows the relationship between azimuthal phase Φa and θ for different t. As t increases from 0.5 to 1, the opening of the HCBs also decreases but does not completely close alongside the strengthening in symmetry, which can be inferred from the evolutionary tendency in Fig. 2(b). The phase distribution for the case of chirped-oscillation type in the initial plane is expressed as ΦC(r,θ)=l[θ+θ2π·c·cos(mθ)](K−r/r0),(7)where c is the chirp amplitude factor and m is the periodical factor. As shown in Figs. 2(c1) and 2(c2), larger c leads to more intense phase oscillations. The factor m determines the number of cycles, for it is set to 3 in Fig. 2(c1) and 5 in Fig. 2(c2). In the case of the chirped-oscillation type, the curve trajectories of HCBs will be folded into m segments, and with a larger value of c, its effect would become more obvious. In summary, the HCB pattern could be reconfigured by means of modulating the azimuthal phase term, especially as for the curve trajectories, opening size, and degree of symmetry, etc.

In the above sections, we have discussed about the reconfiguration of a single HCB. In fact, we are also able to change the position of the patterns through geometric transformation operations such as translation, rotation, and flipping^[7,16]. According to the properties of the Fourier transform, rotation and flipping could be easily realized. The translation in the observation plane needs to be accomplished by introducing a phase shift in the initial plane, which is given by $$\begin{gathered} Et[ρ,ϕ]=F[Ψt(r,θ)] \\ =F{Ψo(r,θ)·exp[i2π(SXX+SYY)]}, \end{gathered}$$(8)where X and Y refer to Cartesian coordinates in the initial plane. The translation factors SX and SY are defined as the displacement in the focal plane divided by the wavelength and focal length of the lens in the X and Y directions, respectively^[7]. Furthermore, one can also combine the same or different HCBs to further enrich the variety of patterns.

To verify the above theoretical analysis, numerical simulation about the pattern distributions of interference-pattern HCBs could be achieved based on the angular spectrum propagation algorithm. The experimental setup for the generation and observation of HCBs is schematically illustrated in Fig. 3. A Gaussian 1064 nm laser beam is launched through a half-wave plate (HWP) and a polarizer to make the optical polarization state match the spatial light modulator (SLM) for the maximal modulation efficiency. The beam is adjusted to have a suitable transversal size determined by the combination of L1 and L2, and then it passes through a reflection plane mirror to reach a beam splitter. One Gaussian-mode branch is modulated by phase hologram loaded onto a reflective phase-only liquid-crystal SLM. A 4-f system consisting of L3 and L4 filters out the first-order diffraction beams assisted by an iris on the Fourier plane. Lens L5, whose front focus coincides with the back focus of L4, performs as a Fourier lens. The desired HCB field distribution could be observed and captured by a CCD camera.

Let us first focus on ordinary interference-pattern HCBs consisting of the same-sign TCs (type-S for simplicity, the same below) and opposite-sign TCs (type-O for simplicity, the same below), as shown in Fig. 4. Unless specifically stated, K is set to be 1 and w0=1mm. Figures 4(a1)–4(a3) illustrate the phase superposition in the case of type-S, and the topological charge difference Δl=l1−l2 equals 2, 3, and 4, respectively. The corresponding intensity patterns are shown in Figs. 4(b1)–4(b3), while the number of fringes is labeled by red dots at the start point of the outmost interference fringes. One can see that the fringe number appears to be exactly the same as the value of Δl, which is similar to the traditional OAM mode interference patterns. Additionally, the interference fringes in the case of type-S tend to be vague, where a series of “breaking points” seem to be present in the curve. Similarly, the phase and intensity distributions of the HCB interference-pattern in the case of type-O are illustrated in Figs. 4(a4)–4(a6) and 4(b4)–4(b6), respectively. Here Δl is set to 4, 5, and 6, respectively, which exactly correspond to the number of fringes as well. Compared with the former, fringes in the case of type-O appear to be finer. The experimental results presented in Figs. 4(c1)–4(c6) agree well with the above simulation results.

It is necessary to give explicit reasons causing the distinction in interference fringe characteristics of type-S and type-O beams. For better comparison, the TC difference is set to 4 for both of the above cases, as shown in Fig. 5. In terms of the interference regarding the HCBs with the TCs of the same sign, the gradients of Φa of the components contributing to the optical interference are both aligned in the same direction, as shown in Figs. 5(a1) and 5(a2). Otherwise, the gradients would orient in opposite directions, as shown in Figs. 5(b1) and 5(b2). In terms of optical interference, the interference phase of the type-S beam results in the emergence of singularity points, while that of the type-O beam produces curvilinear phase splitting, as shown in Figs. 5(a3) and 5(b3), respectively.

In addition to the characteristics of fringes, more complex reconfigurable interference-pattern HCBs with special curve trajectories are demonstrated by adjusting the expression of azimuthal phase terms Φa. The simulation and experimental results are shown in Fig. 6. First we consider the case of exponential type. Figures 6(a1)–6(a4) and 6(c1)–6(c4) illustrate interference-pattern HCBs in the cases of type-S (l1=3, l2=6) and type-O (l1=1, l2=−3), respectively, and the factor n from the left to the right columns are set to be 0.9, 0.3, 0.1, and 0, respectively. The corresponding experimental results are shown in Figs. 6(b1)–6(b4) and 6(d1)–6(d4), respectively. One can see that as n decreases, the openings of HCBs become narrow till they completely close. It should be noted that the number of fringes would also decrease in this process. More interestingly, when the patterns finally degenerate into standard ring-like beams, the type-S case leaves no fringes at all, as shown in Fig. 6(a4). However, one fringe could still be found in the case of type-O, as shown in Fig. 6(c4). The above characteristics could be explained as follows: the descent of n enhances similarity between initial HCB components, which eliminates the discontinuity of the composite phase. Nonetheless, when n=0, the initial composite phase degenerates into a conical phase and a radial periodic-inverse for the type-S and type-O beams, respectively. The latter leads to the emergence of a ring-like fringe in the interference pattern, as shown in Figs. 6(c4) and 6(d4). When it comes to the sine-type case, let us take the case of type-S interference as an example (l1=3, l2=6). Figures 6(e1)–6(e4) and 6(f1)–6(f4) give the simulation and experimental results, and the factor t from the left to the right columns are set to be 0.5, 0.6, 0.8, and 1, respectively. One can see that as t increases, the openings of HCBs would become narrow but cannot completely close. Moreover, new fringes emerge as the tails of curve trajectories curl up inward. This pattern finally acts as a letter “C,” while t=1 due to the phase symmetry according to Fig. 2(b). Results of the chirped-oscillation type case are shown in Figs. 6(g1)–6(g4) and 6(h1)–6(h4), respectively, and type-O interference are taken as an example in this case (l1=1, l2=−3). Here the chirp-related factors are as follows: c=0.4, m=3 in Figs. 6(g1)–6(h1), c=0.8, m=3 in Figs. 6(g2)–6(h2), c=0.4, m=5 in Figs. 6(g3)–6(h3), and c=0.8, m=5 in Figs. 6(g4)–6(h4). It can be concluded that the curve trajectories of HCBs contain (m−1) knee points, due to the local perturbation introduced by the periodic oscillation term. In addition, with the increment of parameter c, the phase perturbation would become stronger according to Fig. 2(c). Logically, the curve trajectories turn out to be more obviously bending.

Last but not least, more complex HCB patterns could be obtained through a flexible combination of light fields described previously, as shown in Fig. 7. Figures 7(a1) and 7(b1) show a “cloud” pattern by translation and flipping operations. In addition, arrangements consisting of various transforms and components are also available, such as the “sea wave” pattern in Figs. 7(a3) and 7(b3), and the “butterfly” pattern in Figs. 7(a4) and 7(b4).

In summary, we have proposed and experimentally generated one kind of interference-pattern HCBs with reconfigurable characteristics in different degrees of freedom. We make use of optical interference to produce delicate fringes on the curve trajectories of the HCBs, and the difference between the interference of the same or the opposite-sign TCs has been investigated. Moreover, we have demonstrated three kinds of special patterns with nonlinear azimuthal phase terms and discussed the impact of the curve trajectory reconfiguration on optical intricate structures. Finally, various multiple HCBs have been achieved through geometric transformation operations. The HCBs with flexible light-field structures may find applications in the areas of optical metrology, multi-target particle manipulation, and nanostructure fabrication.