Caustic methods have been proposed for wavefront design to enable light beams propagating along curved trajectories, namely accelerating beams. Here we elaborate the complete construction, remarkable characteristics, and hidden constraints of these methods. It is found that accelerating beams based on the caustic design have not only a well-known curved intensity distribution but also a linear phase distribution along the caustic proportional to the curved length, as if light field indeed moved along the caustic. Moreover, with this characteristic, further light-ray analyses are implemented to illustrate the constraints of caustic design in different cases. We expect our work will clarify some confusion on the effectiveness and applicability of caustic methods, and thus facilitate the design of accelerating beams for various applications.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Structured light  with elaborate design of initial field distribution may possess some extraordinary properties, such as self-accelerating [2,3], nondiffracting , and even carrying orbital angular momentum , which reveals the abundance of a light beam in the space or space-time  domain. In particular, self-accelerating beams are a kind of light beams propagating along curved trajectories in free space [2,3,7–10], which have been studied intensively in recent years. This is not only because such beams are fascinating as they seemingly violate the axiom of light propagation along straight lines, but also they can be useful in a variety of applications using light beams for energy, substance or information delivery along arbitrary trajectories, including micro-machining , particle-manipulation [12,13], optical communications [14,15], imaging [16,17], and laser assisted guiding of some processes such as filamentation  and electric discharges .
Since the above-mentioned applications take advantage of light beams propagating along curved trajectories, it is critical to enable the possibility of tailoring accelerating beams with various propagating trajectories, beyond the parabolic type of the first discovered accelerating Airy beam in the paraxial regime [2,3]. To this end, caustic methods  that associate the desired propagating trajectory with a caustic, namely the envelope of a family of light rays , have been proposed. This method was first demonstrated with real-space design in the paraxial regime and achieved on-demand convex propagating trajectories for two-dimensional (2D) light fields . Later it has been widely extended, from real-space design to Fourier-space design  and even phase-space design , from paraxial regime to nonparaxial regime , from convex trajectories to nonconvex trajectories [26,27], from 2D light fields to 3D light fields [28,29], and from the phase-only modulation to the combined phase and amplitude modulation [30,31]. Such scheme does greatly extend the available propagating trajectories, and therefore there is often a claim in some papers that arbitrary propagating trajectories are obtainable, which is actually not the case. In this work, we elaborate the complete construction, remarkable characteristics, and hidden constraints of accelerating beams that are designed based on caustic methods, which may clarify some confusion about the applicability of these methods in the design of accelerating beams.
2. Construction and characteristics of accelerating beams from light-ray perspective
2.1 Light-ray model based on stationary phase approximation
In general, the propagation of a light beam in free space can be characterized based on the plane-wave (angular spectrum) decomposition. In the case of 2D light field discussed in this work, (i.e. light field distribution independent of y direction), the angular spectral integral can be expressed as32], suppose the phase distribution varies much faster than the amplitude distribution , the contribution to the above integral mainly comes from the stationary point of , namely the point satisfyingEq. (1) at the stationary point , the field distribution can be simplified asEq. (3) can be intuitively interpreted as the superposition of the light rays described by Eq. (2) as shown in Fig. 1(a), with a normal phase evolution due to the propagation of the light ray. However, it is noted that there is an additional phase shift in Eq. (3), with a value of or depending on the sign of , which would have some significant effects as discussed in the following.
2.2 Convex type of caustic beams
Based on the above light-ray analysis, a caustic as shown in Fig. 1(b), can be obtained by setting the second derivative of to be zero:Eqs. (2) and (4) can jointly result in a relation after eliminating the variable , and this curve satisfies:Fig. 1(b). Since the light field near the caustic is intensive, accelerating beams can be designed by associating the desired propagating trajectory with a caustic. Once the caustic is determined, the initial phase distribution can be obtained by solving Eqs. (2) and (5), a caustic method that has been widely used for the design of accelerating beams propagating along various trajectories . However, the above method requires a one-to-one correspondence between and in order to be solvable, which limits the available caustics to the convex type (including the concave one), namely light beams bending toward only one direction without bending back as shown in Fig. 1(b).
It is also interesting to find that if we further require the third derivative of to be zero:Fig. 1(c).
2.3 Phase distribution along caustic
Although enhanced intensity around a caustic has been recognized for some time  and recently employed for the construction of accelerating beams [20–31], to the best of our knowledge, the discussion on the phase distribution along the caustic is missing in the literature, which is actually an important and elegant characteristic as pointed out in this work. The phase of the light field on the caustic can be calculated based on the aforementioned analysis (Appendix A), which is found to be
To confirm such theoretical analysis, two kinds of common accelerating beams propagating along a parabola ( is a parameter) and a circle ( are parameters) are constructed based on the above caustic method, with the following designed angular spectra in the initial planeFig. 2. As expected, the main lobes of these beams propagate along the designed trajectories with linearly increased phase accumulation, as if simple phase accumulation due to curved propagation.
The above result will facilitate our understanding of the characteristics of accelerating beams in the perspective of two-ray interference, as illustrated in Fig. 3(a). On the right side of the caustic, there is no light ray passing by, which corresponds to an area with negligible intensity, while on the left side of the caustic, the light field at each point can be regarded as the interference of the two rays passing through this point and tangential to the caustic. The phase difference of these two rays at this point, e.g. point C in Fig. 3(a) will determine whether they interfere constructively or destructively, which can be directly expressed asEq. (3). For a light ray before touching the caustic, so it has a phase, while after it touches the caustic, so it has a phase, namely resulting in a phase difference. To justify the above formula, we have simulated the propagation of a circular beam as shown in Fig. 3(b). By adjusting the point C for corresponding to the white label 1, 2, 3, 4 in Fig. 3(b), respectively, one can see that the point locates exactly at the main lobe, the first side lobe, the fourth dark lobe, and the tenth side lobe as expected, which confirms the validity of treating accelerating beams as a two-ray interference based on Eq. (9). Moreover, it is noted that the main lobe corresponding to is not exactly on the caustic but having a small distance to it. As the point C approaches to point D on the caustic, the two rays no longer interfere completely constructively because of the phase difference.
2.4 Extending caustic beams to nonconvex type
As mentioned in Section 2.2, only the convex type of caustic beams can be constructed based on the above caustic method. In order to extend the caustic beams to the nonconvex type, a superposition caustic method has been proposed [26,27]. The idea is to construct several convex (including concave) caustics based on the original caustic method individually, and then superpose them to form an entire nonconvex trajectory, as illustrated in Fig. 4(a). The superposition of all the initial angular spectra for each segment can be generally expressed asFig. 4(a), so that the two segments can joint smoothly, which, however, is actually not the case as shown in Fig. 4(c). It is found that the two segments form a smooth nonconvex main lobe only if as shown in Figs. 4(b)-4(d), which is also attributed to the additional phase shift as indicated in Eq. (3). In this case, the interference at the joint point E(F) actually occurs between the light ray tangential to a point right before point E and the light ray tangential to a point right after point F, so there is a phase difference between them similar to Fig. 3(a). In order to ensure these two light rays to interfere in phase at the joint point, a phase shift should be introduced to the second segment during the superposition in Eq. (10), which is a key in the design of nonconvex caustic beams, otherwise a manual uncertain shift of the relative transverse position of the two segments is required as discussed in .
3. Restrictions on accelerating beams caustic design
Caustic methods indeed have greatly extended the obtainable propagating trajectories of constructed accelerating beams to most convex and even nonconvex curves, not limited to the original parabolic type of Airy beams, so that there is a claim in some previous works that arbitrary trajectories is obtainable based on this design without any restriction. However, this is not the case and we would like to illustrate the restrictions imposed by such caustic methods in the following.
3.1 Constraint in available convex caustic beams
Firstly, we discuss the limitations of constructing accelerating beams propagating along convex trajectories based on the caustic method. As mentioned in Section 2, the light-ray model is based on the stationary phase approximation, which means the above light-ray analysis including the construction of accelerating beams based on caustic methods works only if this approximation is justified. This approximation essentially requires enough spatial frequency ingredients within the integral of Eq. (1), so that the contributions from all these spatial frequencies except the stationary point are canceled out because of fast phase oscillation. However, there isn’t such a criterion related to the constructed caustic that can be used to judge whether the caustic will satisfy the approximation and thus obtainable, at present.
Note that the caustic beam can be described by the two-ray interference model as shown in Fig. 3(a). As the point C gradually deviates from the point D on the caustic, the phase difference between the two rays interfering at point C will increase and forms successively the main lobe, the first side lobe, the second side lobe, etc. The maximum phase difference corresponds to the point C where the two rays tangential to the start point and the end point of the caustic, respectively. As this value descends for a caustic, the number of side lobes in the accelerating beam will decrease accordingly and the main lobe will gradually break. A specific instance is presented in Fig. 5, where caustics of an arc with different length corresponding to different as demonstrated are constructed. As the value of descends from , the original well-defined main lobe gradually breaks and finally almost reduces to a symmetric intensity distribution when . Therefore, the constraint for an available curved trajectory of accelerating beams designed by caustic methods can be expressed as
3.2 Constraint in the available nonconvex caustic beams
As discussed in Section 2.4, accelerating beams propagating along nonconvex trajectories can be constructed based on the superposition caustic method, which superpose several convex caustics designed individually to form an entire nonconvex trajectory. However, after superposition, there will be interference among these convex caustics since the light rays forming different segments will intersect as well, as shown in Fig. 6(a), which could break up the entire nonconvex main lobe in certain circumstances.
This problem can be described from the light-ray perspective as illustrated in Fig. 6(a), where two segments of convex caustics are connected to form a nonconvex trajectory. One can see that the ray2 forming the caustic of the second segment unavoidably passes through the caustic of the first segment and therefore will interfere with the ray1 and its neighbouring rays. The interference between these two rays will be constructive and destructive alternatively depending on their phase difference , which may break the main lobe into several pieces. In order to ensure a smooth main lobe along the nonconvex caustic, the phase difference should be kept below corresponding to the first position of interfering destructively, and therefore the constraint can be expressed asFigs. 6(b) and 6(c). As the radius of the circular caustic increases, it is noted that the intensity of the main lobe is no longer smooth along the whole nonconvex trajectory, and the breakup position almost matches the point with a phase difference as denoted by the end points of the depicted caustic, which confirms the above constraint for an available smooth nonconvex trajectory of accelerating beams designed by superposition caustic methods.
Apart from the 2D cases discussed in this work, accelerating beams in 3D space are also available by associating the propagating trajectories with 3D caustics, which have been demonstrated recently [24,29]. In the 3D case, the problem of interference or intersection between light rays tangential to a caustic discussed above can be easily avoided and thus a smooth main lobe can be obtained without breaking up into pieces, while there also exists another inherent constraint as discussed in .
In summary, we firstly elaborate the Fourier-space light ray model and caustic method for the design of 2D accelerating beams propagating along both convex and nonconvex trajectories, respectively. It is found that caustic design enables the constructed light beam to possess not simply a curved intensity distribution as well-known, but also a linear phase distribution along the caustic, equivalent to the phase accumulation due to moving along a curved path, which is an interesting and useful characteristic for the light-ray analysis. From this light-ray perspective, we further analyze the restrictions of the above caustic design in constructing different types of accelerating beams and illustrate constraints in each case, including 2D convex and nonconvex caustic beams. We anticipate that the full analysis of the caustic method for the design of accelerating beams including its construction, characteristics, and constraints may clarify, inspire, and thus advance the development of such method, which will certainly benefit a variety of applications based on accelerating beams.
Appendix A Derivation of the phase distribution along the caustic
Based on Eqs. (2) and (5), we have
National Natural Science Foundation of China (NSFC) (11774437, U1701661, 61490715); Science and Technology Program of Guangzhou (201804010302); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121); Fundamental Research Funds for the Central Universities of China (SYSU: 17lgzd06).
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