## Abstract

We study both analytically and numerically nonparaxial propagation dynamics of the Chirped Airy vortex (CAiV) beams in uniaxial crystal orthogonal to the optical axis. The propagation trajectory, the intensity, the radiation forces, the Poynting vector and the angular momentum (AM) of the CAiV beams are illustrated by numerical examples. The influences of the ratio of the extraordinary refractive index to the ordinary refractive index, the linear chirp factor and the quadratic chirp factor on the nonparaxial evolution of the CAiV beams are examined in detail. Results show that the linear chirp factor provides an intensity concentration, which is totally different with the influence of the quadratic chirp. Besides, the uniaxial crystals with different refractive index ratios can be used to control the intensity of optical lobes. Moreover, the value and the direction of the radiation forces depend on the propagation distance and the chirp factor. The chirp factor acting on the Poynting vector and the AM mainly occurs in the direction of vectors. The nonparaxial propagation characteristics of the CAiV beams provide a convenient method to the intensity modulation and the optical manipulation of micro particles.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The original theoretical research and corresponding experimental demonstration to the finite energy Airy beams were carried out by Siviloglou and Christodoulides in 2007 [1, 2]. With the novel properties of self-acceleration [1, 2], self-healing [3] and diffraction-free [4], the Airy beams have attracted significant attentions in applications such as soliton pairs and light bullet generation [5–7], optical clearing of micro particles [8], curved plasma channel generation [9,10] and etc. In the last few decades, the propagation of the Airy beams in linear medium [3, 4], harmonic potential [11], and the nonparaxial self-accerating beams in an atomic vapor [12] were intensely investigated. With a spiral phase, optical beams carrying the phase singularities and the orbital angular momentum are known as the vortex beams [13]. Due to unique properties such as spatial propagation invariance, carrying the orbital angular momentum and helical wavefront, several applications of the vortex beams have been found, including optical trapping [14, 15], information transfer and communications in free space [16], astrophysics [17] and so on. In practice, the Airy vortex (AiV) beams could be obtained by multiplying a vortex factor to the Airy beams in theory and by the Airy beams passing through a spiral phase plate in experiment.

On the other hand, laser beams propagating in uniaxial crystal could be used effectively to determine the crystal structure and investigate available optical phenomena of uniaxial crystal. Furthermore, uniaxial crystal plays a key role in the design of the polarizer, the compensator, and the amplitude-and the phase-modulation devices [18, 19]. Particularly, when the waist radius of the beams is comparable with the wavelength in uniaxial crystal or the far-field divergence angle becomes large, the paraxial theory is not accurate enough and in this case, the beam’s nonparaxial effect should be considered seriously. Up to now, several studies on the nonparaxial propagation of various kinds of laser beams in uniaxial crystal have been reported [20–22].

The foregoing researches on the laser beams have only been limited to the unchirped case. In reality, the chirp parameter is regarded as a powerful tool in the generation and the manipulation of optical beams. In optical field, the initial finite energy Airy beam with a chirp parameter was reported by Zhang et al., disclosing that a linear chirp introduces a transverse displacement of the beam at the phase transition point, but does not change the location of the point and a quadratic chirp moves the phase transition point, but does not affect the beam profile [23].

Here, we propose a kind of Chirped Airy vortex (CAiV) beams and accordingly investigate nonparaxial propagation properties of a Linearly Chirped Airy vortex (LCAiV) beam and a Quadratically Chirped Airy vortex (QCAiV) beam in uniaxial crystal orthogonal to the optical axis, respectively. Afterwards, the intensity, the energy flow, the angular momentum and the radiation forces of the CAiV beams mentioned above are analyzed in detail. Some useful and interesting results such as the adjustability of the intensity, the focusing behavior in the far field and the unique tunable radiation forces have been found in our investigation, which could be applied availably in the information transfer and the optical trapping particle.

The organization of the paper is as follows. In Sec. 2, we introduce the theoretical model for the nonparaxial propagation in uniaxial crystal orthogonal to the optical axis and derive analytical solutions of the CAiV beams; in Sec. 3, the nonparaxial evolutions of a LCAiV beam and a QCAiV beam are analyzed specifically; and in Sec. 4, the nonparaxial propagation characteristics of the CAiV beams in uniaxial crystals with different chirp parameters and ratios of the extraordinary refractive index to the ordinary refractive index are briefly discussed. Finally, some extraordinary and useful results are summarized in Sec. 5.

## 2. Analytic solution of the CAiV beams in uniaxial crystal orthogonal to the optical axis

In the Cartesian coordinate system, the z-axis is set as the propagation axis. The optical axis of the uniaxial crystal coincides with the x-axis. The observation plane is taken to be *z* and the input plane is *z* = 0. The relative dielectric tensor *ε* of uniaxial crystal is described by

*n*and

_{e}*n*are the extraordinary and the ordinary refractive indices, respectively. We suppose that the CAiV beam linearly polarized in the x-direction is incident on uniaxial crystal at the initial plane

_{o}*z*= 0. Thus the electric field distribution of the CAiV beams in the input plane takes the form

*w*

_{0}represents the initial beam waist size;

*a*is the truncation factor of the CAiV beams; Ai· denotes the first kind of the Airy function;

*β*

_{1}

*and*

_{x}*β*

_{1}

*are the linear chirp parameters, while*

_{y}*β*

_{2}

*and*

_{x}*β*

_{2}

*are the quadratic chirp parameters. When*

_{y}*β*

_{1}

_{x}_{,}

*=*

_{y}*β*

_{2}

_{x}_{,}

*= 0, Eq. (2) is the AiV beam; when*

_{y}*β*

_{1}

_{x}_{,}

*≠ 0,*

_{y}*β*

_{2}

_{x}_{,}

*= 0, Eq. (2) is the LCAiV beam; when*

_{y}*β*

_{1}

_{x}_{,}

*= 0,*

_{y}*β*

_{2}

_{x}_{,}

*≠ 0, Eq. (2) is the QCAiV beam.*

_{y}According to the theory of nonparaxial propagation in uniaxial crystal orthogonal to the optical axis, the electric field of a beam in uniaxial crystal orthogonal to the optical axis reads as [24]

*k*and

_{ez}*k*being defined by

_{oz}*λ*being the wavelength of the incident beam in vacuum. ${d}^{2}{\overrightarrow{k}}_{\perp}=d{k}_{x}d{k}_{y}$ is the transverse vector surface element in the spatial-frequency domain. ${\tilde{E}}_{j}\overrightarrow{{k}_{\perp}}$ is the two-dimensional Fourier transform of the transverse components of the optical field in the plane

*z*= 0.

By using the property of the Fourier transforms [25], Eq. (3) can also be rewritten as the following three components of the nonparaxial field in uniaxial crystal orthogonal to the optical axis [24]

*A*,

_{e}r*r*

_{0}and

*A*,

_{o}r*r*

_{0}being given by

Inserting Eq. (2) into Eqs. (5)–(7) and performing the integrals, the three components of the nonparaxial propagation of the CAiV beams in uniaxial crystal orthogonal to the optical axis are found to be

*K*,

_{n}*L*and

_{n}*M*(

_{n}*n*=1, 2, 3, 4) are given by

## 3. The numerical calculations and analysis

Based on the analytical solutions obtained in Eqs. (10)–(12), we concentrate on analyzing their properties in the following. To get a further understanding of the influences of the linear chirp and the quadratic chirp coefficient on the CAiV beams, the nonparaxial propagation evolutions of a LCAiV beam and a QCAiV beam in uniaxial crystal orthogonal to the optical axis are discussed respectively. In our simulation, some parameters are chosen as *n _{o}* = 2.616,

*a*= 0.2,

*w*

_{0}= 0.1

*mm*,

*λ*= 633

*nm*and the Rayleigh distance can be written as ${Z}_{R}=\frac{k{w}_{0}^{2}}{2}=4.9630cm$. Hereafter, the parameters are the same as those aforesaid except other stated.

#### 3.1. Nonparaxial evolution of a LCAiV beam in uniaxial crystal orthogonal to the optical axis

Figure 1 represents a side-view in the x-direction of the nonparaxial propagation trajectory of a LCAiV beam with different *β*_{1}_{x}_{,}* _{y}* in uniaxial crystal orthogonal to optical axis, in the case that the propagation distance is 8

*Z*. In Fig. 1(a), both

_{R}*β*

_{1}

*and*

_{x}*β*

_{1}

*are 0, which suggests that the beam is the AiV beam. Compared with Fig. 1(a), when nonparaxially propagating, the LCAiV beam has a greater initial velocity than the AiV beam in uniaxial crystal orthogonal to the optical axis. Moreover, the bigger*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*are, the greater the initial velocity is. When the AiV beam propagates through the medium, the intensity of the main lobe and the side lobes decreases continuously with the increase of the propagation distance. However, it will happen that the re-focusing effect of the intensity when the LCAiV beam propagates, which is enhanced with the increase of*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*.*

_{y}Figure 2 depicts the transverse intensity distribution of a LCAiV beam in uniaxial crystal orthogonal to optical axis. During the propagation process of the LCAiV beam, the main lobe shape of the intensity distribution changes from the initial triangle-like one to the ellipse. In the propagation vicinity, the intensity distributes in the main lobe, as well as the side lobes closer to the main lobe. From Figs. 2(a1)–2(c1), one can see that the intensity of the main lobe is relatively small in the propagation vicinity, but it will not weaken during the propagation and the intensity concentration will happen in the distance. As shown in Figs. 2(a1)–2(a3), the changes of *β*_{1}* _{x}* and

*β*

_{1}

*mainly affect the position of the intensity concentration in the distance of the propagation.*

_{y}Subsequently, we investigate the gradient force and the scattering force of the LCAiV beam, which are associated with momentum changes of the electromagnetic wave. We assume that a micro particle with refractive index *n*_{1} is struck by the LCAiV beam. When the particle experiences a steady state, the time-average gradient force and the scattering force can be expressed as [26]:

*n*

_{2}is the refractive index of a surrounding medium,

*r*

_{0}is the radius of the micro particle,

*c*is the light velocity and

*ε*

_{0}is the permittivity of vacuum. Here we assume that

*n*

_{1}= 1.592,

*r*

_{0}= 50

*nm*, and the emergent light is extraordinary ray

*n*

_{2}=

*n*.

_{e}The transverse gradient force distribution of an extraordinary LCAiV beam in uniaxial crystal orthogonal to the optical axis is displayed in Fig. 3. The background is the pattern of the gradient force, and the arrow represents the direction. Different from the intensity distribution, the gradient force generally distributes in both sides of the optical lobes. The gradient force mainly distributes in the side lobes at the positions 0.1*Z _{R}* and 2

*Z*, shown in Figs. 3(a1)–3(a2), while it centralizes in the main lobe at 8

_{R}*Z*, shown in Fig. 3(a3). The gradient force vectors of the LCAiV beam in the y-direction are approximately parallel to the x-axis symmetrically, and most of the arrows near the main lobe point down. When the LCAiV beam propagates to the distance, bilateral arrows in the main lobe are parallel to the x-axis symmetrically, and the middle part of the arrows is parallel to the negative y-axis.

_{R}In Fig. 4, we plot the scattering force of the LCAiV beam in uniaxial crystal orthogonal to the optical axis *a* = 0.5. When keeping *β*_{1}* _{x}* and

*β*

_{1}

*constant, the longer the propagation distance is, the more slowly the scattering force reaches a maximum in the x-direction, and the smaller the maximum value of the corresponding scattering force is. When keeping the propagation distance constant, the general distribution of the scattering force can be basically consistent, but the smaller*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*are, the more advanced the scattering force distribution is. Therefore,*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*play a delaying role in the modulation of the scattering force for the LCAiV beam.*

_{y}Next, we consider the Poynting vector and the angular momentum (AM) of a LCAiV beam when nonparaxially propagating through uniaxial crystal orthogonal to the optical axis. The Poynting vector $\overrightarrow{S}$ and the time-averaged AM density $\overrightarrow{J}$ can be written as [27, 28]

Based on Eq. (19) and Eq. (20), the energy flow and the AM of an extraordinary LCAiV beam are intuitive in Fig. 5 and Fig. 6, respectively. The Poynting vector of the LCAiV beam almost entirely distributes in the side lobes, as can be seen in Fig. 5. From Figs. 5(a2)–5(c2), we can find that the effect of *β*_{1}* _{x}* and

*β*

_{1}

*on the nonparaxial propagation of the LCAiV beam lies in changing the maximum position of the energy flux density. With the increase of*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*, the maximum position of the Poynting vector is closer to the main lobe. In Figs. 5(a3)–5(c3), as the LCAiV beam propagates to the distant, the distribution of the side lobes is more concentrated and continuous with a better focusing effect. The Poynting vector generally bends anticlockwise towards the positive y-axis direction. In addition, the changes of*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*will also cause altering the direction of the energy flow, as it is shown in Figs. 5(a3)–5(c3) in more detail. As we can see, with increasing the value of*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*, the Poynting vector bends clockwise to the positive x-axis gradually at 8*

_{y}*Z*.

_{R}Figure 6 presents the AM distribution of an extraordinary LCAiV beam in uniaxial crystal orthogonal to the optical axis. It can be seen from the image that the AM density mostly distributes in the side lobes, and the focusing effect is obvious in the distance during the propagation of the LCAiV beam. The AM density flow principally points at the positive y-axis direction. As shown in Figs. 6(a3)–6(c3), when *β*_{1}* _{x}* and

*β*

_{1}

*increase, the pattern of the AM moves up towards the positive y-axis and the bending degree of the arrows strengthens.*

_{y}#### 3.2. Nonparaxial evolution of a QCAiV beam in uniaxial crystal orthogonal to the optical axis

Figure 7 represents the intensity distribution of a QCAiV beam in uniaxial crystal orthogonal to the optical axis. It can be seen from the figure that the intensity of the QCAiV beam distributes in the both main lobe and side lobes but the intensity of the side lobes is larger relatively. As the propagation distance increases, the distribution of the intensity in the lobes parallel to the y-axis is dispersed, while in the lobes parallel to the x-axis has a more concentrated effect. *β*_{2}* _{x}* and

*β*

_{2}

*play a part in weakening the intensity of the QCAiV beam in uniaxial crystal orthogonal to the optical axis. The larger*

_{y}*β*

_{2}

*and*

_{x}*β*

_{2}

*are, the smaller the intensity of the QCAiV beam is. Compared with Fig. 2,*

_{y}*β*

_{1}

_{x}_{,}

*and*

_{y}*β*

_{2}

_{x}_{,}

*have distinct effects to the propagation properties of the CAiV beam.*

_{y}To further validate the intensity distribution of the QCAiV beam, we also discuss the peak intensity distribution in Fig. 8. The shape of the peak intensity seems like a hyperbola. As the propagation distance increases, the peak intensity decreases rapidly, especially at the distance of 1*Z _{R}*. It can be seen that the larger

*β*

_{2}

*and*

_{x}*β*

_{2}

*are, the smaller the peak intensity of the QCAiV beam is, and the faster the tendency to decrease of the peak intensity is.*

_{y}Figure 9 depicts the transverse gradient force distribution of an extraordinary QCAiV beam in uniaxial crystal orthogonal to the optical axis. Similar to the LCAiV beam, the gradient force mainly distributes in the sides of the optical lobes. It shows a great amount of the arrows point down in the main lobe and the gradient force direction of the side lobes parallel to the y-direction distributes symmetrically, parallel to the x-axis. The farther the propagation distance is, the smaller the gradient force of the side lobes parallel to the x-axis is. At the same time, the shape of the side lobes parallel to the x-axis changes from the U-shape to the semicircle. In conclusion, *β*_{2}* _{x}* and

*β*

_{2}

*have an impact on the dispersion degree of the gradient force. The greater*

_{y}*β*

_{2}

*and*

_{x}*β*

_{2}

*are, the more dispersed the gradient force distribution is.*

_{y}The scattering force of an extraordinary QCAiV beam with different *β*_{2}_{x}_{,}* _{y}* and at different propagation distance in uniaxial crystal orthogonal to the optical axis is indicated in Fig. 10. It can be obviously known that the larger

*β*

_{2}

*and*

_{x}*β*

_{2}

*are, the smaller the scattering force is. Likewise, as the propagating distance increases, the corresponding scattering force decreases drastically. However, compared with the former, the latter has the weaker effect.*

_{y}Figure 11 shows the energy flow density distribution of an extraordinary QCAiV beam nonparaxially propagating in uniaxial crystal orthogonal to the optical axis. It can be seen that the energy density mainly distributes in the side lobes, and its vectors approximately point to the positive x-axis. The larger *β*_{2}* _{x}* and

*β*

_{2}

*are, the smaller the energy density is, the greater the number of the side lobes is, the farther the corresponding position of the energy density distribution is.*

_{y}Figure 12 presents the AM distribution of an extraordinary QCAiV beam in uniaxial crystal orthogonal to the optical axis. It is not difficult to find from Figs. 12(a1)–12(c1) that with the increase of the propagation distance, the number of the side lobes increases vehemently. Meanwhile, the arrows mainly point along the positive y-axis direction at the distance of 0.1*Z _{R}*, but they drift along the positive x-axis direction slowly during the QCAiV beam propagating.

## 4. The influences of the chirp factor and the refractive index ratio for uniaxial crystal on the propagation properties of the CAiV beams

To gain a deeper insight into the effects of the linear chirp and the quadratic chirp on the evolution of the intensity for the CAiV beams, the intensity distribution as a function of *x* for the CAiV beams in uniaxial crystal orthogonal to the optical axis is elucidated in Fig. 13. When *β*_{1}_{x}_{,}* _{y}* and

*β*

_{2}

_{x}_{,}

*are smaller enough, they play the same role in the intensity distrisbution at the distance of 0.1*

_{y}*Z*, shown in Fig. 13(a1). With the futher increases of the chirp parameter and the propagation distance, the intensity of the QCAiV beam and the CAiV beam with the linear chirp and the quadratic chirp simultaneously dicreases remarkably and distributes dispersedly. By contrast, the intensity of the LCAiV beam is much larger and more concentrated than that of the QCAiV beam and the CAiV beam during the propagation. Moreover, the intensity distributions of the QCAiV beam and the CAiV beam are quite similar, demonstrating that

_{R}*β*

_{2}

*and*

_{x}*β*

_{2}

*play a leading role in the evolution of the intensity for the CAiV beams.*

_{y}The transverse intensity of the CAiV beams carrying both the linear and the quadratic chirp parameters *β*_{1}_{x}_{,}* _{y}* =

*β*

_{2}

_{x}_{,}

*= 1 in uniaxial crystal orthogonal to the optical axis is investigated in Fig. 14. It is easy to see from Figs. 14(a1)–14(a3) that when*

_{y}*n*= 0.5

_{e}*n*, the transverse intensity of the side lobes parallel to the x-axis is more dispersed as the propagation distance increases, and on the contrary, the intensity of the main lobe and the side lobes along the y-axis direction is more concentrated. But in the case of $\frac{{n}_{e}}{{n}_{o}}>1$, it is opposite that with the increase of the propagation distance, the intensity of the side lobes along the x-axis is more concentrated, and the intensity of the main lobe and the side lobes along the y-axis direction is more dispersed. Due to the fantastic characteristics of the uniaxial crystal, it is available for optical field modulation.

_{o}## 5. Conclusion

In conclusion, analytical expressions for the three components of nonparaxial propagation of the CAiV beams in uniaxial crystal orthogonal to the optical axis are derived, and the corresponding propagation trajectory, the intensity, the radiation forces, the Poynting vector and the AM of the LCAiV beam and the QCAiV beam have been analyzed. The LCAiV beam has a greater initial velocity than the AiV beam, and the bigger *β*_{1}* _{x}* and

*β*

_{1}

*are, the greater the initial velocity is. The intensity of the main lobe strengthens during the propagation of the LCAiV beams and the intensity concentration will happen in the distance. Whereas,*

_{y}*β*

_{2}

*and*

_{x}*β*

_{2}

*play a part in weakening the intensity of the QCAiV beam in uniaxial crystal orthogonal to the optical axis. Besides,*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*play a delaying role in the modulation of the scattering force, and*

_{y}*β*

_{2}

*and*

_{x}*β*

_{2}

*have an impact on the dispersion degree of the gradient force. Interestingly, the Poynting vector of the LCAiV beam generally bends anticlockwise towards the positive y-axis but with increasing the value of*

_{y}*β*

_{1}

*and*

_{x}*β*

_{1}

*, it bends clockwise to the positive x-axis gradually at 8*

_{y}*Z*. As for the QCAiV beam, the number of the side lobes of the Poynting vector and the AM increases with

_{R}*β*

_{2}

*and*

_{x}*β*

_{2}

*increasing. Lastly, we find that the intensity of the CAiV beams is mainly determined by the quadratic chirp and changes with the ratio of the extraordinary refractive index to the ordinary refractive index flexibly. We believe that our investigation of the CAiV beams will be available for the application of self-accelerated beams in several branch of fields including the optical trapping and the manipulation of microparticles.*

_{y}## Funding

National Natural Science Foundation of China (11374108, 11775083); the National Training Program of Innovation and Entrepreneurship for Undergraduates (201710574046).

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