# Propagation Characteristics of Circular Airy Vortex Beams in a Uniaxial Crystal along the Optical Axis

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## Abstract

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## 1. Introduction

_{e}/n

_{o}ratio has an important effect on its propagation characteristics [35]. Taking advantage of the electro-optical effect, one can enhance or weaken the “abruptly autofocusing” effect of a CAB [36]. To the best of our knowledge, there is no research that discusses the propagation of CABs with OVs along the optical axes of uniaxial crystals. In the following, we numerically investigate the propagation properties of CABs imposed different OVs in a uniaxial crystal.

## 2. Theory Model

**E**(r) of the light field

**E**(r) = Re[

**E**(r)exp(-iωt)] propagating in an anisotropic medium obeys the following equation:

_{0}= ω/c is the wave number in the vacuum and ε is the relative dielectric tensor. In the Cartesian coordinate system, z-axis is taken to be the optical axis of the uniaxial crystal. We assume that the light wave propagates along the optical axis, or the z-axis. The relative dielectric tensor of the uniaxial crystal is:

_{o}and n

_{e}are the ordinary and extraordinary refractive indices of the uniaxial crystal, respectively. Based on the angular spectrum theory, Ciattoni A. proposed a method to deal with the problem of light propagation in uniaxial crystals [25]. The main result of this method is that the transverse component of an input light field at z = 0 gives rise to a light field inside the crystal that is a linear superposition of two parts, the ordinary and the extraordinary [25]:

**r**is the position vector at any transverse location, and ${A}_{o}(r,z)$ and ${A}_{e}(r,z)$ are given by:

**k**is the angular frequency vector. In order to obtain a more suitable solution representation for a CP incident field, two complex unit vectors, ${\widehat{e}}_{+}=\sqrt{2}/2({\widehat{e}}_{x}+i{\widehat{e}}_{y})$ and ${\widehat{e}}_{-}=\sqrt{2}/2({\widehat{e}}_{x}-i{\widehat{e}}_{y})$, corresponding to left- and right-hand CP light waves, are introduced. The left- and right-hand CP components of the whole light field in the crystal, ${A}_{+}(r,z)$ and ${A}_{-}(r,z)$, are given as [25]:

## 3. Numerical Study

_{0}is the initial radius of the CAB, w is the radial scale coefficient, a is the decay parameter, (${r}_{k}$, ${\phi}_{k}$) denotes the location of the OV, and l represents the TC number of the OV. Although the closed-form approximation of the FT of the CAB is given by relying on a suitable plane wave angular spectrum representation of the beam [5], there is no analytic expression for the FT of the CAB with OVs. Therefore, we employ the discrete Fourier transform to obtain the FT of the initial beam using a fast Fourier transform algorithm. In our numerical study, the beam parameters are given as follows: r

_{0}= 0.5 mm, w = 25µm, a = 0.1, and the wavelength $\lambda $ = 632.8 nm, and the ordinary and extraordinary refractive indices of the uniaxial crystal are n

_{o}= 2.616 and n

_{e}= 2.903, respectively.

#### 3.1. CAB with On-Axis OV

_{m}/I

_{0m}is introduced to study the “abruptly autofocusing” effect, where I

_{0m}is the maximum intensity at the initial plane and I

_{m}is the maximum intensity at an arbitrary transverse plane. The ratio I

_{m}/I

_{0m}vs. propagation distance, z, is provided in Figure 3. Figure 3 also shows that the “abruptly autofocusing” effect appears twice, focal planes of which are at z = 150 mm and z = 192 mm. Because the portion of the energy from the incident light is coupled to the RHCP component, the ratio I

_{m}/I

_{0m}is lower than 10, which is much lower than that of common CABs. Next, we studied the case of the RHCP component. Figure 4 provides the intensity pattern and phase distribution of the RHCP component at z = 100 mm and z = 180 mm. From Figure 4, one can see that the RHCP component also exhibits an “abruptly autofocusing” effect. Figure 4e,f shows that the RHCP component carries a vortex phase whose TC number is 3. This is because the RHCP component acquires a vortex phase with a TC number of 2 due to the spin reversal when the LHCP component is converted to an RHCP component [23,24]. Figure 5 shows the propagation dynamics of the RHCP component. As can be seen in Figure 5, near the focal plane, the beam is hollow, and the hollow region is greater than that of the LHCP component due to the larger TC number. The “abruptly autofocusing” property of the RHCP component is shown in Figure 6. From Figure 6, we can see that the maximum light intensity of RHCP appears at z = 180 m, which is different from that of the LHCP component.

#### 3.2. CAB with Off-Axis OVs

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The intensity pattern and phase distribution of the left-hand circular polarized (LHCP) component at different distances. (

**b**,

**c**) show the intensity pattern for z = 100 mm and z = 200 mm, respectively; (

**e**,

**f**) show the phase distribution for z = 100 mm and z = 200 mm, respectively. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**d**).

**Figure 4.**The intensity pattern and phase distribution of the right-hand circular polarized (RHCP) component at different distances. (

**b**,

**c**) show the intensity pattern for z = 100 mm and z = 180 mm, respectively; (

**e**,

**f**) show the phase distribution for z = 100 mm and z = 180 mm, respectively. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**d**).

**Figure 7.**The intensity pattern and phase distribution of the LHCP component at different distances. (

**b**–

**d**) show the intensity pattern for z = 50 mm, z = 100 mm, and z = 200 mm, respectively; (

**f**–

**h**) show the phase distribution for z = 50 mm, z = 100 mm and z = 200 mm, respectively. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**e**). The center of the OV is marked with a red circle.

**Figure 8.**The intensity pattern and phase distribution of the RHCP component at different distances. (

**b**,

**c**) show the intensity pattern for z = 100 mm and z = 180 mm, respectively; (

**e**,

**f**) show the phase distribution for z = 100 mm and z = 180 mm, respectively. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**d**).

**Figure 9.**The intensity patterns of the LHCP component (

**a**–

**c**) and RHCP component (

**d**–

**f**) near the focal plane. (

**a**,

**c**), (

**b**,

**e**) and (

**c**,

**f**) show the intensity patterns for z = 140 mm, z = 145 mm, and z = 150 mm, respectively.

**Figure 10.**The intensity pattern and phase distribution of the LHCP component at different distances. (

**b–d**) show the intensity pattern for z = 50 mm, z = 100 mm, and z = 200 mm; (

**f**–

**h**) show the phase distribution for z = 50 mm, z = 100 mm, and z = 200 mm. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**e**). The center of the optical vortex (OV) is marked with a red circle.

**Figure 11.**The intensity pattern and phase distribution of the RHCP component at different distances. (

**b**,

**c**) show the intensity pattern for z = 100 mm and z = 180 mm, respectively. (

**e**,

**f**) show the phase distribution for z = 100 mm and z = 180 mm, respectively. The intensity pattern and phase distribution of the initial beam are also shown in (

**a**,

**d**).

**Figure 12.**The intensity patterns of the LHCP component (

**a**–

**c**) and RHCP component (

**d**–

**f**) near the focal plane. (

**a**,

**c**), (

**b**,

**e**) and (

**c**,

**f**) show the intensity patterns for z = 140 mm, z = 145 mm, and z = 150 mm, respectively.

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**MDPI and ACS Style**

Zheng, G.; Wu, Q.; He, T.; Zhang, X.
Propagation Characteristics of Circular Airy Vortex Beams in a Uniaxial Crystal along the Optical Axis. *Micromachines* **2022**, *13*, 1006.
https://doi.org/10.3390/mi13071006

**AMA Style**

Zheng G, Wu Q, He T, Zhang X.
Propagation Characteristics of Circular Airy Vortex Beams in a Uniaxial Crystal along the Optical Axis. *Micromachines*. 2022; 13(7):1006.
https://doi.org/10.3390/mi13071006

**Chicago/Turabian Style**

Zheng, Guoliang, Qingyang Wu, Tiefeng He, and Xuhui Zhang.
2022. "Propagation Characteristics of Circular Airy Vortex Beams in a Uniaxial Crystal along the Optical Axis" *Micromachines* 13, no. 7: 1006.
https://doi.org/10.3390/mi13071006