1. Introduction

The Pearcey beams, based on the Pearcey function, were firstly studied by Pearcey in 1946^[1]. In 2012, Ring et al. discovered experimentally that Pearcey beams include these propagation properties: autofocusing, form-invariance, and self-healing^[2]. Due to the fact that Pearcey beams contain infinite energy^[2], scientists tend to add the Gaussian factor to derive the more physical Pearcey Gaussian beams (PGBs), which contain finite energy. The unique properties of PGBs and Pearcey Gaussian Vortex Beams (PGVBs) such as their propagation properties and vectorial structures have been revealed by some researchers. For instance, in 2019, Deng et al. introduced a kind of PGBs with an astigmatic phase (APPGBs)^[3], which reveals the rotating factor can rotate the transverse intensity distribution of the APPGBs on propagation in a chiral medium. Afterward, the effects of the multi-order and off-axis vortex on the propagation of PGVBs in a chiral medium were illustrated by Deng et al. in 2021^[4]. As indicated in Refs. [5-8], Pearcey beams can be useful in multi microparticle manipulation^[5], human tissue medicine, particle guiding^[6], optical imaging, and optical trapping^[7].

It is also interesting to investigate the propagation of beams in uniaxial crystals. As we know, laser beams propagating in uniaxial crystals can be applied effectively to determine the crystal structure and investigate the available optical phenomena of uniaxial crystals. Uniaxial crystal plays a significant role in the optimal design of polarizers, compensators, and amplitude and phase-modulation devices^[8-9]. According to the current resources available, there are some research reports about Airy Gaussian beams^[10], Airy vortex beams^[11], and Pearcey beams^[12] in uniaxial crystals, which show the influence of the anisotropic and refractive index on laser beam propagation.

Cosh-Airy beams, which have more manipulation degrees of freedom than the corresponding Airy beams, have been proposed, and the self-healing ability of the cosh-Airy beam was found be higher than that of the corresponding Airy beam in 2019^[13]. To the best of our knowledge, energy flux distributions could be changed by cosh parameters and we cannot find more details about the vector properties of Cosh-Pearcey-Gaussian Vortex (CPeGV) beams. We introduce the cosh function to Pearcey-Gaussian vortex optical field in this paper, and it is natural to ask whether CPeGV beams can show more distinctive properties propagating in uniaxial crystals than in corresponding PeGV beams.

The purpose of this paper is to discuss the longitudinal and transverse Poynting vector and Angular Momentum Density (AMD) of CPeGV beams propagating in uniaxial crystal. We explore the influence of the topological charge on the Poynting vector in the far-field. Furthermore, we find the Fourier spectrum, the Poynting vector and how the energy flux distributions can be changed by properly selecting the cosh parameter during the propagating distance. Therefore, the propagation properties of the CPeGV beams in uniaxial crystals are more abundant than those of the corresponding PeGV beams. This research is beneficial to the practical applications of the CPeGV beams in many fields such as information storage and optical trapping particles.

2. Cosh-Pearcey-Gauss vortex beams

The input plane is z=0 and the observation plane is z. The ordinary and extraordinary refractive indices of the uniaxial crystal are n_o and n_e, respectively. The relative dielectric tensor of the uniaxial crystal reads as^[12]

In the Cartesian coordinate system, the transverse electric field of a CPeGV beam with off-axis optical vortices in the input plane z =0 is written as

The Pearcey-Gauss background beam in Eq. (2) is expressed as [14]

where Pe(x/x₀, y/y₀) is the Pearcey function with scaling lengths x₀ and y₀ along the x- and y-axes, respectively. To ensure the finite energy in real space, the Pearcey function is also modulated by a Gaussian factor with a waist width of w₀, m is the topological charge or order number of the embedded optical vortex at z=0, sgn (m) is the sign function, the complex parameter Q is the noncanonical strength of the embedded optical vortex and for Q=±1 the Eq. (2) is simplified to the canonical or symmetric vortex. Ω denotes the modulation parameter related to the cosh part.

The two-dimensional Fourier transforms is given by^[15].

where j is x or y. Substituting Eq. (2) into Eq. (4), we obtain the two-dimensional Fourier transform of the initial electric field of CPeGV beams as

where

and

Within the framework of the paraxial approximation, the propagation of the CPeGV beams in uniaxial crystals orthogonal to the optical axis obeys the following equations^[10]:

where k = 2/λ is the wavenumber and λ is the optical wavelength, z_e = 2ky₀² and z is the propagation distance. Substituting Eqs. (1) and (2) into Eq. (6), and using the integral formulas

Therefore, the propagating form of the CPeGV beams in uniaxial crystals is

where

and

Figure 1 shows the dependence of the Fourier spectrum of the CPeGV beam on cosh modulation parameter Ω and topological charge m. One can see that their spectra present parabola structures and they can be partitioned by cosh parameter Ω and topological charge m. The maximal values are located on two sides rather than the vertex of a parabola as shown in the i and iii figures in Fig.1 (a) (color online). A larger Ω or m can lead to a growth in the number of partitioned side-lobes, and an increase in their maximal values in the Fourier spectrum as shown in iv of Fig.1 (a) and 1 (b) (color online), by comparing ii in Fig.1 (a) and in Fig.1 (b), it can be seen that the number of partitioned side-lobes in Ω=5 is significantly greater than that of Ω=2 in the case of equal m, which indicates that the cosh parameter Ω has an advantage over topological charge m in the ability of partitioned side-lobes.

Figure  1.  The dependence of Fourier spectrum of CPeGV beams on cosh modulation parameter Ω and topological charge m; (a) the topological charge m= 1; (b) the cosh modulation parameter Ω=5.

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3. The Poynting vector and angular momentum density of CPeGV

In the paraxial domain, the time-averaged Poynting vector^[16] of the optical wave field in a Cartesian coordinate is defined as

where E and H represent the electric and magnetic field vectors, respectively, c is the speed of light, w stands for circular frequency k=2π/λ is the wavenumber, and the asterisk denotes complex conjugation. The S_x, S_y, and S_z are the x-, y- and z components of the time-averaged Poynting vector, respectively, and their values are real. The Poynting vector describes the rate of electromagnetic energy flow per unit area, and the z-component of the Poynting vector is proportional to the light intensity.

The AMD of the electromagnetic field J is related to the linear momentum density P by J= r × P^[17], where r is the position vector and the linear momentum density is proportional to the Poynting vector. Therefore, the time-averaged AMD can be expressed as^[18]

3.1 The transverse and longitudinal Poynting vector

Figure 2 (color online) shows the evolution of the longitudinal and transverse Poynting vectors of the CPeGV beams in uniaxial crystals, where the direction and length of the arrows denote the direction and magnitude of the transverse Poynting vector. It is found that auto-focusing and inversion are still maintained, and the far-field pattern presents parabolic structures with dark lines. The original energy of CPeGV beams concentrates on side lobes due to the vortex core and the cosh parameter, which is different from the concentrated energy on the main lobes of the Pearcey beams. The direction of the transverse Poynting vector moves from side-lobes to the main lobes, but their direction is opposite beyond the auto-focusing plane. Moreover, the energy flux flows from the vertex to the two sides of the parabola along the optical intensity channels, and the deviation of the optical bright spot can be found owing to anisotropy.

Figure  2.  The intensity (backgrounds) and the transverse Poynting vector (arrows) of the CPeGV beams propagating in uniaxial crystal for different propagation distances z with different cosh modulation parameters Ω and n_e=1.2n_o; (a): Ω=1; (b): Ω=2 and m=1.

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Figure 3 (color online) depicts the longitudinal and transverse Poynting vectors of the CPeGV beams for different cosh modulation parameters Ω and topological charges m at z=300 z_e. One can see that the number of partitioned side-lobes with a parabolic structure increases as cosh modulation parameter Ω and topological charge m increase. The main lobe near the origin gradually disappears with an increase of cosh modulation parameter Ω and further presents four-lobe structures rather than the usual parabolic curves. For the embedded optical vortex with topological charge m=+2, the far-field optical vortex can be also found and marked by a white “×”. Specifically, it can be seen their locations are (0.11,15.91) for Ω=3, and (2.91,−1.73), (4.88,−0.88) for Ω=5 and (−2.35,0.2), (−1.6,−0.5) and (0.06,−1.9) for Ω=9 respectively.

Figure  3.  The longitudinal and transverse Poynting vectors of the CPeGV beams for different cosh modulation parameters Ω and topological charges m at z=300z_e. (a): m=0; (b): m=2.

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3.2 The transverse and longitudinal angular momentum density

To further describe the evolution of AMD, the dependencies of the z-component, x-component, and y-component of the AMD of the beam on cosh modulation parameter Ω, topological charge m, and propagation distance z are shown in Figures 4 to 6 (color online).

Figure  4.  The longitudinal AMD of the CPeGV beams propagating in uniaxial crystal for different propagation distances z with different cosh modulation parameters Ω and n_e=1.2n_o, m=1. (a): Ω=1; (b): Ω=2

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Figure 4 gives the evolution of the longitudinal AMD of the CPeGV beams in uniaxial crystal. Comparing the two figures in the first column of Fig. 4, we find that a greater Ω leads to the maximal rate of AMD increase. On the other hand, by observing the numerical value of the color bars, we find that the maximal value of AMD firstly increases and then decreases as z evolves, reaching a peak at the focusing plane. The longitudinal AMD of Fig.4 exhibits parabolic shapes (see the last column) at z=50 z_e. Furthermore, due to the embedded optical vortex, the splits in the positive direction are greater than those in the negative direction, which leads to the net AMD being positive. For example, the maximum values of the positive and negative directions are 0.0442 and −0.04 at the input plane, respectively. This is because J=L+S and the linear polarization leads to S=0, then the J will be positive or negative depending on the AMD. In other words, it's related to vortex properties. The net AMD is still positive in the far-field.

More details on the evolution of the maximum AMD of CPeGV beams in uniaxial crystals are shown in Fig. 5 (color online). Their maxima increase and reach peak values at the autofocusing plane, and a higher peak value is accompanied by a larger Ω. For example, their maxima in AMD densities are 0.083, 0.02, 0.03 for Ω=1, 2, and 3, respectively. The results indicate that the autofocusing properties not only appear in the evolution of the Poynting vector but also display in AMD densities.

Figure  5.  Maximum of angular momentum density of CPeGV beams in uniaxial crystal for different propagation distances z with different cosh modulation parameters Ω

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Figure  6.  The normalized AMD of the CPeGV beams in uniaxial crystal at z=300z_e for different cosh modulation parameters Ω with different topological charges and n_e=1.2n_o; (a): m=0; (b): m=2

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Figure 6 (color online) describes the normalized longitudinal AMD of the CPeGV beams for different cosh modulation parameters Ω and topological charges m at z=300z_e. One can find that the AMD J_z tends to split more branches from the positive and negative direction of the z-axis with increases in the absolute value of topological charge m. The splits in the positive direction are greater than those in the negative direction, which leads to the appearance of a positive AMD. However, a greater Ω leads to AMD presenting four side-lobe structures in the far-field rather than a parabolic structure. Compared with PeGV beams, CPeGV beams have one more regulatory parameter Ω. Therefore, it is easier to regulate and control the Poynting vector and AMD of the optical beam in uniaxial crystals. The results obtained here are beneficial to the design of a more reasonable compensator and polarizer.

4. Conclusion

The propagation properties of CPeGV beams are numerically demonstrated in uniaxial crystals. The effects of Ω variation and topological charge variation on Poynting vector and AMD in uniaxial crystal propagation and far-field stability are investigated. Although all parameters have effects on the intensity pattern and the energy distribution of CPeGV beams, the main roles they play are different. The results show that the CPeGV beams still exhibit self-focusing and reversal properties with an increase in propagation distance during the propagation of uniaxial crystal. The energy distribution of the CPeGV beams can be changed by Ω, where energy might shift to a side lobe. Moreover, the longitudinal Poynting vector first presents a parabolic shape in the far-field, which can be explained by the direction of the arrow of the transverse Poynting vector. With an increase in Ω, the longitudinal Poynting vector will converge along the direction of the transverse Poynting vector, and finally, become four-lobes structures. Along with the propagation distance, the trend of change to the maximum AMD is to first increase and then decrease. Moreover, as the Ω increases, the peak value of AMD also increases significantly. Our investigation will provide a better understanding of the state of CPeGV beams propagating in uniaxial crystal and be useful for applications in information transmission.