Spiraling elliptic hollow beams with cross phase

We introduced a class of spiraling elliptic hollow beams with the cross phase. Due to the cross phase, the spiraling elliptic hollow beams exhibit three key characteristics, having the elliptic peak ring, carrying the orbital angular momentum (OAM), and performing rotations. We investigated both linear and nonlinear evolutions of the spiraling elliptic hollow beams, and found they can propagate stably, thanks to the cross phase. Especially, we obtained the breather states of spiraling elliptic hollow beams in nonlocally nonlinear medium, and could handily control the rotation by changing optical powers. We discussed both the OAM property and optical force property. By using the spiraling elliptic hollow beams, we can achieve a jointly multiple manipulation on particles at the same time. In one step, we can trap and simultaneously rotate the particles.


Introduction
Besides a linear momentum, angular momentum (AM) is also one of important characteristics of light [1], and has a variety of applications, such as optical manipulations [2,3], quantum information [4][5][6], optical communications [7] and imaging [8,9]. AM can be decomposed into two parts: one is spin angular momentum (SAM) related to the polarization state of the field, and the other is orbital angular momentum (OAM) caused by the helical phase [1]. The SAM can result in the rotation of a particle around its own axis [10], whereas the OAM can cause the rotation of the particles around the optical axis [11]. The two AM components may not be independent of each other, and spin-orbit interaction occurs under certain situations. Spin-to-orbit AM conversion could occur for a circularly polarized Gaussian beam when tightly focused [12] and via light intensity gradient [13].
The optical beams carrying the OAM are usually associated with optical vortices and related ring-shaped beams with the phase-singularity described by exp(inϕ) in polar coordinate system, such as the Laguerre-Gauss beams [14] and the Bessel beams [15], as well as the other hollow beams [16]. In fact, there is another kind of optical beam carrying the OAM [17], which was called as the 'astigmatic beam' [18]. Such a beam is obtained by making an elliptical Gaussian beam with a plane phase pass through a tilted cylindrical lens [17]. Therefore, the 'astigmatic beam' is vortex-free and without phase-singularity. The physical nature of such a beam is that the beam has the cross-phase term expressed by exp(iΘxy) in rectangular coordinate system or exp[iΘr 2 sin(2ϕ)/2] in polar coordinate system [19]. Soon afterwards, it was found that the cross-phase induced OAM can generate an anisotropic diffraction [20]. At the special valve of the OAM, that is P 0 (b 2 − c 2 ) 2 /4b 2 c 2 , an elliptic beam of major semiaxis b, minor semiaxis c and optical power P 0 can keep its initial elliptic degree b/c changeless during linear propagations. The special OAM is named the critical OAM [20]. Besides, the cross phase can also lead to the rotation of the elliptic beams. Then the elliptical Gaussian beam with the cross phase is also called the spiraling elliptical beam [21]. In the nonlinear media, the cross phase can make the elliptical Gaussian beams propagate stably as the elliptic solitons even though the saturable nonlinearity [22] or the nonlocal nonlinearity of media is isotropic [19,23].
Compared with the optical vortices with phase-singularity, the spiraling elliptical beam exhibits more degrees of freedom in adjusting and controlling the light OAM. The OAM of optical vortices is only determined by the topological charge. While, the OAM of the spiraling elliptical beam is proportional to [21,23], where the cross phase coefficient Θ, the major semiaxis b and the minor semiaxis c can all be continuously tuned in experiments. In this paper, based on the spiraling elliptic beam with the cross phase, we can construct the hollow beams with faint intensities at the center. Such optical beams with low central intensity have attracted much attention due to their wide and attractive applications in the fields of modern optics, atomic optics, and plasmas [24][25][26][27]. Specially, due to the flexibility in tuning the OAM of spiraling elliptical beams, both the OAM and the propagation properties can be controlled conveniently for spiraling elliptic hollow beams. It should be noted that the spiraling elliptic hollow beams resemble the elliptic Gaussian optical vortices obtained in [28], but only is similar in intensity shapes. The OAM property as well as the linear and the nonlinear propagations are quite different for the two optical patterns. Furthermore, we found that the spiraling elliptic hollow beams can evolve as breathers in nonlocally nonlinear media, which keep the elliptic hollow shape changeless, but only oscillate in beam width due to the competition between the diffraction and the self-focusing effects.

Optical OAM and force properties
We consider the elliptic beam in the form of where A is the amplitude, b and c are the major and the minor semiaxis if d = 0. It should be noted that equation (1) as well as the following calculations are all in the dimensionless system after the variable where XYZ are the coordinates in laboratory system, and Z R = kw 2 0 is the diffraction length of paraxial beam. Therefore, z = 1 in the paper in fact represents one diffraction length. When d ̸ = 0, the maximal intensity is away from the axis, which is shown in figure 1. Although in laboratory system dw 0 is the offset distance between the intensity maximum and the z axis, it will not cause confusions if we refer to d as the offset distance in the dimensionless system. Θ is the cross phase coefficient, and the cross phase exp(iΘxy) contributes the OAM to the elliptic beams. The OAM is proportional to Θ, and also depends on the offset distance d, the two semi-axis b and c (or the ellipticity b/c, equivalently). We can numerically calculate the OAM (per optical power) via the following integration being the error function. It is noted that the OAM M will be equal to zero if b = c. Figure 1 shows the intensity (a)-(c) and phase (d)-(f) distribution for different offset distances d. As d increases, the bright ring tends to thinner. In shapes, the patterns shown here seem the elliptic Gaussian vortices found in [28]. However, the most important difference between the two kinds of optical patterns lies in their phase structure. Due to the phase structure, the OAM properties of the two patterns are also different. The Poynting vector component, revealing the optical energy flow, (1) is also drawn, which is along the normal direction of the equiphase surface. It is found that the direction of the optical energy flow does not depend on the offset distances d, but only relates to the ellipticity, and makes an angle of arctan(b/c) with respect to x axis. Then, the elliptic beam will reshape and rotate. The rotation direction is determined by the sign of Θ(b − c). The spiraling elliptic hollow beam will rotate clockwise and anticlockwise when Θ(b − c) is negative and positive, respectively. On this basis, we can propose a method of the phase-controlled rotations for optical beams.
In the presence of an optical field, a particle experiences the gradient force [29] where κ is a dimensionless quantity related to the effective polarizability of the particle. In general, a particle displays a positive polarizability (PP) when its refractive index exceeds that of the background medium, while, conversely, negative polarizability (NP) is associated with the opposite scenario. Under the action of gradient force, PP particles are attracted towards the bright ring of the elliptic beam, whereas their NP counterparts are repelled. In the following calculations, we assume that κ = 1 for simplicity, and the gradient force will be reversed in the case of κ = −1. From equation (2), we calculated the maximal force components, where . As the offset distance increases, the bring ring will become thinner. The force ratio F x /F y is inversely proportional to the ellipticity. While, the radii ratio H x /H y is proportional to the square root of the ellipticity. Besides, the OAM can increase with the ellipticity. Particularly, when b = c (i.e. the ellipticity b/c = 1), the OAM will be zero. The optical beam with the OAM can be used as optical tweezers to manipulate particles [1], bacterium [30], human red blood cells [31] and so on. Particularly, in manipulating the rod-shaped objects, this kind of elliptic hollow beam has an advantage over the Gaussian one.

Linear and nonlinear evolution properties
To fully demonstrate the role of the cross phase, we simulate the propagation of the spiraling elliptic hollow beam (1) in free space, which is governed by the paraxial wave equation [32] i∂φ/∂z + (1/2)∇ 2 ⊥ φ = 0 in the dimensionless form. We conduct the numerical simulation according the method raised in [33]. The process consists of three major steps: (i) transform input beam φ| z=0 to the angular spectrum domain and obtaiñ φ(k x , k y ) z=0 ; (ii) add a phase exp − i k 2 x +k 2 y 2 z + iz on the angular spectrumφ(k x , k y ) z=0 and obtaiñ φ(k x , k y , z); and (iii) inversely transform the angular spectrumφ(k x , k y , z) to the spatial domain and obtain φ(x, y, z). The numerical simulation of spiraling elliptic hollow beams is given in figure 2. The elliptic beams rotate during propagation due to the cross phase induced OAM. However, the OAM does not have much effect on the rotational speed overall, which can be confirmed by comparing the beam evolution for different Θ in figure 2. We can explain this in the following. On the one hand, the OAM induces the rotation, then larger OAM will make the beam rotate faster if the beam size is fixed. On the other hand, the OAM enhances the diffraction, and the beam's expanding will slow down the rotation [34].
The beams in figures 2(c) and (f) have been in far-field, and their distributions are inverted with the input beams in figures 2(a) and (d). We can make some explanations on this point in the following. It is known that for the input of any profile, the output after a linear propagation can be expressed by the Fresnel diffraction integral [32] φ(x, y, z) where φ(x, y, 0) is the input beam at z = 0. In the limit of Fresnel diffraction for large distances, in order to obtain the far field, we can take an approximation in (5) and ignore the term (x ′2 + y ′2 )/2z. Therefore, the approximation is valid if the beam width w 0 (the magnitude of x ′2 + y ′2 ) and the propagation distance z meets the following condition In this limit The integral over x ′ and y ′ produces the Fourier transform of φ(x ′ , y ′ , 0), denoted byφ, then equation (8) becomes Therefore, the far-field distribution of an input beam can be expressed by its Fourier transform, which can be confirmed by figure 3. We can conclude that there are two main differences in propagations of the elliptic hollow beam compared with the Gaussian one. The first difference results from the cross phase exp(iΘxy). When Θ = 0, a Gaussian beam will diffract more quickly in the minor-axis direction than in the major-axis direction. Then, the elliptic Gaussian beam will evolve to a circular shape, and eventually become an inverted elliptic shape. It should be noted that no rotation takes place for Gaussian beams in the case of Θ = 0. If Θ ̸ = 0, the elliptic beam will carry the OAM, then can rotate during propagations. Besides, Θ can have a significant influence on the beam's ellipticity [20]. The second difference results from the offset distance d. We have obtained the radii of the peak ring along x− and y− axis for the elliptic hollow beam, which closely depend on d given by equations (3) and (4). As the offset distance increases, the bring ring will become thinner, and the beam will diffract more quickly during propagations.
The beams' diffraction can be balanced by the self-focusing effect in nonlinear media. As an example, we assume that the medium is of nonlocal nonlinearity, the nonlinear refractive index of which can be expressed by a convolution ∆n = R |φ| 2 . We employed the split-step Fourier method [35] to simulate the nonlinear evolution of the spiraling elliptic hollow beam governed by the nonlocal nonlinear Schrödinger equation i∂φ/∂z + (1/2)∇ 2 ⊥ φ + ∆nφ = 0 [36,37]. The well-known breathing phenomenon for optical breathers in nonlocally nonlinear media can be found in figure 4. The spiraling elliptic hollow solitons can be obtained by numerical iterations, which will be reported elsewhere due to length limitations.
As described above, the spiraling elliptic hollow beams exhibit three key characteristics, having the bright ring, carrying the OAM, and performing rotations. Due to these characteristics, the spiraling elliptic hollow beams not only can trap the PP particles, restrict them within the bright ring, but also can rotate them at the same time. While, the NP particles can be trapped within the dark hollow and be rotated. Therefore, we can use the spiraling elliptic hollow beams to realize the jointly multiple manipulation on particles. Furthermore, at the given distance (the actual case in experiment) the rotation can be controlled by the optical power as shown in figure 5. By increasing the optical power, we can enhance the beam's rotation.
It should be noted that the novel characteristics of the spiraling elliptic hollow beams are attributed to the cross phase. We show the linear evolution the linear evolutions of the elliptic hollow beam without the cross phase in figure 6. The elliptic hollow structure can not maintain any more during propagation. The light energy flows towards the beam center, and forms the peak at the center in the far field.
We can use the spiraling elliptic hollow beams as the 'building blocks' to construct complex rotating patterns carrying the OAM via the linear superposition, which are shown in figure 7. In figure 7(a), the rotating patterns is constructed by Similarly, we can construct the complex rotating pattern as shown in figure 7(d) by the linear superposition of four spiraling elliptic hollow beams.

Conclusion
In summary, we have obtained a novel class of spiraling elliptic hollow beam, which have an offset bright ring and a dark elliptic hollow. Due to the cross phase, the spiraling elliptic hollow beams carry the OAM and can rotate during linear and nonlinear evolutions. The breather states exist when a spiraling elliptic hollow beam propagates in the nonlocally nonlinear medium. We analyzed the influences of the offset distance and the ellipticity on the gradient force and the OAM of the spiraling elliptic hollow beams. By using the spiraling elliptic hollow beams, we can achieve a jointly multiple manipulation on particles at the same time, can trap and simultaneously rotate the particles. Furthermore, the spiraling elliptic hollow beams can serve as the 'building blocks' to construct complex rotating patterns carrying the OAM. Compared to the Gaussian beams, the spiraling elliptic hollow beams have advantages in trapping and rotating the rod-shaped objects, and the particles of negative polarizability.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.