Paraxial propagation of the first-order chirped Airy vortex beams in a chiral medium

Jintao Xie; Jianbin Zhang; Junran Ye; Haowei Liu; Zhuoying Liang; Shangjie Long; Kangzhu Zhou; Dongmei Deng

doi:10.1364/OE.26.005845

1. Introduction

The Airy beams have attracted extensive attention since they were predicted by Berry and Balazs [1], and in 2007, the finite energy Airy beams were intensively studied and experimentally demonstrated by Siviloglou and Christodoulides [2, 3]. Subsequently, the Airy beams were studied extensively owing to the unique advantages such as self-acceleration [2, 3], diffraction-free [4] and self-healing [5]. To date, the Airy beam is still a research hotspot and has been used in many applications such as optical cleaning of the micro particles [6], curved plasma channel generation [7, 8], light bullet generation [9, 10] and etc. The vortex beams are laser beams with continuous spiral phase distribution, and the propagation dynamics of the Airy vortex beams have been analyzed theoretically [11] and experimentally [12].

The first experimental demonstration of using simple linear chirped laser pulses was carried by Melinger et al. in the early 1990s [13]. Afterwards, a chirp carried by the Airy beams was introduced by Zhang et al., which led to a transverse displacement of the beams at the phase transition point, but did not change the location of the point in the optical field [14]. The propagation properties of the chirped Bessel waves were studied in 2015 [15]. In 2017, the propagation properties of the chirped Airy beams through left-handed and right-handed material slabs [16] and the gradient-index medium [17] have been studied.

On the other hand, the chiral medium which can be found in biological materials such as human tissue, medicinal materials and certain types of terrestrial vegetation layer has many properties different from the ordinary optical medium [18]. As is well known that when a linearly polarized vortex beam incidents upon a chiral medium, it will divide into a left circularly polarized vortex (LCPV) beam and a right circularly polarized vortex (RCPV) beam, with different phase velocities in the chiral medium [18, 19]. Over recent years, the Airy-Gaussian beams [20], the Airy-Gaussian vortex beams [21] and the Vortex Airy beams [22] in a chiral medium have been studied by researchers. But there is no report on the propagation of the first-order chirped Airy vortex beams (FCAiV) in a chiral medium so far. Hence, here we will mainly investigate the influences of the chirped factor and the chiral factor on the propagation properties of the FCAiV beams in a chiral medium under the paraxial propagation.

The paper is organized as follows. Firstly, based on the Huygens diffraction integral formula, we obtain the propagation expressions of the FCAiV beams in a chiral medium under the paraxial propagation in Sec. 2. Then in Sec. 3, the effects of the chirped parameter and the chiral parameter on different typical properties of the FCAiV beams are analyzed and described. Finally, the paper is concluded in Sec. 4.

2. Analytic solution of the FCAiV beams in a chiral medium

In the coordinate system, the electric field distribution of the FCAiV beams at the initial plane can be expressed as

E (x_{0}, y_{0}, 0) = A_{0} A_{i} (\frac{x_{0}}{w_{1}}) A i (\frac{y_{0}}{w_{2}}) \exp [\frac{a x_{0}}{w_{1}} + \frac{b y_{0}}{w_{2}} + i β (\frac{x_{0}}{w_{1}} + \frac{y_{0}}{w_{2}})] (\frac{x_{0}}{w_{1}} + i \frac{y_{0}}{w_{2}}),

where A₀ represents the complex constant amplitude, a and b are exponential truncation factors ranging from 0 to 1, w₁ and w₂ denote arbitrary transverse scales in the x-direction and y-direction, β is the chirped parameter, and Ai(.) is the Airy function. With the Huygens diffraction integral, the FCAiV beams propagating through an optical ABCD system under the paraxial approximation can be determined as [23]

\begin{array}{l} E (x, y, z) = \frac{i k}{2 π B} \int \int_{- \infty}^{+ \infty} E_{0} (x_{0}, y_{0}, 0) \\ \times \exp {- \frac{i k}{2 B} [A (x_{0}^{2} + y_{0}^{2}) - 2 (x_{0} x + y_{0} y) + D (x^{2} + y^{2})]} d x_{0} d y_{0}, \end{array}

where A, B and D are the compositions of the transfer matrix and the wave number k = 2π/λ (λ is the wavelength of beams in the free space). Substituting Eq. (1) into Eq. (2), we can obtain the expression of the FCAiV beams propagating through an optical ABCD system as

E (x, y, z) = - \frac{A_{0}}{A} \exp [Q (x, y, z)] (K_{1} + K_{2} + K_{3}),

where

\begin{array}{l} Q (x, y, z) = - \frac{i k D}{2 B} (x^{2} + y^{2}) - \frac{1}{4 M} (N_{1}^{2} + N_{2}^{2}) + \frac{1}{8 M^{2}} (\frac{N_{1}}{w_{1}^{3}} + \frac{N_{2}}{w_{2}^{3}}) - \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}}), \\ K_{1} = A i [f (x)] A i [g (y)] [(- \frac{N_{1}}{2 w_{1} M} + \frac{1}{8 w_{1}^{4} M^{2}}) + i (- \frac{N_{2}}{2 w_{2} M} + \frac{1}{8 w_{2}^{4} M^{2}})], \\ K_{2} = - \frac{i}{2 M w_{2}^{2}} A i [f (x)] A i^{'} [g (y)], K_{3} = - \frac{1}{2 M w_{1}^{2}} A i [g (y)] A i^{'} [f (x)], \\ M = - \frac{i k A}{2 B}, N_{1} = \frac{a + i β}{w_{1}} + \frac{i k x}{B}, N_{2} = \frac{b + i β}{w_{2}} + \frac{i k y}{B}, \\ f (x) = \frac{1}{16 w_{1}^{4} M^{2}} - \frac{N_{1}}{2 w_{1} M}, g (y) = \frac{1}{16 w_{2}^{4} M^{2}} - \frac{N_{2}}{2 w_{2} M} . \end{array}

Then, the ABCD matrix of the propagation system in the chiral medium can be expressed as

(\begin{matrix} A_{(L)} & B_{(L)} \\ C_{(L)} & D_{(L)} \end{matrix}) = (\begin{matrix} 1 & z / n_{(L)} \\ 0 & 1 \end{matrix}) a n d (\begin{matrix} A_{(R)} & B_{(R)} \\ C_{(R)} & D_{(R)} \end{matrix}) = (\begin{matrix} 1 & z / n_{(R)} \\ 0 & 1 \end{matrix}),

where n₍_L₎ = n/(1 + nkγ) and n₍_R₎ = n/(1 − nkγ) indicate the refractive indices of the LCPV beams and the RCPV beams, respectively. n is the original refractive index in the chiral medium and γ denotes the chiral parameter. Substituting Eq. (5) into Eq. (3), we can get the analytical expressions of the two components of the FCAiV beams through the chiral medium as E₍_L₎(x, y, z) and E₍_R₎(x, y, z). The total electric field is

E = E_{(L)} (x, y, z) + E_{(R)} (x, y, z) .

The total intensity expression of the FCAiV beams in a chiral medium can be written as

I = {| E_{(L)} (x, y, z) |}^{2} + {| E_{(R)} (x, y, z) |}^{2} + I_{i n t},

with

I_{i n t} = E_{(L)} (x, y, z) E_{(R)}^{*} (x, y, z) + E_{(R)} (x, y, z) E_{(L)}^{*} (x, y, z),

where I_int is the interference term and * represents the complex conjugate.

As the FCAiV beams propagate, the parabolic deflections of the LCPV beams and the RCPV beams are

X_{1, (L, R)} = \frac{z^{2}}{4 k^{2} A w_{1}^{3} n_{(L, R)}^{2}} - \frac{z β}{k w_{1} n_{(L, R)}},

and the velocity of the beams can be described as

V_{1, (L, R)} = \frac{z}{2 k^{2} A w_{1}^{3} n_{(L, R)}^{2}} - \frac{β}{k w_{1} n_{(L, R)}},

where X_1,(_L,R₎ and V_1,(_L,R₎ represent the propagation trajectory and the velocity of the main lobes for the LCPV beams and the RCPV beams in the x-direction or the y-direction, respectively. Furthermore, the propagation trajectory of the optical vortex also can be expressed as

X_{2, (L, R)} = \frac{z^{2}}{2 k^{2} A w_{1}^{3} n_{(L, R)}^{2}} - \frac{z β}{k w_{1} n_{(L, R)}},

and the velocity of the optical vortex can be represented as

V_{2, (L, R)} = \frac{z}{k^{2} A w_{1}^{3} n_{(L, R)}^{2}} - \frac{β}{k w_{1} n_{(L, R)}},

where X_2,(_L,R₎ and V_2,(_L,R₎ denote the propagation trajectory and the velocity of the optical vortex for the LCPV beams and the RCPV beams in the x-direction or the y-direction, respectively.

3. Numerical simulation and discussion

With the analytical results of the propagation expression, we further investigate the propagation properties of the FCAiV beams in a chiral medium.

In the following simulations, we choose the calculation parameters as A₀=1, λ=633nm, w₁=w₂=100µm, a=b=0.1, n =3 and $Z r = k w_{1}^{2} / 2$ is the Rayleigh range. Next, we are going to compare the propagation properties of the LCPV beams, the RCPV beams and the total beams by changing different parameters.

The propagation of the FCAiV beams through a chiral medium with γ=0.16/k, β=0.05 is shown in Fig. 1. From Fig. 1 we can see that the propagation trajectory and the intensity focusing position of the LCPV beams and the RCPV beams are apparently different. The intensity focusing position (18Zr) of the LCPV beams which is shown in Fig. 1(b2) is much closer than that (52Zr) of the RCPV beams which can be seen in Fig. 1(b4). Moreover, it is obvious that the acceleration of the LCPV beams is much bigger than that of the RCPV beams and the total intensity is complicated because of the interference.

Fig. 1 Intensity distributions of the FCAiV beams propagating through the chiral medium with γ=0.16/k, β=0.05. (a1)–(a3) represent the LCPV, RCPV and total intensity, respectively (x = y). (b1–b6) are the transverse intensity patterns taken by the dotted lines in (a1)–(a3).

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In order to investigate the influence of the chiral parameter, the propagation of the FCAiV beams with γ=0.28/k, β=0.05 is depicted in Fig. 2. Comparing with Fig. 1, we can see that when the chiral parameter increases, the acceleration of the LCPV and the RCPV beams changes oppositely, i.e., the acceleration of the LCPV beams increases but that of the RCPV beams decreases. More importantly, the intensity focusing position of the LCPV beams is closer (from 18Zr to 15Zr) but that of the RCPV beams is further (from 52Zr to 160Zr) with the increase of the chiral parameter.

Fig. 2 All are the same as those in Fig. 1 except the intensity positions of choice and the chiral parameter γ= 0.28/k.

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The propagation trajectory of the LCPV, the RCPV and the total beams with different chirped parameter β is presented in Fig. 3. In the Fig. 3, we can easily find that with the first-order chirped parameter increasing, the propagation trajectories of the LCPV beams, the RCPV beams and the total beams decline. Meanwhile, the degressive extent of the LCPV beams is bigger than that of the RCPV beams with the first-order chirped parameter increasing.

Fig. 3 Intensity distributions of (a1)–(a4) the LCPV beams, (b1)–(b4) the RCPV beams and (c1)–(c4) the total beams in the chiral medium with γ=0.16/k, β=0.05, 0.5, 1, 2.

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In Figs. 4 and 5, we mainly discuss the influences of the chiral parameter on the relative position between the main lobe and the optical vortex and the location of the intensity focusing. The intensity of the LCPV beams with different chiral parameters and at different propagation distances is shown in Fig. 4. It is interesting that the LCPV beams are affected by the optical vortex so that there is no intensity in the main lobe of the LCPV beams in Figs. 4(a1) and 4(b1). Because the relative position between the main lobe and the optical vortex becomes more and more further, the intensity of the main lobe will be less affected by the optical vortex so that the intensity of the main lobe will recover gradually in Figs. 4(a2) and 4(b2).

Fig. 4 Intensity distribution of the LCPV beams with β=0.05, (a1)–(a4) γ=0.16/k, (b1)–(b4) γ=0.28/k (x = y).

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Fig. 5 All are the same as those in Fig. 4 except the RCPV beams and the intensity positions of choice (x = y).

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More interestingly, the intensity of the side lobes still flows into the main lobe after recovering in Figs. 4(a3) and 4(b3). Then, when the intensity of the main lobe gathers to a certain extent, it will move into the side lobes with the further increase of the propagation distance and the intensity of the main lobe becomes weak in Figs. 4(a4) and 4(b4).

In addition, comparing Fig. 4(a2) with Fig. 4(b2), we find that as the chiral parameter increases, the intensity of the main lobe will recover faster and that is to say the relative position between the main lobe and the optical vortex will be further and their relative velocity will increase. Therefore, the intensity refocusing position of the LCPV beams with a bigger γ becomes closer.

The intensity of the RCPV beams with different chiral parameters and at different propagation distances is illustrated in Fig. 5. From Figs. 5(a1)–5(a2) and 5(b1)–5(b2), the effect of the optical vortex on the intensity of the RCPV beams is the same as that of the LCPV beams. But quite surprisingly, we find the chiral parameter has an opposite influence on the intensity refocusing position. Comparing Fig. 5(a2) with Fig. 5(b2), it is evident that the propagation distance of the intensity recovering in the main lobe becomes much further when increasing the chiral parameter because the relative position and the relative velocity between the main lobe and the optical vortex becomes much closer and smaller. And obviously, the refocusing distance of the RCPV beams is much further than that of the LCPV beams.

Following Figs. 4 and 5, we study the propagation of the total beams with the same parameters, which is shown in Fig. 6. From Figs. 6(a1), 6(b1), 6(a2) and 6(b2), it is easy to find that when the propagation distance is less than about 4Zr, the intensity of the total beams generated by the interference is similar to the intensity of the LCPV beams and the RCPV beams. With the increase of the propagation distance, the pattern of the total intensity will divide into two Airy-like patterns, as shown in Figs. 6(a3), 6(b3), 6(a4) and 6(b4), due to the relative velocity of the LCPV beams and the RCPV beams. As the propagation distance further increases, the intensity of the total beams mainly distributes in the lobe of the RCPV pattern in Figs. 6(a4) and 6(b4) Comparing Figs. 6(a3) and 6(a4) with Figs. 6(b3) and 6(b4), it is not difficult to discover that the interference term can be affected by chiral parameter γ. When the chiral parameter γ increases, the distance of the interference decreases.

Fig. 6 All are the same as those in Fig. 4 except the total beams and the intensity positions of choice (x = y).

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In order to study further on the interference term, the 3D figure is shown in Fig. 7. From Fig. 7, it is easy to find that when the chiral parameter γ increases, the distance of the interference decreases. Moreover, comparing the three diagrams of Figs. 7(a1)–7(a3), when γ=0.16/k, it is obvious that with the increase of the chirped parameter β, the intensity at z = 20Zr becomes stronger and stronger.

Fig. 7 The interference terms with different chirped parameters and chiral parameters, (a1)–(a3) γ=0.16/k, (b1)–(b3) γ=0.28/k, (a1) and (b1) β=0.05, (a2) and (b2) β= 0.5, (a3) and (b3) β=1 (x = y).

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Based on the Eqs. (9) – (12), we create the trajectory and the velocity diagrams of the LCPV beams and the RCPV beams with their optical vortex in Fig. 8. From Fig. 8, we can apparently see that with the increase of γ, the relative position and the relative velocity between the main lobe and the vortex of the LCPV beams become further and bigger, respectively, but those of the RCPV beams are opposite. It also means that the intensity focusing position of the LCPV beams becomes closer and that of the RCPV beams becomes further.

Fig. 8 The propagation trajectory (a, b) and the velocity (c, d) of the FCAiV beams in a chiral medium with β=0.05, (a) and (c) γ=0.16/k, (b) and (d) γ=0.28/k.

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We can also learn about the influence of the chiral parameter on the trajectory and acceleration of the LCPV beams and the RCPV beams clearly. The propagation trajectory and the acceleration of the LCPV beams ascends and increases respectively with the increase of the chiral parameter γ, which can be seen in the lines with circle from Figs. 8(a)–8(d). However, those properties of the RCPV beams are opposite by comparing the lines with triangle from Figs. 8(a)–8(d). Moreover, it is easy to see that the acceleration of the LCPV beams is much bigger than that of the RCPV beams by comparing the lines with circle and the lines with triangle in Figs. 8(c) or 8(d). It is interesting that the propagation trajectory and the velocity of the optical vortex are always higher and bigger than those of the corresponding LCPV and the RCPV beams.

At last, we investigate the gradient force and the scattering force of the LCPV beams and the RCPV beams, which are relative to momentum changes of the electromagnetic wave. Assuming a micro particle with refractive index n₁ is in a stable state, the gradient force and the scattering force can be expressed as [24]:

{\vec{F}}_{g r a d} (x, y, z) = \frac{2 π n_{2} r_{0}^{3}}{c} (\frac{m^{2} - 1}{m^{2} + 2}) \nabla I (x, y, z),

{\vec{F}}_{s c a t} (x, y, z) = \frac{8 π n_{2} k^{4} r_{0}^{6}}{3 c} {(\frac{m^{2} - 1}{m^{2} + 2})}^{2} I (x, y, z) {\vec{e}}_{z},

with

I (x, y, z) = c n_{2} ε_{0} \frac{{| E (x, y, z) |}^{2}}{2},

where

m = \frac{n_{1}}{n_{2}}

is the relative refractive index of the particle, n₂ is the refractive index of a surrounding medium, r₀ is the radius of the micro particle, c is the light velocity and ε₀ is the permittivity of vacuum.

Here we assume that n₁ = 1.592, r₀ = 50nm. The maximum gradient force distributions of the LCPV beams and the RCPV beams with different chirped parameter β and chiral parameter γ at different propagation distance are displayed in Fig. 9. Comparing Figs. 9(a1) and 9(b1) with Figs. 9(a2) and 9(b2) respectively, it is apparent that the maximum gradient forces of the LCPV and RCPV beams won’t be affected by the chirped parameter. Moreover, comparing Fig. 9(a2) with Fig. 9(a3) and Fig. 9(b2) with Fig. 9(b3), we find that the maximum gradient force of the LCPV beams will entirely decrease but that of the RCPV beams is opposite with the increase of γ. Meanwhile, it is easily to see that the maximum gradient force of the LCPV beams is always smaller than that of the RCPV beams in the same situation.

Fig. 9 Maximum gradient force distribution of the LCPV beams (a1)–(a3) and the RCPV beams (b1)–(b3) during the propagation. (a1) and (b1) with β = 2 and γ = 0.16/k, (a2) and (b2) with β = 1 and γ = 0.16k, (a3) and (b3) with β = 1 and γ = 0.28/k.

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In Fig. 10, we plot the maximum scattering force distribution of the LCPV beams and the RCPV beams with different chirped parameter and chiral parameter at different propagation distances. During the entire propagation process, there is always a peak maximum scattering force which can be seen in all pictures of Fig. 10. Adopting the same analytic methods as the Fig. 9, we find that the maximum scattering forces of the LCPV beams and the RCPV beams are also influenced by the chiral parameter γ but the chirped parameter β. Moreover, with the chiral parameter increasing, the maximum scatting force of the LCPV beams becomes smaller but that of the RCPV beams becomes bigger. Meanwhile, the position of the peak maximum scattering force of the LCPV beams becomes closer but that of the RCPV beams becomes further which can be seen in Figs. 10(a2)–10(a3) and 10(b2)–10(b3).

Fig. 10 All are the same as those in Fig. 9 except the maximum scattering force distribution.

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4. Conclusions

Since the evolution of the intensity and the phase distributions for the Vortex Airy beams have been investigated in Ref. [22], in this paper we mainly study the effects of the first-order chirped and the chiral parameters on the propagation trajectory, the position of intensity focusing, the distance of the interference and the radiation force of the FCAiV beams. Based on the Huygens diffraction integral, the expression of the FCAiV beams in a chiral medium is derived. Our results show that the acceleration of the LCPV beams increases but that of the RCPV beams decreases with the increase of the chiral parameter γ, and the acceleration of the LCPV beams is always much higher than that of the RCPV beams in the same situation. Moreover, when increasing the chiral parameter, the relative position between the main lobe and the optical vortex of the LCPV beams becomes further and the relative velocity becomes bigger but those of the RCPV beams are opposite. Thus, the intensity focusing position of the LCPV beams becomes closer but that of the RCPV beams becomes much further. Quite interestingly, the propagation trajectory of the optical vortex is always higher and the velocity is always bigger than those of their corresponding beams. Besides, with the chiral parameter increasing, the maximum gradient and scattering forces of the LCPV beams will decrease but those of RCPV beams will increase and the position of the peak maximum scattering force of the LCPV beams becomes closer but that of the RCPV beams becomes further during the propagation. When increasing the chirped parameter β, the propagation trajectory of the FCAiV beams declines and the degressive extent of the LCPV beams is bigger than that of the RCPV beams. It is significant that the propagation trajectory, the refocusing position and the radiation force of the FCAiV beams can be changed by varying the value of the chiral parameter γ and the chirped parameter β. And we believe that these propagation properties of the FCAiV beams in a chiral medium may have potential applications in areas such as optical micromanipulation and optical sorting.

Funding

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

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Paraxial propagation of the first-order chirped Airy vortex beams in a chiral medium

Abstract

1. Introduction

2. Analytic solution of the FCAiV beams in a chiral medium

3. Numerical simulation and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (10)

Equations (15)

Optics Express