Electro-optical coupling of a circular Airy beam in a uniaxial crystal

Guoliang Zheng; Shixiang Xu; Qingyang Wu; Qiong Wang; Zhengbiao Ouyang

doi:10.1364/OE.25.014654

1. Introduction

A circular Airy beam (CAB), i.e., a circularly symmetric beam with an Airy radial profile, has stirred wide interest in the past few years because of the abruptly autofocusing property [1–5]. The abruptly autofocusing property allows CABs to focus their intensity by an increase of orders of magnitude immediately before a target, which is applicable to biomedical treatment [6] and nonlinear optical processes [7]. CABs can also be used to trap and guide micro-particles [8,9] and to form special beams such as “bottle” beams [10], optical “needles” [11], optical “cages” [11], optical dark “channels” [11], and optical “bullets” [7]. Therefore, controlling the properties of CABs is a powerful way to develop the new applications of CABs such as modulating and switching. 1D and 2D Airy beams can be generated from a broad Gaussian beam through a Fourier Transform (FT) for a imposed cubic phase, which can be realized by several methods such as using spatial light modulator (SLM) and binary phase mask [12,13]. Fortunately, a CAB can also be generated in Fourier space by using a highly accurate closed-form expression for the FT of a CAB [10]. By modifying the Fourier spectrum of CABs, one can enhance the abruptly autofocusing property [14] and produce some interesting beams, such as the optical “needle” and optical “bottle” [10,15]. Recent research has shown that the propagation trajectory of CABs can be controlled and manipulated by potentials [16–18]. But there is still lack of a way in controlling the propagation properties of Airy beam. As we know, linear electro-optic (LEO) effect is extensively used to control the properties of light, such as polarization, phase and intensity. Therefore, it is expected that LEO effect can play its role in controlling a CAB.

On the other hand, most wave coupling theories describing the LEO effect are based on the plane-wave model [19,20]. This treatment is valid only when the crystal length is much shorter than the confocal parameter of the light beam such that the beam width keeps approximately constant within the crystal. This condition is not fulfilled in the Airy beam, where the beam is often micron sized or even smaller. In this case, the transverse distribution of the optical field should be taken into account. In fact, LEO effect for a Gaussian or Gaussian-like beam considering transverse distribution in a uniaxial crystal [21–25] and an optical superlattice [26] have been numerically or experimentally studied. In addition, a simpler approach that addresses paraxial propagation in a uniaxial crystal by FT was recently developed by Ciattoni et al [27–29]. With the inspiration of [27], Airy beams [30], Airy vortex beams [31], Airy Gaussian beams [32], Airy Gaussian vortex beams [33], and CABs [34] propagating through uniaxial crystals have been investigated. However, to our best knowledge, the LEO effect for CABs has never been reported. In the remainder of this paper, we will focus on the EOC of a CAB in uniaxial crystals. We present the evolution of a CAB with LEO effect considering the transverse distribution by means of 2D FT. We find that the abrupt autofocusing of the CAB can be enhanced or weakened by EOC. The exchange between SAM and OAM, the polarization state and intensity pattern can also be controlled electrically. Since LEO effect usually has very short response time (tens of nanoseconds), we believe that our work presents an efficient and fast way to control the propagation characteristics of a CAB, which has potential application in micromanipulation.

2. Theory

Supposing that all other second-order nonlinear effects are so weak because of phase mismatch that only the LEO effect must be considered, we can derive from Maxwell’s equations that

\nabla^{2} E (r) - \nabla [\nabla \cdot E (r)] + k_{0}^{2} \overset{\leftrightarrow}{ε} E (r) + μ_{0} ω^{2} P^{E O} (r) = 0,

where k₀ = ω/c, ω is the angular frequency of light, μ₀ and c are the magnetic susceptibility and the speed of light in vacuum,

\overset{\leftrightarrow}{ε}

is the relative dielectric tensor, given by

\overset{\leftrightarrow}{ε} =diag(n_{o}^{2}, n_{o}^{2}, n_{e}^{2})

for uniaxial crystals,

P^{E O} = ε_{0} χ^{(2)} (- ω, ω, 0) : E E_{0}

is the nonlinear polarization corresponding to the LEO effect, E₀ is the external electric field, and

χ^{(2)} (- ω, ω, 0)

is the second-order susceptibility tensor of the LEO effect. To observe the EOC effect, the incident light propagates along the optical axis (z-axis) for the phase matching. To obtain the exact solution of Eq. (1), we expand the optical fields E and P^EO into the two-dimensional Fourier integrals

E (r) = E (r_{⊥}, z) = \int d^{2} k_{⊥}^{2} \exp (i k_{⊥} \cdot r_{⊥}) \tilde{E} (k_{⊥}, z),

and

P (r) = P (r_{⊥}, z) = \int d^{2} k_{⊥}^{2} \exp (i k_{⊥} \cdot r_{⊥}) \tilde{P} (k_{⊥}, z) .

Substituting Eqs. (2) and (3) into Eq. (1) and inverting the obtained Fourier integrals, we obtain [21, 27]

(\frac{\partial^{2}}{\partial z^{2}} + k_{0}^{2} n_{o}^{2} - k_{y}^{2}) {\tilde{E}}_{x} + k_{x} k_{y} \tilde{E} {}_{y}- i k_{x} \frac{\partial}{\partial z} {\tilde{E}}_{z} + μ_{0} ω^{2} {\tilde{P}}_{x}^{E O} = 0,

(\frac{\partial^{2}}{\partial z^{2}} + k_{0}^{2} n_{o}^{2} - k_{x}^{2}) {\tilde{E}}_{y} + k_{x} k_{y} \tilde{E} {}_{x}- i k_{y} \frac{\partial}{\partial z} {\tilde{E}}_{z} + μ_{0} ω^{2} {\tilde{P}}_{y}^{E O} = 0,

(k_{0}^{2} n_{e}^{2} - k_{⊥}^{2}) {\tilde{E}}_{z} - i k_{x} \frac{\partial}{\partial z} {\tilde{E}}_{x} - i k_{y} \frac{\partial}{\partial z} {\tilde{E}}_{y} + μ_{0} ω^{2} {\tilde{P}}_{z}^{E O} = 0.

Here,

{\tilde{E}}_{⊥} (k_{⊥}, z) = {(2 π)}^{- 2} \int d^{2} r_{⊥} \exp (- i k_{⊥} \cdot r_{⊥}) E_{⊥} (r_{⊥}, z)

, and

{\tilde{P}}_{i}^{E O} = ε_{0} \sum_{j k} χ_{i j k}^{(2)} (- ω, ω, 0) {\tilde{E}}_{j} (ω) \otimes {\tilde{E}}_{k} (0)

(

\otimes

denotes the convolution operator). The solution expressions of Eq. (4) depend on the specific expression of

{\tilde{P}}_{i}^{E O}

because

{\tilde{P}}_{i}^{E O}

may involve one, two or all of

{\tilde{E}}_{i}

(i = x, y, z). For the uniaxial electro-optical SBN crystal, n_o = 2.3117, n_e = 2.2987, and the non-vanishing electro-optic coefficients γ₁₃ = γ₂₃ = 67, γ₃₃ = 1340, and γ₄₂ = γ₅₁ = 42 (in pm/V) at 632.8 nm [35]. To take advantage of the largest electro-optic coefficient γ₃₃, we apply the external electric field along the z-axis. In this case, we have

{\tilde{P}}_{x}^{E O} = - ε_{0} n_{0}^{2} n_{e}^{2} γ_{13} E_{0} {\tilde{E}}_{x}

,

{\tilde{P}}_{y}^{E O} = - ε_{0} n_{0}^{2} n_{e}^{2} γ_{13} E_{0} {\tilde{E}}_{y}

and

{\tilde{P}}_{z}^{E O} = - ε_{0} n_{e}^{4} γ_{33} E_{0} {\tilde{E}}_{z}

. We can see that

{\tilde{P}}_{i}^{E O}

only involves the optical field component

{\tilde{E}}_{i}

. Therefore, the wave coupling equations shown in Eq. (4) can be simplified as

\frac{\partial^{2}}{\partial z^{2}} {\tilde{E}}_{x} - i k_{x} \frac{\partial}{\partial z} {\tilde{E}}_{z} + k_{x} k_{y} {\tilde{E}}_{y} + [k_{0}^{2} n_{0}^{2} (1 - n_{e}^{2} γ_{13} E_{0}) - k_{y}^{2}] {\tilde{E}}_{x} = 0,

\frac{\partial^{2}}{\partial z^{2}} {\tilde{E}}_{y} - i k_{y} \frac{\partial}{\partial z} {\tilde{E}}_{z} + k_{x} k_{y} {\tilde{E}}_{x} + [k_{0}^{2} n_{0}^{2} (1 - n_{e}^{2} γ_{13} E_{0}) - k_{x}^{2}] {\tilde{E}}_{y} = 0,

[k_{0}^{2} n_{e}^{2} (1 - n_{e}^{2} γ_{33} E_{0}) - k_{⊥}^{2}] {\tilde{E}}_{z} - i k_{x} \frac{\partial}{\partial z} {\tilde{E}}_{x} - i k_{y} \frac{\partial}{\partial z} {\tilde{E}}_{y} = 0.

Because the initial boundary of the CAB is circularly symmetric, i.e., $E (r, 0) = E (r, 0)$ , the Fourier transform can be expressed as a zeroth-order Hankel transform,

\tilde{E} (k) = \frac{1}{2 π} \int_{0}^{\infty} d r r J_{0} (k r) E (r, 0),

where J₀ stands for the zeroth order Bessel function of the first kind. The complex amplitude E(r) of the CAB at the input plane can be expressed as

E (r, 0) = C \cdot A i (\frac{r_{0} - r}{w}) \exp (a \frac{r_{0} - r}{w}),

where C is a constant, while Ai, r₀, r, w and a stand for the Airy function, initial radius of the main ring, radial distance, radially scaled coefficient, and decay parameter, respectively. By relying on a suitable plane wave angular spectrum representation of the beam, the closed-form approximation of the CAB spatial FT can be expressed as [10]

\tilde{E} (k) = C \cdot w^{2} (\frac{r_{0}}{w} + k^{2} w^{2}) \exp (- a k^{2} w^{2}) \sqrt{\frac{3 k r_{0} + k^{3} w^{3}}{3 k r_{0} + 3 k^{3} w^{3}}} J_{0} (k r_{0} + \frac{k^{3} w^{3}}{3})

where k is the radial spatial frequency.

On the other hand, Ciattoni et al have developed a simple approach to deal with paraxial propagation along the optical axis of a uniaxial crystal [27–29]. They directly derived the plane wave angular spectrum of the superimposed ordinary and extraordinary monochromatic plane waves that are present inside a uniaxial crystal. The approach allows us to solve the boundary-value problem, that is, to obtain the field of beam propagating in the uniaxial crystal (z>0) when its distribution is defined on the initial plane (z = 0). The approach is still valid here because the external electric field applied along the optical axis does not change the symmetry of SBN crystal. For convenience to find the solution of Eq. (5), we express r_⊥ and k_⊥ by $r_{⊥} = r_{⊥} ({\hat{e}}_{x} \cos ϕ + {\hat{e}}_{y} \sin ϕ)$ and $k_{⊥} = k ({\hat{e}}_{x} \cos θ + {\hat{e}}_{y} \sin θ)$ . Following the approach presented in [21,27–29], it is very easy to obtain the solution of Eq. (5)

E (r, ϕ, z) = \exp (i k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z) [Α^{(0)} (r, z) + R (ϕ) Α^{(2)} (r, z)],

with

A^{(n)} (r, z) = π \int_{0}^{\infty} d k [\exp (- \frac{i z k^{2}}{2 k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}}}) + \exp (- \frac{i z n_{0} k^{2} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}}}{2 k_{0} n_{e}^{2} (1 - n_{e}^{2} γ_{33} E_{0})})] k J_{n} (k r) \tilde{E} (k),

and

R (ϕ) = [\begin{matrix} \cos 2 ϕ & \sin 2 ϕ \\ \sin 2 ϕ & - \cos 2 ϕ \end{matrix}]

Here, J_n in Eq. (10) stands for the n-th order Bessel function of the first kind. From Eqs. (9)-(11), we can see that

E (r, ϕ, z)

is controlled by the external electric field E₀ and that A⁽ⁿ⁾ plays the key role here. Unlike the symmetry of the first term A⁽⁰⁾ of Eq. (9), the second term

R (ϕ) Α^{(2)}

is azimuth angle (ϕ)-dependent, which means the distribution of the optical electric field is asymmetric during the propagation.

Suppose that the initial beam is linearly polarized, that is, $E (r, 0) = E (r, 0) \hat{t} (α)$ with $\hat{t} (α) = \cos α {\hat{e}}_{x} + \sin α {\hat{e}}_{y}$ , which is a unit vector that forms an angle α with the x-axis. Inserting $E (r, 0)$ into Eq. (6) and the corresponding spectrum $\tilde{E} (k) = \tilde{E} (k) \hat{t} (α)$ into Eq. (10) and Eq. (9), we can get the two Cartesian components of the beam in the polar coordinate system [29],

\begin{array}{l} E_{x} (r, ϕ, z) = \exp (i k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z) \cdot [A^{(0)} (r, z) \cos α + A^{(2)} (r, z) \cos (2 ϕ - α)], \\ E_{y} (r, ϕ, z) = \exp (i k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z) \cdot [A^{(0)} (r, z) \sin α + A^{(2)} (r, z) \sin (2 ϕ - α)] . \end{array}

From Eq. (12), we can see that E_x and E_y are coupled through A⁽⁰⁾ and A⁽²⁾, and both of them can be controlled by the external electric field. Additionally, E_x and E_y are asymmetric due to their dependences of the azimuth angle ϕ. Interestingly, Eq. (12) reveals that the beam can be regarded as the superposition of two beams with different polarizations: one is linear polarization while the other is vortex polarization with a topological charge of 2 [34].

To obtain a more suitable solution representation for a circularly polarized incident field, we introduce two complex unit vectors, ${\hat{e}}_{+} = \sqrt{2} / 2 ({\hat{e}}_{x} + i {\hat{e}}_{y})$ , and ${\hat{e}}_{-} = \sqrt{2} / 2 ({\hat{e}}_{x} - i {\hat{e}}_{y})$ , corresponding to left-handed and right-handed circularly polarized light waves, respectively. Without loss of generality, we assume that the incident field is a left-handed circularly polarized beam without a vortex, namely, $E (r, 0) = E (r, 0) {\hat{e}}_{+}$ , so the solution of Eq. (5) can be expressed as [21,28]:

\begin{array}{l} E_{+}^{} (r, ϕ, z) = \exp (i k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z) \cdot A^{(0)} (r, z), \\ E_{-} (r, ϕ, z) = \exp [i (k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z + 2 φ)] A^{(2)} (r, z) . \end{array}

Here, A⁽⁰⁾ (r, z) and A⁽²⁾(r, z) are the same as those in Eq. (12). From Eq. (13), one can see that the intensity distributions of both the left-handed and the right-handed component are symmetric, meanwhile, the right-handed component is a vortex beam of topological charge 2 [28]. As we know, a left- and right-handed circularly polarized photons possess their SAMs of

\pm ℏ

, respectively. The OAM depends on the helical phase structure exp(ilϕ) of light, which carries an eigenvalue of

l ℏ

per photon [36]. From Eq. (13), we can see that the right-handed component is with an OAM of

2 ℏ

per photon. That is to say, the exchange of the two angular momentums happens during propagation, and its efficiency can be controlled by electro-optic effect. To investigate the efficiency of the angular momentum exchange, we evaluate the total power carried by the two fields [28]

W_{\pm} (z) = \frac{1}{2} W_{+} (0) \pm 4 π \int_{0}^{\infty} d k k \cos (\frac{z Δ}{2 k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}}} k^{2}) {| {\tilde{E}}_{+} (k) |}^{2},

where

Δ = n_{o}^{2} (1 - n_{e}^{2} γ_{13} E_{0}) / n_{e}^{2} (1 - n_{e}^{2} γ_{33} E_{0}) - 1

. We can see that

Δ

plays the key role in the power evolution of the two field, however, it can be adjusted by LEO effect. Especially, we have

W_{+} (z) = W_{+} (0)

and

W_{-} (z) = 0

when

Δ = 0

, which means that the power exchange between the two fields is forbidden. Correspondingly, the fluxes of SAM

Φ_{s}

and OAM

Φ_{o}

carried by the light field turn out to be [28]

\begin{array}{l} Φ_{s} (z) = n_{o} \frac{ε_{0} c}{2 ω} [W_{+} (z) - W_{-} (z)], \\ Φ_{o} (z) =2 n_{o} \frac{ε_{0} c}{2 ω} W_{-} (z) . \end{array}

Unfortunately, it is almost impossible to present an analytical description of the power dynamics for a CAB beam as that for a Gaussian beam because the FT of CAB is much complicated than that of Gaussian beams.

3. Numerical Calculations

Taking advantage of Eqs. (12) and (13), the propagation characteristics of the beam can be analysed. However, the most important factor in Eqs. (12) and (13) A⁽ⁿ⁾, presented in Eq. (10), does not allow an analytic treatment. We use an accurate numerical method to perform the integral related to oscillating function $\tilde{E} (k)$ and obtain the numerical solution. Similarly, the numerical integrals are also needed for calculating SAM $Φ_{s}$ and OAM $Φ_{o}$ using Eqs. (14) and (15). Here, we adopt adaptive Simpson quadrature to compute the integrals and use MATLAB software to perform the simulations.

Here, we investigate numerically the EOC of a circularly and linearly polarized CAB in a SBN crystal. The beam propagates along the optical axis (z-axis) of the crystal. The beam parameters are selected as follows: r₀ = 0.2 mm, w = 8 µm, a = 0.05, wavelength $λ$ = 632.8 nm. We assume that I_m is the maximum intensity of the incident beam at the initial plane. To see the change of the intensity during the propagation, all intensities are normalized by I_m in our calculations. First, we focus on the situation where the input CAB is left-handed circularly polarized. Since the intensity distributions of E₊ and E_- are cylindrically symmetric, the intensity distributions can easily be shown by displaying the dependence of the intensities of E₊ and E_- on the radial distance r. Figure 1 shows the intensity distributions of E₊ and E_- at the propagation distance z = 10mm with different external electric field. From Fig. 1, we can see that E_- is much weaker than E₊ with absence of an external electric field. When E₀ = 5 kV/mm, the maximum intensity of the E_-’ main lobe is above 10 times larger than that without external electric field, and become comparable with that of the E₊’ main lobe. If E₀ increases further to10 kV/mm, the first lobe of E₊ weakens, while the second one becomes the main lobe, which is stronger than the main lobe of E_-. For E₀ = 15 kV/mm, the second lobe of E_- becomes dominant, whereas all lobes of E₊ turns to diminish. That is to say, the exchange between the two components and the intensity pattern can be controlled by the external electric field.

Fig. 1 Intensity distributions (I/I_m) of E₊ (dark solid line) and E_- (red dash line) with different external electric fields: (a) E₀ = 0 kV/mm; (b) E₀ = 5 kV/mm; (c) E₀ = 10 kV/mm; (d) E₀ = 15 kV/mm.

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To observe the influence of EOC on the beam’ propagation dynamics, Figs. 2 and 3 provide the detailed plots of the evolutions of E₊ and E_- with 0 kV/mm and 5kV/mm external electric field, respectively. Figures 2 and 3 obviously shows that the radii of both E₊ and E_- become increasingly smaller with the propagation distance, whereas the intensities rapidly increases at the focal point, which means the abruptly autofocusing happens in both E₊ and E_-. From Fig. 2(a), we can see that the width of beam E₊ becomes very small from the focal point (about z = 15mm) while the depth of focal spot lasts about 10 mm long along the propagation axis. Since the beam E₊ is solid near the focal point, it becomes a “needle-like” beam [8, 10, 11, 15, 37]. Similarly, we can see from Fig. 2(b) that the beam E_- becomes narrow but still hollow near the focal point, and forms a “cage” or “bottle” beam [11, 38]. This can be interpreted that the vortex phase of E_- does not allow the beam to focus into a spot. Moreover, the intensities of optical “needle” and “cage” can be controlled electrically, which is useful in micromanipulation. In the absence of external electric field, one can see from Figs. 2(a) and 2(b) that abruptly autofocusing is strong for E₊, while E_- is weak due to the weak medium anisotropy [28]. When the external electric field is applied, we can see that from Fig. 3(a) and 3(b) that the abruptly autofocusing of E₊ is weakened while that of E_- is enhanced because E₊ couple its power to E_- through EOC during propagation.

Fig. 2 Propagation dynamics of E₊ (a) and E_- (b) with 0 kV/mm external electric field.

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Fig. 3 Propagation dynamics of E₊ (a) and E_- (b) with 5 kV/mm external electric field..

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Figure 4 shows the intensity patterns of the beam at the propagation distance z = 10 mm when the external electric field is E₀ = 5 kV/mm. As comparison, we plot the distribution of the beam at the initial plane (z = 0) as shown in Fig. 4(a). Due to the autofocusing, the radii of E₊ and E_- at z = 10 mm are smaller than those at initial plane. The maximum intensity of E_- is stronger than that of E₊, which agrees with the result in Fig. 1(b). To visualize the topological charge carried by the left-handed component E_-, Fig. 4(b) presents the interference pattern between E_- ( $E_{-} (r, φ, z) / \max (| E_{-} (r, φ, z) |)$ ) and the plane wave ( $\exp [i (k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z)]$ ). One can see that the left-handed component carries a vortex phase with a topological charge of 2.

Fig. 4 Intensity patterns (I/I_m) of the beam when the external electric field E₀ = 5kV/mm at z = 10 mm: (a) initial distribution of the beam; (b) interference pattern of ${| E_{-} (r, φ, z) / \max (| E_{-} (r, φ, z) |) + \exp [i (k_{0} n_{0} \sqrt{1 - n_{e}^{2} γ_{13} E_{0}} z)] |}^{2}$ ; (c) intensity distributions of E₊; (d) intensity distributions of E_-.

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Further, we obtain the evolutions of SAM and OAM using Eqs. (14) and (15) as shown in Fig. 5, where Φ_s gradually decreases while Φ_o symmetrically increases. This can be interpreted that the left-handed circularly polarized component with initial zero OAM and SAM of $+ ℏ$ per photon is gradually converted into the right-handed component with OAM of $2 ℏ$ and SAM of $- ℏ$ per photon. However, the total angular momentum Φ_s + Φ_o keeps constant because the electro-optic effect here does not break the rotational invariance around the optical axis [21]. Interestingly, Fig. 5 also indicates that the efficiency of angular momentum exchange can be controlled by external electric field. Especially, when E₀ = −1.687kV/mm, EOC forbids the exchange between SAM and OAM, which is similar to the case of Gaussian beam [21].

Fig. 5 Plots of Φ_s(z)/Φ_s(0) (solid line) and Φ_o(z)/Φ_s(0) (dash line) of a CAB vs propagation distance with different external electric field.

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In the following, we study the other situation where the input CAB is linearly polarized along the x-axis of the crystal. Figures 6 and 7 respectively show the intensity distributions of E_x and E_y with different external electric fields at the propagation distance z = 10 mm. Similar to the circularly polarized CAB, the autofocusing also occurs and make the radius of the beam smaller than that at initial plane. From Fig. 6, we can see that the intensity pattern of E_x is no longer concentric rings when the external electric field is applied; instead, it varies with external electric field. From Fig. 7, we can see that the intensity of E_y is low without an electric field but becomes higher with the external electric field because of the EOC. In addition, we can see that the intensity patterns of E_x and E_y change with the external electric field.

Fig. 6 Intensity patterns of I_x/I_m with different external electric fields at z = 10 mm: (a) E₀ = 0 kV/mm; (b) E₀ = 5 kV/mm; (c) E₀ = 10 kV/mm; (d) E₀ = 15 kV/mm.

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Fig. 7 Intensity patterns of I_y/I_m with different external electric fields at z = 10 mm: (a) E₀ = 0 kV/mm; (b) E₀ = 5 kV/mm; (c) E₀ = 10 kV/mm; (d) E₀ = 15 kV/mm.

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Figure 8 shows the polarization distribution of beam with different external electric field at the propagation distance z = 10mm. As expected, the polarization state becomes spatially inhomogeneous, and possesses vortex with a topological charge of 2. This result agrees with Eq. (12), which shows obviously that the beam contains one component of vortex polarization [34]. More importantly, the polarization distribution of beam can be controlled by external electric field.

Fig. 8 Polarization distribution with different external electric fields at z = 10 mm: (a) E₀ = 0 kV/mm; (b) E₀ = 5 kV/mm. The size of the ellipse denotes the beam intensity.

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4. Conclusion

In summary, as far as we know, this paper reports firstly the theoretical investigation on the EOC of a circular Airy beam in a uniaxial crystal. Basing on the deduction of the wave coupling equations considering EOC, we simulate numerically the propagations of CABs for a left-handed circularly and a linearly polarized incident CABs in a SBN crystal. Our results show EOC causes a left-handed circularly polarized beam to couple toward a right-handed circularly polarized beam with a vortex topological charge of 2, which opens a novel way to generate vortex light beam. It is worth noting that the EOC can be used to enhance or weaken, thereby manipulate the abrupt autofocusing, the most important property for CAB during their propagation. Interestingly, EOC also plays a role in forming and controlling an optical “needle” and “cage” for both left-handed and right-handed components near the focal point, which can find its application in micromanipulation. In addition, EOC can influence or even forbid the exchange between SAM and OAM. For the case of linearly polarized CAB, EOC can be applied to control the powers exchange of the two Cartesian field components, as well as the intensity pattern and polarization distribution of the CAB. Excitingly, the polarization state can become spatially inhomogeneous, and possess vortex with a topological charge of 2 during propagation. Anyway, our work finds a very useful method to control an Airy beam.

Funding

National Natural Science Foundation of China (NSFC) (11404220, 61275101, 61605128); Shenzhen Science and Technology Development Fund (JCYJ20160308093947132).

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Electro-optical coupling of a circular Airy beam in a uniaxial crystal

Abstract

1. Introduction

2. Theory

3. Numerical Calculations

4. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (19)

Optics Express