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Propagation of the first order annular Bessel Gaussian beams in a uniaxial crystal along the optical axis governed by the Pockels effect

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Abstract

We investigate the first order annular Bessel Gaussian beams propagating in a strontium barium niobate (SBN) crystal governed by the Pockels effect. The left-hand circularly polarized incident waves propagating along the optical axis in the crystal give rise to the right-hand circularly polarized vortex fields with a topological charge of 2. If an external dc field is applied along the optical axis of the SBN crystal, the anisotropy strength can be controlled rapidly, and the rotational invariance around the optical axis is still maintained. In this case, the normalized intensity in the focal region, the power exchange between left-hand and right-hand components, and the exchange between the spin and orbit contributions of the angular momentum flux can be manipulated. We also consider the case where the incident waves are linearly polarized along the x axis. The cylindrical symmetry of two Cartesian components is broken during propagation due to the anisotropy, and the intensity distribution as well as the polarization state of the beams can be regulated electrically.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The abruptly autofocusing (AAF) waves were first introduced in 2010 that tend to focus in an abrupt fashion [1]. The maximum intensity of the beams remains almost constant during propagation, whereas they suddenly increase by orders of magnitude right before the focus. Different from the self-focusing effect due to nonlinear medium, this autofocusing behavior is a result of the optical field structure itself [2]. Such beams have been experimentally demonstrated in [35] and could be utilized in creating ablation spots [3], optical manipulation [4,6,7], and generating non-linear intense light bullets [5]. In recent years, different classes of AAF beams have been suggested [2,810]. Recently, we introduced a new class of AAF beams called the annular Bessel Gaussian beams (ABGBs) which can autofocus both in the near field and the far field.

As known, the anisotropic media affect the polarization state of radiation greatly and can be used to generate optical vortices [1114]. In this perspective, the study of light propagating in anisotropic media has attracted a good deal of attention. A simple method was proposed to describe the beams which possess cylindrical symmetry propagating along the optical axis of a uniaxial crystal [11]. Especially when the incident vortex fields with a topological charge of $m$ are left-hand circularly polarized, the right-hand circularly polarized vortex fields are generated with a topological charge of $m+2$ [12].

The angular momentum of the radiation has a spin momentum (SAM) part, related to the polarization state, and an orbit momentum (OAM) part, associated with the spatial profile of light intensity and the phase [15]. In 2002, Ciattoni demonstrated that the angular momentum flux along the z axis is conserved during propagation due to the rotational invariance of the crystal around the optical axis, and it can be decomposed into the sum of the intrinsic and orbital contributions [16]. In 2008, Chen proposed a method to control the spin-to-orbital angular momentum conversion due to the Pockels effect in a single strontium barium niobate (SBN) [17]. The anisotropy can be enhanced or weakened by the Pockels effect which offers a rapid way to control the light. Later, the Bessel Gaussian beams, the Laguerre-Gaussian beams and the circular Airy beams propagating in the uniaxial crystal under the Pockels effect were investigated [1820].

In this paper, we investigate the first order ABGBs with autofocusing property in the SBN crystal along the optical axis by means of Pockels effect. The cases for the input beams with circularly polarization and linearly polarization are discussed. Besides, we perform a numerical experiment for the ABGBs in the initial input plane to demonstrate the correctness of the analytical results.

2. Theory

Here we consider the Pockels effect of a vectorial beam under an applied electric field $E_{0}$. For the uniaxial crystal SBN, $\sqrt {\varepsilon _{1}}=\sqrt {\varepsilon _{2}}=n_{o}=2.3117$, $\sqrt {\varepsilon _{3}}=n_{e}=2.2987$; and the nonvanishing electro-optic (EO) coefficients are $\gamma _{13}=\gamma _{23}=67$, $\gamma _{33}=1340$, and $\gamma _{42}=\gamma _{51}=42$ (in picometers per volt) at 632.8nm [21]. The electrical displacement vector can be expressed as $\textbf {D}=\varepsilon _{0}\overset {\leftrightarrow }{\varepsilon }_{r}\textbf {E}+\textbf {P}^{EO}$, where $\varepsilon _{0}$ is the vacuum permittivity, $\overset {\leftrightarrow }{\varepsilon } _{r}=\textrm{diag}(n_{o}^{2},n_{o}^{2},n_{e}^{2})$ is the relative dielectric tensor, and $\textbf {P}^{EO}=2\varepsilon _{0}\chi ^{(2)}(\omega ,0):\textbf {E}\textbf {E}_{0}$ is the polarization contribution of the EO effect. To make good use of $\gamma _{33}$, we assume that the incident beams propagate along the optical axis of the crystal and the applied electric field is applied along the same direction. Using the well-known relationship between the EO tensor elements $\gamma _{ijk} (i,j,k=1,2,3)$ and the second-order nonlinear susceptibilities $\chi _{ijk}^{(2)}(\omega ,0)=-\frac {1}{2}(\varepsilon _{ii}\varepsilon _{jj})\gamma _{ijk}$, one can obtain the expressions: $P_{x}^{EO}=-\varepsilon _{0}n_{o}^{4}\gamma _{13}E_{0}E_{x}$, $P_{y}^{EO}=-\varepsilon _{0}n_{o}^{4}\gamma _{13}E_{0}E_{y}$, and $P_{z}^{EO}=-\varepsilon _{0}n_{e}^{4}\gamma _{33}E_{0}E_{z}$.

According to [17], the wave equation is

$$\triangledown ^{2}\textbf{E}-\triangledown(\triangledown\cdot \textbf{E})+k_{0}^{2}\overset{\leftrightarrow }{\varepsilon }_{r}\textbf{E}+\mu _{0}\omega ^{2}\textbf{P}^{EO}=0,$$
where $k_{0}=2\pi / \lambda$ and $\mu _{0}$ is the magnetic susceptibility in vacuum.

The initial first order ABGBs can be expressed as

$$E(r, z=0) = CJ_{1}(\frac{r_{0}-r}{w_{0}})\textrm{exp}[-a(\frac{r_{0}-r}{w_{0}})^{2}],$$
where $C$ is a constant, $J_{1}$ is the first order Bessel function, $r_{0}$ is related to the beam size, $w_{0}$ is the scale parameter, and $a$ is the decay factor which controls the side rings of the incident beams. Here, we follow Ciattoni’s angular spectrum representation method to solve Eq. (1) [22]. The main idea is to express the fields using two-dimensional Fourier transformation, i.e.,
$$\begin{aligned} \textbf{E}(\textbf{r}_{\perp },z) & =\int \textrm{d}^{2}\textbf{k}_{\perp }\textrm{exp}(i\textbf{k}_{\perp }\cdot \textbf{r}_{\perp })\widetilde{\textbf{E}}(\textbf{k}_{\perp },z),\\ \textbf{P}^{EO}(\textbf{r}_{\perp },z) & =\int \textrm{d}^{2}\textbf{k}_{\perp }\textrm{exp}(i\textbf{k}_{\perp }\cdot \textbf{r}_{\perp })\widetilde{\textbf{P}}^{EO}(\textbf{k}_{\perp },z), \end{aligned}$$
where $\textbf {r}_{\perp }=x\hat {\textbf {e}}_{x}+y\hat {\textbf {e}}_{y}$ and $\textbf {k}_{\perp }=k_{x}\hat {\textbf {e}}_{x}+k_{y}\hat {\textbf {e}}_{y}$ are vectors in the real and frequency space.

Substituting Eq. (3) into Eq. (1), we can arrive the coupling equations that the light field components satisfy,

$$\begin{aligned} \frac{\partial^2 \widetilde{E}_{x}}{\partial z^2}-ik_{x}\frac{\partial \widetilde{E}_{z}}{\partial z}+k_{x}k_{y}\widetilde{E}_{y}+[k_{0}^{2}n_{o}^{2}(1-n_{o}^{2}\gamma _{13}E_{0})-k_{y}^{2}]\widetilde{E}_{x} & =0,\\ \frac{\partial^2 \widetilde{E}_{y}}{\partial z^2}-ik_{y}\frac{\partial \widetilde{E}_{z}}{\partial z}+k_{x}k_{y}\widetilde{E}_{x}+[k_{0}^{2}n_{o}^{2}(1-n_{o}^{2}\gamma _{13}E_{0})-k_{x}^{2}]\widetilde{E}_{y} & =0,\\ [k_{0}^{2}n_{e}^{2}(1-n_{e}^{2}\gamma _{33}E_{0})-k_{\perp }^{2}]\widetilde{E}_{z}-ik_{x}\frac{\partial \widetilde{E}_{x}}{\partial z}-ik_{y}\frac{\partial \widetilde{E}_{y}}{\partial z} & =0. \end{aligned}$$

Here, we first assume the incident field is left-hand circularly polarized. To better describe the circularly polarized beams, we consider the unit vectors, i.e., $\hat {\textbf {e}}_{+}=\frac {1}{\sqrt {2}}(\hat {\textbf {e}}_{x}+i\hat {\textbf {e}}_{y}), \hat {\textbf {e}}_{-}=\frac {1}{\sqrt {2}}(\hat {\textbf {e}}_{x}-i\hat {\textbf {e}}_{y}),$ which are commonly employed to describe left-hand and right-hand circularly polarized states, respectively [23]. An arbitrary transverse vector $\textbf {E}_{\perp }$ can be projected onto both the Cartesian and the circular basis, i.e., $\textbf {E}_{\perp }=E_{x}\hat {\textbf {e}}_{x}+E_{y}\hat {\textbf {e}}_{y} =E_{+}\hat {\textbf {e}}_{+}+E_{-}\hat {\textbf {e}}_{-}$, and we can achieve $E\pm =\frac {1}{\sqrt {2}}(E_{x}\mp iE_{y})$ [19]. Following [12], we can obtain the solution of Eq. (4) with the initial condition $\textbf {E}(\textbf {r}, z=0)=E(r, z=0)\hat {\textbf {e}}_{+}$ as

$$\begin{aligned} E_{+}(r ,z) & =A^{(0)}(r ,z),\\ E_{-}(r ,z) & =\textrm{exp}(i2\phi )A^{(2)}(r ,z), \end{aligned}$$
where
$$\begin{aligned} A^{(0)}(r, z) & =\textrm{exp}(ik_{0}{n_{o}}'z)\pi \int_{0}^{\infty }\textrm{d}kk[\textrm{exp}(-\frac{ik^{2}z}{2k_{0}{n_{o}}'}) + \textrm{exp}(-\frac{ik^{2}{n_{o}}'z}{2k_{0}{ n_{e}'^{2}}})]J_{0}(kr)\widetilde{E}(k),\\ A^{(2)}(r, z) & =\textrm{exp}(ik_{0}{n_{o}}'z)\pi \int_{0}^{\infty }\textrm{d}kk[\textrm{exp}(-\frac{ik^{2}z}{2k_{0}{n_{o}}'}) - \textrm{exp}(-\frac{ik^{2}{n_{o}}'z}{2k_{0} {n_{e}}'^{2}})]J_{2}(kr)\widetilde{E}(k), \end{aligned}$$
$$\begin{aligned} {n_{o}}' & =n_{o}\sqrt{1-n_{o}^2\gamma _{13}E_{0}},\\ {n_{e}}' & =n_{e}\sqrt{1-n_{e}^2\gamma _{33}E_{0}}. \end{aligned}$$

Note that ${n_{o}}'$ and ${n_{e}}'$ in Eq. (7) stand for the efficient refractive indices for ordinary and extraordinary waves, respectively, and

$$\widetilde{E}(k)=\frac{1}{2\pi }\int_{0}^{\infty }\textrm{d}rrJ_{0}(kr)E(r,0)$$
is the zeroth-order Hankel transform.

As we can see, these two fields do not depend on the azimuth angle, so they possess cylindrical symmetry during propagation. The generated right-hand circularly polarized component is a vortex field possessing a topological charge 2. Besides, the electric fields of the left-hand and right-hand components are affected by the applied dc field.

We also consider the situation where the first order ABGBs are linearly polarized, that is $\textbf {E}(\textbf {r} ,z=0) = E(r, z=0)\hat {\textbf {t}}(\alpha )$, where $\hat {\textbf {t}}(\alpha )=\textrm{cos}(\alpha )\hat {\textbf {e}}_{x}+\textrm{sin}(\alpha )\hat {\textbf {e}}_{y}$, a unite vector forming an angle $\alpha$ with the $x$ axis. According to [11], we obtain

$$\textbf{E}(r,\phi ,z)=A^{(0)}(r,z)\hat{\textbf{t}}(\alpha ) + A^{(2)}(r,z)\textbf{R}(\phi )\cdot \hat{\textbf{t}}(\alpha ),$$
where
$$\textbf{R}(\phi )=\begin{bmatrix} \textrm{cos}(2\phi ) & \textrm{sin}(2\phi ) \\ \textrm{sin}(2\phi ) & -\textrm{cos}(2\phi ) \end{bmatrix}.$$

For simplicity, we assume the incident beams are linearly polarized along the $x$ axis, i.e., $\alpha =0$. Therefore, the two Cartesian components of the beams in the polar coordinate system can be expressed as

$$\begin{aligned} E_{x}(r,\phi ,z) & =A^{(0)}(r,z) + A^{(2)}(r,z)\textrm{cos}(2\phi ),\\ E_{y}(r,\phi ,z) & =A^{(2)}(r,z)\textrm{sin}(2\phi ). \end{aligned}$$

From the above equations, one can see that the first order ABGBs lose the circularly symmetry during propagation since $E_{x}$ and $E_{y}$ are dependent on the azimuth angle.

3. Results

Although we can not derive the analytical expression for the Hankel transform of the first order ABGBs due to the complexity of the initial field, we adopt the quasi-discrete Hankel transform method proposed in [24] to calculate Eq. (8), and the result is shown in Fig. 1(a). Using the trapezoid rule, we then calculate Eq. (6) numerically. In our numerical calculations, some parameters are chosen as: $C=1$, $r_{0}=0.2$ mm, $w_{0}=3\mu$ m, $a=0.001$, and $\lambda =632.8$ nm. If there is no other statement, these parameters remain unchanged.

 figure: Fig. 1.

Fig. 1. (a) Hankel transform of the first order ABGBs; (b) analytical result of the intensity distribution; (c) interference pattern of the initial generated beam and a plane wave; (d) normalized intensity versus the $x$ axis obtained from the analytical and numerical results; (e) transverse intensity distribution of the numerical experiment; (f) computer-generated hologram.

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In practice, a spatial light modulator (SLM) which carries a two-dimensional computer-generated hologram (CGH) is placed at the front focal plane of the a 4f system, and at the back focal plane, the ABGBs are obtained [25]. In Figs. 1(c)–1(f), we perform a numerical experiment of the ABGBs at the initial input plane to verify the theoretical analytical results in Fig. 1(b). By calculating the interference between the ABGBs and tilted plane wave [Fig. 1(c)], we obtain the computer-generated hologram via a typical 4f system with numerical spatial filtering in Fig. 1(f). Subsequently, we get the intensity distribution of the numerical experiment in Fig. 1(e). We plot the normalized intensity versus the $x$ axis obtained from the analytical and the numerical results in Fig. 1(d). From Figs. 1(b), 1(d), and 1(e), we can see that the intensity pattern of the numerical experiment agrees well with the analytical result but the magnitude of the latter one is much higher because we only choose the information of the first order bright interference fringe in our numerical experiment. Accordingly, the numerical experiment can certify the correctness of the analytical results.

Figure 2 shows the normalized intensity distributions of the left-hand and the right-hand components in the side view plane when $E_{0}=0$, where the normalized constant is the maximum intensity of the input beams. As we can see, these two components autofocus toward the beam center, and the focal spot size is much smaller than the incident beams. The left-hand component has a main lobe in the beam center, whereas the right-hand component is a vortex field. Owing to the autofocusing property, the generated right-hand component of the first order ABGBs has higher normalized intensity at the focal region than that of the Gaussian beam [12].

 figure: Fig. 2.

Fig. 2. The normalized intensity distribution of (a) $E_{+}$ and (b) $E_{-}$ in the side view plane.

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We plot the maximum normalized intensity versus the propagation distance when $E_{0}=0$ and $E_{0}=3$ kV/mm as shown in Fig. 3, where $I_{max}/I_{0}$ represents the maximum intensity in an arbitrary transverse plane during propagation normalized by the maximum intensity of the input beams. When there is an applied dc field, the normalized intensity of the left-hand component decreases and that of the right-hand component increases. Also note that, if $E_{0}$ is too large, due to the high anisotropy strength, the autofocusing property is weakened and it is difficult to modulate the focus position.

 figure: Fig. 3.

Fig. 3. The maximum normalized intensity of (a) $E_{+}$ and (b) $E_{-}$ versus the propagation distance when $E_{0}=0$ and $E_{0}=3$ kV/mm.

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For a crystal without net electric charge but with nonlinearity, we obtain the continuity equation for optical angular momentum as $\textrm{d}\textbf {L}_{f}/\textrm{d}t=\int _{\sum }^{}F\hat {\textbf {n}}\textrm{d}S+\int _{\tau }^{}\textbf {g}\textrm{d}\textbf {r}$ [16], where $\textbf {L}_{f}$ is the angular momentum of the electromagnetic field stored in the crystal, $F$ is the optical angular momentum flux, and $\textbf {g}$ is the angular momentum density transferred and stored in the crystal. Under the Pockels effect, we have

$$\begin{aligned} \textbf{g} & =\begin{bmatrix} g_{x}\\ g_{y}\\ g_{z} \end{bmatrix} =\begin{bmatrix} T_{yz}-T_{zy}\\ T_{zx}-T_{xz}\\ T_{xy}-T_{yx} \end{bmatrix} =\varepsilon _{0}({{n_{o}}'}^{2}-{{n_{e}}'}^{2})E_{z}\begin{bmatrix} -E_{y}\\ E_{x}\\ 0 \end{bmatrix}\\ & =\varepsilon _{0}[(n_{o}^{2}-n_{e}^{2})+E_{0}(n_{e}^{4}\gamma _{33}-n_{o}^{4}\gamma _{13})]E_{z}\begin{bmatrix} -E_{y}\\ E_{x}\\ 0 \end{bmatrix}. \end{aligned}$$
For the SBN crystal, $n_{o}>n_{e}$. Because of the large $\gamma _{33}$, the anisotropy caused by the Pockels effect [the term $E_{0}(n_{e}^{4}\gamma _{33}-n_{o}^{4}\gamma _{13})$] plays an important role here. Thus, when $E_{0}$ grows, the anisotropy strength increases. In other words, the anisotropy can be controlled rapidly by the applied dc field. Especially when $(n_{o}^{2}-n_{e}^{2})+E_{0}(n_{e}^{4}\gamma _{33}-n_{o}^{4}\gamma _{13})=0$, namely, $E_{0}=-1.6883$ kV/mm, the anisotropy in the crystal will vanish. From Eq. (12), one can see that the applied dc field does not break the rotational invariance around the optical axis, because the second-order susceptibility tenser $\chi ^{(2)}$ of the SBN crystal possesses the same rotational invariance. Owing to this property of the SBN crystals, we can investigate the angular momentum dynamics along the $z$ axis. The spin and the orbital angular momentum fluxes can be described as [16]
$$\begin{aligned} \Phi _{S}(z) & ={n_{o}}'\frac{\varepsilon _{0}c}{2\omega }[W_{+}(z)-W_{-}(z)],\\ \Phi _{O}(z) & ={n_{o}}'\frac{\varepsilon _{0}c}{\omega }W_{-}(z), \end{aligned}$$
where
$$W_{\pm }(z)=\frac{1}{2}W_{+}(0)\pm 4\pi ^{3}\int_{0}^{\infty }\textrm{d}kk\textrm{cos}(\frac{z\Delta }{2k_{0}{n_{o}}'}k^{2})\left | \widetilde{E}_{+}(k) \right |^{2}$$
is the total power carried by the two fields, and $\Delta ={{n_{o}}'}^{2}/{{n_{e}}'}^{2}-1$ is a measure of the anisotropy of the crystal governed by the Pockels effect.

$E_{-}$ is zero initially and increases gradually during propagation because there is an energy coupling between $E_{+}$ and $E_{-}$. To investigate the efficiency of the power exchange between these two fields, we plot Fig. 4 according to Eq. (14). Since the intensity is low after the focus, we just focus on the energy exchange before and in the focal region. Note that when $\Delta =0$, i.e., $E_{0}=-1.6883$ kV/mm, the power exchange stops. One can see that, the Pockels effect with positive external dc field leads to higher exchange efficiency.

 figure: Fig. 4.

Fig. 4. The power exchange between $E_{+}$ (solid lines) and $E_{-}$ (dash lines) with different external dc fields versus the propagation distance. (The unit for $E_{0}$ is kV/mm.)

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Besides, the normalized SAM and OAM fluxes carried by the first order ABGBs are plotted in Fig. 5 using Eq. (13). $\Phi _{o}(z)/\Phi _{s}(0)$ is zero originally, but it grows during propagation because of the anisotropy. Like the power exchange in Fig. 4, the exchange between the fluxes of OAM and SAM is affected by the external field, and the exchange efficiency can be enhanced by applying the proper external field.

 figure: Fig. 5.

Fig. 5. Plots of $\Phi _{s}(z)/\Phi _{s}(0)$ (solid lines) and $\Phi _{o}(z)/\Phi _{s}(0)$ (dash lines) versus the propagation distance with different external dc fields. (The unit for $E_{0}$ is kV/mm.)

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In the following, we will discuss the situation where the incident first order ABGBs are linearly polarized along the $x$ axis. $A^{(2)}$ has the maximum value near $z=16$ mm when $E_{0}=0$ (obtained from Fig. 3). We show the normalized intensity distribution of two Cartesian components in the transverse plane when $z=16$ mm in Fig. 6, where Fig. 6(a) and Fig. 6(b) represent the cases when $E_{0}=0$ and $E_{0}=3$ kV/mm, respectively. It demonstrates that the two Cartesian components lose the circularly symmetry during propagation, and the y-component has four main lobes in the focal region. Besides, the normalized intensity of $E_{y}$ can be increased by applying a proper external field.

 figure: Fig. 6.

Fig. 6. The normalized intensity distribution of (a1)-(b1) $E_{x}$ and (a2)-(b2) $E_{y}$ in the transverse plane when $z=16$ mm. (a) $E_{0}=0$; (b) $E_{0}=3$ kV/mm.

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The polarization states when $E_{0}=0$ and $E_{0}=3$ kV/mm at $z=16$ mm are plotted in Fig. 7. Note that the polarization state in the $x$ and $y$ axes is linearly polarized along the $x$ direction because $E_{y}$ has no contribution in these two axes. Besides, the polarization state can be changed by $E_{0}$, and it is symmetric about the origin.

 figure: Fig. 7.

Fig. 7. Polarization distribution in the transverse plane with (a) $E_{0}=0$ and (b) $E_{0}=3$ kV/mm when $z=16$ mm.

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4. Conclusion

In summary, we have investigated the first order ABGBs propagating in the SBN crystal along the optical axis with an applied dc field $E_{0}$ along the same direction. Under this condition, the left-hand circularly polarized first order ABGBs give rise to the right-hand circularly polarized component which is a vortex field with a topological charge of 2. Meanwhile, the numerical experiment of the ABGBs at the initial input plane has been presented, which is in good agreement with the theoretical analytical result. Due to the autofocusing property of the first order ABGBs, the maximum intensity during propagation normalized by the maximum intensity in the input plane is high compared to the Gaussian beams. The applied dc field can enhance or depress the anisotropy of the crystal, which offers a rapid way to control the power exchange between the left-hand and right-hand components and the exchange of the angular momentum flux between the spin and orbit contributions. However, if $E_{0}$ is too large, the autofocusing dynamic will be weakened. The case where the first order ABGBs are linearly polarized along the $x$ axis is also discussed. There exist four main lobes in the focal region of $E_{y}$, and the normalized intensity of two Cartesian components as well as the polarization state can be modulated by applying an external dc field.

Funding

National Natural Science Foundation of China (11374108, 11775083); Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation (pdjh2020a0149).

Disclosures

The authors declare no conflicts of interest.

References

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References

  • View by:

  1. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [Crossref]
  2. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011).
    [Crossref]
  3. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
    [Crossref]
  4. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
    [Crossref]
  5. P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
    [Crossref]
  6. W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
    [Crossref]
  7. Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing airy beams on a rayleigh particle,” Opt. Express 21(20), 24413–24421 (2013).
    [Crossref]
  8. X. Chen, D. Deng, J. Zhuang, X. Peng, D. Li, L. Zhang, F. Zhao, X. Yang, H. Liu, and G. Wang, “Focusing properties of circle pearcey beams,” Opt. Lett. 43(15), 3626–3629 (2018).
    [Crossref]
  9. J. Zhuang, D. Deng, X. Chen, F. Zhao, X. Peng, D. Li, and L. Zhang, “Spatiotemporal sharply autofocused dual-airy-ring airy gaussian vortex wave packets,” Opt. Lett. 43(2), 222–225 (2018).
    [Crossref]
  10. X. Chen, D. Deng, G. Wang, X. Yang, and H. Liu, “Abruptly autofocused and rotated circular chirp pearcey gaussian vortex beams,” Opt. Lett. 44(4), 955–958 (2019).
    [Crossref]
  11. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
    [Crossref]
  12. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003).
    [Crossref]
  13. A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94(2), 235–244 (2003).
    [Crossref]
  14. M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
    [Crossref]
  15. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  16. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67(3), 036618 (2003).
    [Crossref]
  17. L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33(7), 696–698 (2008).
    [Crossref]
  18. W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express 20(23), 25876–25883 (2012).
    [Crossref]
  19. W. Zhu and W. She, “Electro-optically generating and controlling right- and left-handed circularly polarized multiring modes of light beams,” Opt. Lett. 37(14), 2823–2825 (2012).
    [Crossref]
  20. G. Zheng, S. Xu, Q. Wu, Q. Wang, and Z. Ouyang, “Electro-optical coupling of a circular airy beam in a uniaxial crystal,” Opt. Express 25(13), 14654–14667 (2017).
    [Crossref]
  21. A. Yariv and P. Yeh, Optical waves in crystals (Wiley, 1984).
  22. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001).
    [Crossref]
  23. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
    [Crossref]
  24. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21(1), 53–58 (2004).
    [Crossref]
  25. X. Zhang, P. Li, S. Liu, B. Wei, S. Qi, X. Fan, S. Wang, Y. Zhang, and J. Zhao, “Autofocusing of ring airy beams embedded with off-axial vortex singularities,” Opt. Express 28(6), 7953–7960 (2020).
    [Crossref]

2020 (1)

2019 (2)

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

X. Chen, D. Deng, G. Wang, X. Yang, and H. Liu, “Abruptly autofocused and rotated circular chirp pearcey gaussian vortex beams,” Opt. Lett. 44(4), 955–958 (2019).
[Crossref]

2018 (2)

2017 (1)

2013 (2)

Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing airy beams on a rayleigh particle,” Opt. Express 21(20), 24413–24421 (2013).
[Crossref]

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

2012 (2)

2011 (3)

2010 (1)

2008 (1)

2006 (1)

2004 (1)

2003 (4)

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67(3), 036618 (2003).
[Crossref]

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003).
[Crossref]

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94(2), 235–244 (2003).
[Crossref]

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

2002 (1)

2001 (1)

1992 (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Allen, L.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

Chen, H.

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

Chen, L.

Chen, X.

Chen, Z.

Chremmos, I.

Christodoulides, D. N.

Ciattoni, A.

Cincotti, G.

Couaironand, A.

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

Crosignani, B.

Deng, D.

Dennis, M. R.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

Efremidis, N. K.

Fadeeva, T.

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94(2), 235–244 (2003).
[Crossref]

Fan, X.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Huang, K.

Jiang, Y.

Li, D.

Li, P.

Lin, Z.

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

Liu, H.

Liu, S.

X. Zhang, P. Li, S. Liu, B. Wei, S. Qi, X. Fan, S. Wang, Y. Zhang, and J. Zhao, “Autofocusing of ring airy beams embedded with off-axial vortex singularities,” Opt. Express 28(6), 7953–7960 (2020).
[Crossref]

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

Lu, W.

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

Lu, X.

Mills, M. S.

Ouyang, Z.

Palma, C.

Panagiotopoulos, P.

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

Papazoglou, D. G.

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

Peng, X.

Porto, P. D.

Prakash, J.

Qi, S.

She, W.

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Sun, X.

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

Tzortzakis, S.

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

Volyar, A.

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94(2), 235–244 (2003).
[Crossref]

Wang, G.

Wang, Q.

Wang, S.

Wei, B.

Woerdman, J.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Wu, Q.

Xu, S.

Yang, X.

Yariv, A.

A. Yariv and P. Yeh, Optical waves in crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical waves in crystals (Wiley, 1984).

Zhan, Q.

Zhang, L.

Zhang, P.

Zhang, X.

Zhang, Y.

Zhang, Z.

Zhao, F.

Zhao, J.

Zheng, G.

Zhu, W.

Zhuang, J.

J. Opt. Soc. Am. A (4)

Nat. Commun. (1)

P. Panagiotopoulos, D. G. Papazoglou, A. Couaironand, and S. Tzortzakis, “Sharply autofocused ring-airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622–2627 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (10)

W. Zhu and W. She, “Electro-optically generating and controlling right- and left-handed circularly polarized multiring modes of light beams,” Opt. Lett. 37(14), 2823–2825 (2012).
[Crossref]

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
[Crossref]

L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33(7), 696–698 (2008).
[Crossref]

X. Chen, D. Deng, J. Zhuang, X. Peng, D. Li, L. Zhang, F. Zhao, X. Yang, H. Liu, and G. Wang, “Focusing properties of circle pearcey beams,” Opt. Lett. 43(15), 3626–3629 (2018).
[Crossref]

J. Zhuang, D. Deng, X. Chen, F. Zhao, X. Peng, D. Li, and L. Zhang, “Spatiotemporal sharply autofocused dual-airy-ring airy gaussian vortex wave packets,” Opt. Lett. 43(2), 222–225 (2018).
[Crossref]

X. Chen, D. Deng, G. Wang, X. Yang, and H. Liu, “Abruptly autofocused and rotated circular chirp pearcey gaussian vortex beams,” Opt. Lett. 44(4), 955–958 (2019).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref]

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref]

Opt. Spectrosc. (1)

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94(2), 235–244 (2003).
[Crossref]

Phys. Rev. A (2)

W. Lu, X. Sun, H. Chen, S. Liu, and Z. Lin, “Abruptly autofocusing property and optical manipulation of circular airy beams,” Phys. Rev. A 99(1), 013817 (2019).
[Crossref]

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and transformation of laguerre gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Phys. Rev. E (1)

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67(3), 036618 (2003).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

Other (1)

A. Yariv and P. Yeh, Optical waves in crystals (Wiley, 1984).

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Figures (7)

Fig. 1.
Fig. 1. (a) Hankel transform of the first order ABGBs; (b) analytical result of the intensity distribution; (c) interference pattern of the initial generated beam and a plane wave; (d) normalized intensity versus the $x$ axis obtained from the analytical and numerical results; (e) transverse intensity distribution of the numerical experiment; (f) computer-generated hologram.
Fig. 2.
Fig. 2. The normalized intensity distribution of (a) $E_{+}$ and (b) $E_{-}$ in the side view plane.
Fig. 3.
Fig. 3. The maximum normalized intensity of (a) $E_{+}$ and (b) $E_{-}$ versus the propagation distance when $E_{0}=0$ and $E_{0}=3$ kV/mm.
Fig. 4.
Fig. 4. The power exchange between $E_{+}$ (solid lines) and $E_{-}$ (dash lines) with different external dc fields versus the propagation distance. (The unit for $E_{0}$ is kV/mm.)
Fig. 5.
Fig. 5. Plots of $\Phi _{s}(z)/\Phi _{s}(0)$ (solid lines) and $\Phi _{o}(z)/\Phi _{s}(0)$ (dash lines) versus the propagation distance with different external dc fields. (The unit for $E_{0}$ is kV/mm.)
Fig. 6.
Fig. 6. The normalized intensity distribution of (a1)-(b1) $E_{x}$ and (a2)-(b2) $E_{y}$ in the transverse plane when $z=16$ mm. (a) $E_{0}=0$; (b) $E_{0}=3$ kV/mm.
Fig. 7.
Fig. 7. Polarization distribution in the transverse plane with (a) $E_{0}=0$ and (b) $E_{0}=3$ kV/mm when $z=16$ mm.

Equations (14)

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2 E ( E ) + k 0 2 ε r E + μ 0 ω 2 P E O = 0 ,
E ( r , z = 0 ) = C J 1 ( r 0 r w 0 ) exp [ a ( r 0 r w 0 ) 2 ] ,
E ( r , z ) = d 2 k exp ( i k r ) E ~ ( k , z ) , P E O ( r , z ) = d 2 k exp ( i k r ) P ~ E O ( k , z ) ,
2 E ~ x z 2 i k x E ~ z z + k x k y E ~ y + [ k 0 2 n o 2 ( 1 n o 2 γ 13 E 0 ) k y 2 ] E ~ x = 0 , 2 E ~ y z 2 i k y E ~ z z + k x k y E ~ x + [ k 0 2 n o 2 ( 1 n o 2 γ 13 E 0 ) k x 2 ] E ~ y = 0 , [ k 0 2 n e 2 ( 1 n e 2 γ 33 E 0 ) k 2 ] E ~ z i k x E ~ x z i k y E ~ y z = 0.
E + ( r , z ) = A ( 0 ) ( r , z ) , E ( r , z ) = exp ( i 2 ϕ ) A ( 2 ) ( r , z ) ,
A ( 0 ) ( r , z ) = exp ( i k 0 n o z ) π 0 d k k [ exp ( i k 2 z 2 k 0 n o ) + exp ( i k 2 n o z 2 k 0 n e 2 ) ] J 0 ( k r ) E ~ ( k ) , A ( 2 ) ( r , z ) = exp ( i k 0 n o z ) π 0 d k k [ exp ( i k 2 z 2 k 0 n o ) exp ( i k 2 n o z 2 k 0 n e 2 ) ] J 2 ( k r ) E ~ ( k ) ,
n o = n o 1 n o 2 γ 13 E 0 , n e = n e 1 n e 2 γ 33 E 0 .
E ~ ( k ) = 1 2 π 0 d r r J 0 ( k r ) E ( r , 0 )
E ( r , ϕ , z ) = A ( 0 ) ( r , z ) t ^ ( α ) + A ( 2 ) ( r , z ) R ( ϕ ) t ^ ( α ) ,
R ( ϕ ) = [ cos ( 2 ϕ ) sin ( 2 ϕ ) sin ( 2 ϕ ) cos ( 2 ϕ ) ] .
E x ( r , ϕ , z ) = A ( 0 ) ( r , z ) + A ( 2 ) ( r , z ) cos ( 2 ϕ ) , E y ( r , ϕ , z ) = A ( 2 ) ( r , z ) sin ( 2 ϕ ) .
g = [ g x g y g z ] = [ T y z T z y T z x T x z T x y T y x ] = ε 0 ( n o 2 n e 2 ) E z [ E y E x 0 ] = ε 0 [ ( n o 2 n e 2 ) + E 0 ( n e 4 γ 33 n o 4 γ 13 ) ] E z [ E y E x 0 ] .
Φ S ( z ) = n o ε 0 c 2 ω [ W + ( z ) W ( z ) ] , Φ O ( z ) = n o ε 0 c ω W ( z ) ,
W ± ( z ) = 1 2 W + ( 0 ) ± 4 π 3 0 d k k cos ( z Δ 2 k 0 n o k 2 ) | E ~ + ( k ) | 2

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