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Abstract
Spatial phase modulation has become an important method for the design of new self-accelerating light beams. Based on the transverse-longitudinal mapping of Bessel beam, we propose a method of pure phase modulation to directly convert a zero-order Bessel beam into a self-accelerating beam, of which the propagation trajectories can be flexibly predesigned. We experimentally demonstrate three typical types of curves that the modulated Bessel beam propagates along, and the parabolic, spiral, and teleporting self-accelarating beams are realized. The experimental results match the expected trajectory well. This method is simple to operate, and imposes fewer restrictions on the beam trajectory.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the theoretical prediction of the Airy beams by Berry and Balazs in 1979 [1], self-accelerating beams have attracted a great deal of research interests [2,3]. Due to its unique non-diffraction and self-healing properties, researchers have discovered a deal of novel and diverse applications, including particle manipulation and propelling [4], curved plasma generation [5], single-molecule imaging [6], bending surface plasmons and electrons [7], light-sheet microscopy [8], etc. A great deal of research has been done for the propagation characteristics of Airy beams. Such as acceleration [9] or polarization [10,11] control of Airy beams, the characteristics of propagating in nonlinear medium [12,13], optical Bloch oscillations of Airy beams [14], etc. Similar to the study of Airy beams, many new solutions for self-accelerating beams have been proposed. Such as beams bending into large angles along circular, elliptical, or parabolic trajectories but still retaining nondiffracting and self-healing capabilities [15–20]. The above studies have laid the foundation for the study of self-accelerated beams.
In contrast to the beams mentioned above, Bessel beams (BBs) share diffraction-free properties while maintaining an axisymmetric shape [21], and they are widely used in cylindrically symmetric optical systems and related applications, such as particle capture [22], information encoding [23], and optical imaging [24]. Therefore, the realization of beams which combine the characteristics, including symmetric or approximately symmetric distribution of light intensity, non-diffractive, self-accelerating, has attracted the research interests in recent years. Several works have shown that BBs can be constructed with novel propagation properties by spacial light modulation, for instance, controlling the hollow core and intensity during propagation [25], tailoring the axial intensity distribution of a light field by using a dielectric metalens [26], and realizing the analog of optical activity in free space [27]. For the trajectory control of BBs, researchers have mostly focused on the design of the initial optical field, such as the generation of serpentine or spiral Bessel-like beams by constructing the phase diagram [28–31], and the generation of a periodic self-accelerated beam by amplitude modulation in Fourier space [32]. By modulating the optical wavefront with a pre-specified continuous focal curve, self-accelerating Bessel-like beams are generated [33,34], which can maintain a diffraction-free central aperture and constant orbital angular momentum in bending propagation [35,36].
In these previous works, the self-accelerating Bessel-like beams are generated by modulating the traditional laser beam. With the development of the spatially modulated optical beam, there would be a challenge that whether and how to control the new-type light beams when they have already been generated. Here, especially for the BBs, a natural question arises: Is it possible to control the propagation of an existing BB with a simple modulation?
In this paper, we propose a new method to regulate the propagation trajectory of a zero-order BB merely by modulating its phase. Based on the transverse-longitudinal mapping of BB, a radius-dependent titled phase is constructed to control the BB propagating along a predesigned curve. We experimentally demonstrate three typical types of propagation trajectories of BBs: two-dimensional, three-dimensional, and step curves. The experimental results match the expected trajectory well. This method is simple to operate, and imposes fewer restrictions on the beam trajectory.
2. Theory
For a zero-order BB J0(kr,r), the propagation can be analyzed by the main lobe formed via the focusing of conical wave. As shown in Fig. 1(a), two sets of rays with transverse wave vectors kr and -kr are located on two conical surfaces (blue arrows) of the BB. At a circle of radius r in a cross section of the beam (denoted as z = 0), the rays of the wave vector cone -kr converge to a point at the propagation distance z. While the rays of kr go far away from the propagation axis, and contribute less to the main lobe of the beam. The light in an arbitrary circle r will be concentrated at the corresponding distance in z-axis. There is a one-to-one correspondence between the radius r and the distance z:
where, k0 = 2π/λ denotes the wavenumber of the beam in free space, and kr is the transverse component of k0. There is a transverse to longitudinal mapping for the zero-order BB that the change of complex amplitude on a ring of radius r will be accordingly mapped to the on-axis field at distance z. By attaching an addition phase gradient on the circle r, the field on the ring is tilted, and converges to an off-axis point at distance z, as shown in Fig. 1(b). For the entire cross-section of the beam, if different phase gradients Q(r) are attached at different radiuses r, the corresponding fields would converge to different off-axis spots, of which the transverse shifts and axial positions are determined by Q(r) and r, respectively. As a whole, these off-axis spots, of which the transverse shifts are independent ideally, are strung into a trajectory curve. We assume that the additional phase is represented by the oblique phase term Q(r) exp{-i[KX(r)x + KY(r)y]}, and the trajectory can be represented by the polar coordinates [Δ(z), ψ(z)], where Δ(z) and ψ(z) represent the value and angle of the displacement at z.
Fig. 1. Schematic of trajectory control of a zero-order Bessel beam. (a) Transverse to longitudinal mapping; (b) wave vector rotation induced by a phase gradient; (c) relationships between wave vector and obliquity factors variables in real space and k-space.
Figure 1(c) illustrates the relationship of these variables in real space and k-space. In the wave vector cone, since the axial component kz describes the phase velocity of BB, the propagation direction is indicated with wave vector kz for simplicity. Under the modulation of phase gradient Q(r), wave vector kz is rotated to kz′, and meets kz′=(k02-kr2)1/2. In k-space shown in the bottom of Fig. 1(c), the obliquity factors KX and KY of phase Q can be expressed as
It is important to note that in the above equation, the function argument of Δ(z) and ψ(z) are propagation distance z. It is necessary to transform Δ(z) and ψ(z) into radius-dependent expressions by using Eq. (1). By employing the polar coordinates (r,φ) in the modulation plane, the phase Q can be written as
Theoretically, we can calculate the modulation phase by the parameters (Δ, ψ) from the pre-designed propagation trajectory.
It seems that the proposed trajectory control method can realize the propagation along an arbitrary curve. However, there is a limitation for the transverse-to-longitudinal mapping of Bessel beam: the titling angle of the wave vector cannot be too large. As a result, the displacement of the beam cannot be too large. But under the paraxial condition, the proposed method supports the most types of trajectories.
3. Results and discussion
3.1 Experimental setup
To observe the BBs with different trajectories, we use the experimental setup as shown in Fig. 2. The linearly polarized He-Ne laser beam with wavelength of 633 nm is expanded by a pair of lenses L1 and L2, and then incident on a computer-generated hologram (CGH) loaded on a phase-type spatial light modulator (SLM1) to generate a zero-order BB [37]. After reflecting from the SLM1, the beam passes through SLM2 loading with a pre-set oblique phase Q(r,ψ) given in Eq. (5). Considering that the beam reflected from SLM1 has several diffraction orders, we align the SLM2 with the +1st-order beam, which forms the desited BB. Then, the beam goes through the 4f system consisting of two lenses and an aperture filter, which allows only +1st-order component to pass through. In order to observe the propagation processes of the field, the cross-sectional intensity of the output beam is recorded using a CCD step by step along the propagation axis. It is important to point out that there is a transverse displacement error in every recording step, that would cause the measurement error of the propagation trajectory. Thus, the transverse positions of the recorded beam should be calibrated step by step, by using a standard zero-order BB (setting the phase Q = const).
Fig. 2. Experimental setup. Laser: He-Ne laser of wavelength 633 nm; L1, L2: Lenses with focal lengths of 5 cm and 10 cm; SLM1, SLM2: Spatial light modulator; BS1, BS2: Beam splitters; L3, L4: 4f system of focal length 15 cm; CCD: Camera with 2.2 µm pixel; MDP: Micro-displacement stage with accuracy of 10 µm.
3.2 Parabolic self-accelerating beam
We firstly choose a typical two-dimensional curve, parabola, as the propagation trajectory. The parameter expression of this curve is expressed as
By setting the parameters kr = 5×104m−1, a = 5×10−3m−1, we demonstrate the zero-order Bessel-like beam with a parabolic trajectory as shown in Fig. 3, where 3(a) shows the modulation phase, 3(b) gives the schematic of the trajectory, and 3(c) presents the simulated propagation process in the x-z plane with the pre-designed trajectory marked by the white dashed line. It is evident that the simulation results are in good agreement with the pre-designed trajectory. Figures 3(d)-(h) display the simulation (top) and experimental (bottom) results of the cross-sectional intensity of the beam at different propagating distances, where the white crosses denote the ordinate origins, and the green points remark the centers of the main lobe of the Bessel-like beam. The offset (δ) of the main lobe at different distances reveals that the beam exhibits a self-accelerating propagation with a parabolic trajectory in x-direction.
Fig. 3. Simulation and experiment results of a parabolic Bessel-like beam. (a) Modulation phase; (b) schematic of the trajectory; (c) side view of the simulated propagation process; (d)-(h) simulated (top) and experimental (bottom) transverse beam profiles at different distances.
Figure 4 shows the comparison of the experimentally measured beam trajectory with the pre-designed curve, which fits together perfectly. From the result, it can be found that the experimentally measured displacement has an absolute error less than 5 µm, which might be induced by the mechanical shift error of the MDP. The main lobe of the beam in Fig. 3 is measured about 60 µm in diameter and remains diffraction-free during propagation (compared to a Gaussian beam of the same size), which can also be verified from the side view of the propagation process [Fig. 3(c)]. It is also noted that the ideal diffraction-free distance of the zero-order BB in our experiments is about 50 cm. However, after being transformed into the self-accelerating beam, the diffraction-free distance drops to about 20 cm. This is because that as the wave vector tilted, it locates no longer on a symmetric cone. The banlance of the plane wave components is broken, no more maintaining the stable propagation. The beam would gradually lose its acceleration and diffraction-free properties during long-distance propagation.
Fig. 4. Experimental and theoretical results of the displacement of the parabolic self-accelerating Bessel-like beam vs. propagation distance.
3.3 Spiral beam
Next, we demonstrate the modulated BB propagate along a three-dimensional curve, spiral line, of which the parameter expression can be written as
where R is the radius of the spiral trajectory, L is the period of the spiral line along the z-axis. In the experiment, we set R = 40 µm, L = 16cm. The results are shown in Fig. 5, where 5(a) presents the corresponding modulating phase pattern calculated by Eq. (5), 5(b) gives the schematic of the spiral trajectory, 5(c)-(g) demonstrate the simulated (top) and experimental (bottom) results of the cross-sectional intensity distributions at different propagation distances, with the pre-designed trajectory marked by the white dashed lines. The position of the main lobe at different propagation distances reveals that the beam rotates by one cycle in 16 cm. This is consistent with the pre-designed trajectory.Fig. 5. Simulated and experiment of the spiral Bessel-like beam. (a) Modulation phase; (b) schematic of the trajectory; (c)-(g) simulated (top) and measured (bottom) transverse beam profiles at different distances.
We measure the offsets along the x- and y-axes (δx, δy) of the main lobe to quantitatively analyze the experimental result. Figure 6 displays the comparison of the experimentally measured trajectory with the pre-designed curves and the simulation results. It can be found that as the beam propagates, the measured offsets along the x- and y-axes are basically consistent with the cosinoidal and sinusoidal curves, respectively, of which the maximum differential is about 80 µm. Notably, the experimental result only matches well with the initial section of the pre-designed curve, while having a large error after a certain propagation distance. In contrast to the Bessel-like beam with the parabolic trajectory in the previous section, the error is obviously not induced by the diffraction effect in a long-distance propagation. Compared with the simulation results (dashed lines) which accord closely with the experimental one, it can be concluded that the error does not originate from the experimental measurement, but caused by the theory itself. For the spiral-like propagation, the main lobe of the beam changes its moving direction incessantly, and is more easily to collapse, which can be seen clearly from the beam profiles after a certain propagation distance [Figs. 5(f) and 5(g)]. If the displacement Δ is decreased or the period L is increased, the spiral-like propagation would be more smoother. As shown in Fig. S1 in Supplement 1, a spiral Bessel-like beam with a smaller radius (20 µm) can maintain its spiral-like propagation with two cycles at least, and the main lobe has little distortion.
Fig. 6. Experimental, theoretical, and simulated results of the displacement of the spiral Bessel-like beam vs. propagation distance.
3.4 Teleporting self-accelerating beam
It is intuitively plausible that the propagation trajectories should be curves of continuous and derivable functions, as demonstrated above. But actually, the transverse-longitudinal mapping of BB enables the adjacent points along the propagation axis modulated independently. Thus, we can set the propagation trajectory as a step curve, which is spliced with two different trajectories. To demonstrate a general case, we make the two trajectories accelerate in different directions with different accelerated speeds. The expression of the step curve is expressed as
Fig. 7. Simulated and experiment of teleporting self-accelerating Bessel-like beam. (a) Modulation phase; (b) schematic of the trajectory; (c)-(l) simulated (top) and experimentally generated (bottom) transverse beam profiles at different distances.
We use the offset of the main lobe along the x- and y-axes (δx, δy) for the quantitative analyzation. Figure 8 shows the pre-designed curve and the experimental results. It can be seen that the offset along x- and y-axes has obvious step changes at z = 7cm. The measured offsets match well with the pre-designed trajectory. This proves that the proposed method of control the propagation trajectory of BB can perfectly support the noncontinuous and non-derivable curves.
Fig. 8. Experimental and simulation results of the displacement of the teleporting self-accelerating Bessel-like beam vs. propagation distance.
3.5 Self-acceleration of higher-order Bessel beam
The transverse-to-longitudinal mapping also applies to the higher order BBs, because it has the similar conical wave vector as the zero-order BB. As a result, the proposed method of trajectory control is still valid for the higher order BBs. For example, a second-order BB would perform the self-accelerating along the parabolic curve when modulated by the phase of Fig. 3(a), as shown in Fig. S2 in Supplement 1. While a first-order BB attached the phase of Fig. 5(a) propagates along a spiral line (see Fig. S3).
4. Conclusion
In summary, we have demonstrated a method to convert a zero-order BB into a self-accelerating Bessel-like beam with flexible trajectories by a pure-phase modulation. Simulation and experimental results show that the beam can propagate along the pre-designed parabolic, spiral, and teleporting self-accelerating trajectories merely by designing the modulation phase. Compared with the previous research, this method is simple to operate, and has fewer restrictions on the beam trajectory. In addition, as a general form of Bessel-like beams, the beams obtained in this paper can be used in superposition to produce other Bessel-type self-accelerating beams or other vector beams.
Funding
National Key Research and Development Program of China (2017YFA0303800); National Natural Science Foundation of China (11634010, 12074312, 12074313, 12174309); Fundamental Research Funds for the Central Universities (3102019JC008).
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.
Supplemental document
See Supplement 1 for supporting content.
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