Effects of the cosine complex variable function on the Airy-Gaussian and Airy-Gaussian-vortex beams in a chiral medium

Jintao Xie; Kuangling Guo; Feng Ye; Shijie Chen; Xiaolin Wu; Jianbin Zhang; Dongmei Deng

doi:10.1364/OSAC.2.000424

1. Introduction

In 1979, Berry and Balazs came up with the Airy beams in theory by solving the Schrödinger equation [1]. It was not until 2007 that the finite energy Airy beams were first generated experimentally by Siviloglou and Christodoulides [2, 3]. The Airy beams became a research hotspot owing to their fascinating properties such as self-acceleration [2,3], diffraction-free [4] and self-healing [5]. As of the end of now, the Airy beams have been used in many applications including optical micro-manipulation [6], plasma channel generation [7,8], light bullet generation [9,10] and light-sheet microscopy [11].

For describing the Airy beams more practically, the Airy-Gaussian beams were introduced [12], which could be achieved by letting a finite energy Airy beam pass through the Gaussian aperture. They also carry finite power and still keep the non-diffracting property within a certain propagation distance [12]. In addition, the optical vortex as a factor acting on the Airy beams was well investigated by Dai et al., which showed that the Airy vortex beams also had a continuous spiral phase distribution and the main lobe of the beams could be destroyed because of optical vortex [13,14].

In contrast to ordinary medium, chiral medium, which can be found in biological materials [15], has many important properties in optical field. It is well known that when a linearly polarized beam incidents upon a slab of chiral medium, it will split into a left circularly polarized (LCP) beam and a right circularly polarized (RCP) beam, whose phase velocities are different [15,16].

To date, many works about the effects of the Gaussian factor and the optical vortex on the Airy beams propagating through chiral medium have been introduced [17–20]. For finding out some special intensity distribution beams to explore some new useful results, we have considered many practical functions to be an effective factor acts on some beams. We think the cosine complex variable function is one of them which can control the intensity distributions of some beams and may achieve a single direction intensity distribution. Here are some analyses as below. From the cosh-Airy beams introduced by Li et al we can see that the cosh-Airy beams can be regarded as a superposition of two Airy beams with different decay factors [21, 22]. In addition, it is well known that the cosh function can be replaced by the cosine function through adding an imaginary number on the independent variable. Meanwhile, because of the Euler’s formula, the cosine function can be substituted for a sum of two conjugated exponentials, which may be an important reason to achieve a single direction intensity distribution when we choose the complex variable into the cosine function. Besides, in our view, the cosine complex variable function can be added on the Airy-Gaussian and Airy-Gaussian-vortex beams by using spatial light modulation [23,24].

Based on the thoughts above, we devote our minds to figure out the influences of the cosine complex variable function on the Airy-Gaussian beams and Airy-Gaussian-vortex beams propagating in a chiral medium in this paper.

The paper is organized as follows. In Sec. 2, the analytical expressions for the cos- and Airy-Gaussian beams, cos- and Airy-Gaussian-vortex beams propagating in a chiral medium are derived. In Sec. 3, the effects of the cosine factor on the Airy-Gaussian and Airy-Gaussian-vortex beams propagating in a chiral medium are outstood and the relations of central lobe and vortex positions of the cos- and Airy-Gaussian-vortex beams are analyzed and described in detail. Finally, the paper is concluded in Sec. 4.

2. Theoretical analysis

In the Cartesian coordinate system, the electric fields of cos-Airy-Gaussian-vortex beams at the z=0 plane can be presented as:

\begin{matrix} E (x_{0}, y_{0}, 0) & = A_{0} Ai (\frac{x_{0}}{w_{1}}) Ai (\frac{y_{0}}{w_{2}}) exp (\frac{a x_{0}}{w_{1}} + \frac{a y_{0}}{w_{2}}) exp (- \frac{x_{0}^{2} + y_{0}^{2}}{w_{0}^{2}}) \\ \times {(\frac{x_{0} - x_{1}}{w_{1}} + i \frac{y_{0} - y_{1}}{w_{2}})}^{l} {cos}^{m} (\frac{x_{0}}{w_{1}} + i \frac{y_{0}}{w_{2}}), \end{matrix}

where A₀ is the constant amplitude, a represents an exponential truncation factor ranging from 0 to 1, w₁ and w₂ denote arbitrary transverse scales in the x-direction and y-direction, w₀ is the waist size which is dependent on the equation w₁ = w₂ = χ₀w₀, χ₀ is a distribution factor which controls the beams to tend to cosine-Airy-vortex beams when it is small and cosine-Gaussian-vortex beams when it is large. [(x₀ − x₁)/w₁ + i(y₀ − y₁)/w₂]^l is the vortex term, x₁ and y₁ represent the position of the original vortex and l denotes the order of the vortex. cos^m (x₀/w₁ + iy₀/w₂) is the cosine complex variable function.

When we give the parameters l, m with different values, we can get the electric fields of different kinds of beams. For instance, when l = 0, m = 0, the electric field E indicates that of Airy-Gaussian beams and we mark it as E₀₀. In the same way, E₀₁ is the electric field of cosine-Airy-Gaussian beams, E₁₀ is that of Airy-Gaussian-vortex beams and E₁₁ is that of cosine-Airy-Gaussian-vortex beams actually. In practice, it is hard to derive the general expressions for these beams propagating through an optical ABCD system. For convenience, we just focus on investigating these four kinds of beams. Moreover, it is worth pointing out that the vortices of the Airy-Gaussian-vortex and cos-Airy-Gaussian-vortex beams that are claimed in this paper are only 1 order.

Based on the Huygens diffraction integral, the different kinds of beams propagating through an optical ABCD system under the paraxial approximation can be expressed as

\begin{array}{l} E (x, y, z) & = \frac{i k}{2 π B} \iint_{- \infty}^{+ \infty} E (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 elements of the transfer matrix and k is the wave number equals to 2π/λ (λ is the wavelength of the beams in the free space). Substituting Eq. (1) into Eq. (2), we can derive the expressions of the different kinds of beams propagating in an optical ABCD system as

E_{00} (x, y, z) = \frac{i k A_{0}}{2 BM} exp [Q_{1} (x, y, z)] L_{1},

E_{01} (x, y, z) = \frac{i k A_{0}}{4 BM} {exp [Q_{2} (x, y, z)] L_{2} + exp [Q_{3} (x, y, z)] L_{3}},

E_{10} (x, y, z) = \frac{i k A_{0}}{2 BM} exp [Q_{1} (x, y, z)] (K_{1} + K_{2} + K_{3}),

E_{11} (x, y, z) = \frac{k A_{0}}{4 BM} {exp [Q_{2} (x, y, z)] (P_{1} + P_{2} + P_{3}) + exp [Q_{3} (x, y, z)] (F_{1} + F_{2} + F_{3})},

where

Q_{1} (x, y, z) = - \frac{i k D}{2 B} (x^{2} + y^{2}) + \frac{N_{01}^{2} + N_{02}^{2}}{4 M} + \frac{1}{8 M^{2}} (\frac{N_{01}}{w_{1}^{3}} + \frac{N_{02}}{w_{2}^{3}}) + \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}}),

Q_{2} (x, y, z) = - \frac{i k D}{2 B} (x^{2} + y^{2}) + \frac{N_{11}^{2} + N_{12}^{2}}{4 M} + \frac{1}{8 M^{2}} (\frac{N_{11}}{w_{1}^{3}} + \frac{N_{12}}{w_{2}^{3}}) + \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}}),

Q_{3} (x, y, z) = - \frac{i k D}{2 B} (x^{2} + y^{2}) + \frac{N_{13}^{2} + N_{14}^{2}}{4 M} + \frac{1}{8 M^{2}} (\frac{N_{13}}{w_{1}^{3}} + \frac{N_{14}}{w_{2}^{3}}) + \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}}),

L_{1} = Ai [f_{1} (x)] Ai [g_{1} (y)], L_{2} = Ai [f_{2} (x)] Ai [g_{2} (y)], L_{3} = Ai [f_{3} (x)] Ai [g_{3} (y)],

K_{1} = [\frac{1}{w_{1}} (\frac{N_{01}}{2 M} + \frac{1}{8 w_{1}^{3} M^{2}} - x_{1}) + \frac{i}{w_{2}} (\frac{N_{02}}{2 M} + \frac{1}{8 w_{2}^{3} M^{2}} - y_{1})] L_{1},

K_{2} = \frac{1}{2 w_{1}^{2} M} {Ai}^{'} [f_{1} (x)] Ai [g_{1} (y)], K_{3} = \frac{i}{2 w_{2}^{2} M} Ai [f_{1} (x)] {Ai}^{'} [g_{1} (y)],

P_{1} = [\frac{i}{w_{1}} (x_{1} - \frac{N_{11}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}}) - \frac{1}{w^{2}} (y_{1} - \frac{N_{12}}{2 M} - \frac{1}{8 w_{2}^{3} M^{2}})] L_{2},

P_{2} = - \frac{i}{2 w_{1}^{2} M} {Ai}^{'} [f_{2} (x)] Ai [g_{2} (y)], P_{3} = \frac{1}{2 w_{2}^{2} M} Ai [f_{2} (x)] {Ai}^{'} [g_{2} (y)],

F_{1} = [\frac{i}{w_{1}} (x_{1} - \frac{N_{13}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}}) - \frac{1}{w^{2}} (y_{1} - \frac{N_{14}}{2 M} - \frac{1}{8 w_{2}^{3} M^{2}})] L_{3},

F_{2} = - \frac{i}{2 w_{1}^{2} M} {Ai}^{'} [f_{3} (x)] Ai [g_{3} (y)], F_{3} = \frac{1}{2 w_{2}^{2} M} Ai [f_{3} (x)] {Ai}^{'} [g_{3} (y)],

f_{1} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{01}}{2 w_{1} M}, g_{1} (y) = \frac{1}{16 w_{2}^{4} M^{2}} + \frac{N_{02}}{2 w_{2} M},

f_{2} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{11}}{2 w_{1} M}, g_{2} (y) = \frac{1}{16 w_{2}^{4} M^{2}} + \frac{N_{12}}{2 w_{2} M},

f_{3} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{13}}{2 w_{1} M}, g_{3} (y) = \frac{1}{16 w_{2}^{4} M^{2}} + \frac{N_{14}}{2 w_{2} M},

\begin{array}{l} M = \frac{1}{w_{0}^{2}} + \frac{i k A}{2 B}, N_{01} = \frac{a}{w_{1}} + \frac{i k x}{B}, N_{02} = \frac{a}{w_{2}} + \frac{i k y}{B}, N_{11} = \frac{a + i}{w_{1}} + \frac{i k x}{B}, \\ N_{12} = \frac{a - 1}{w_{2}} + \frac{i k y}{B}, N_{13} = \frac{a - i}{w_{1}} + \frac{i k x}{B}, N_{14} = \frac{a + 1}{w_{2}} + \frac{i k y}{B} . \end{array}

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

(\begin{matrix} A_{(L)} & B_{(L)} \\ C_{(L)} & D_{(L)} \end{matrix}) = (\begin{matrix} 1 & z / n_{(L)} \\ 0 & 1 \end{matrix}) and (\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 LCP beams and RCP beams, respectively. n is the original refractive index in a chiral medium and γ is the chiral parameter. Substituting Eq. (8) into Eq. (2), we can obtain the analytical expressions of the LCP and RCP forms of the different kinds of beams in a chiral medium marked as E_(L)(x, y, z) and E_(R)(x, y, z). Subsequently, the total intensity expression can be derived

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

with

I_{int} = 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 * denotes the complex conjugate.

Besides, the expression for the propagation trajectory of the central lobe of these beams along the z-axis can be obtained as

x_{c} = \frac{A}{16 w_{1}^{3} M M^{*}},

where x_c is the transverse position in the x-direction of the central lobe along the z-axis, and that in the y-direction is the same. If we put an optical vortex on the Airy-Gaussian beams, the propagation trajectory of the optical vortex along the z-axis is

x_{o v} = A x_{1} + \frac{A}{8 w_{1}^{3} M M^{*}},

where x₁ is the original position of the optical vortex, x_ov is the transverse position in the x-direction of the optical vortex along the z-axis, and that in the y-direction is the same. In addition, it is worth indicating that the cosine complex variable function will not change the locations of the central lobe and the optical vortex.

Interestingly, after deriving these expressions for the propagation trajectory of the beams and optical vortex, we discover that there are ultimate transverse displacements when the propagation distance turns to infinity because of the existence of the distribution factor χ₀. The ultimate transverse displacements of the beams and the optical vortex are dependent on the expressions below

x_{cl} = \frac{{Aw}_{1}}{16 χ_{0}^{4}}, (z \to \infty)

x_{ovl} = \frac{{Aw}_{1}}{8 χ_{0}^{4}}, (z \to \infty)

where x_cl is the ultimate transverse displacement of the beams and x_ovl is that of the optical vortex.

Moreover, it is necessary to consider the overlap position of the beams center and the optical vortex when we study on some kinds of vortex beams. Here we present the expression for the overlap position of the Airy-Gaussian-vortex beams as

z_{s (L, R)} = \frac{{An}_{(L, R)}}{\sqrt{- (\frac{w_{1}}{16 x_{1}} + χ_{0}^{4})}} \cdot Zr, (- \frac{w_{1}}{16 χ_{0}^{4}} < x_{1} \leq 0)

where z_s is the propagation distance of the overlap position,

Zr = k w_{1}^{2} / 2

is the Rayleigh range.

3. Numerical simulation

With the analytical expressions for different kinds of beams propagating in a chiral medium from the last section, we concentrate on outstanding the characteristics of the cosine complex variable function by comparing two kinds of beams in different ways.

Unless indicated in captions, the parameters used in simulations are set as follows: A₀ = 1, λ = 632.8nm, w₁ = w₂ = 100μm, x₁ = y₁ = 0, a = 0.1, n = 3 and γ = 0.16/k.

Figure 1 shows the intensity distributions of Airy-Gaussian beams and cos-Airy-Gaussian beams at the source plane z=0 with different distribution factors χ₀. Compared Fig. 1(a1) with Fig. 1(b1), it is distinct that the cosine factor is capable to eliminate the central lobe and the x-direction side lobe of the origin intensity distribution. However, when the χ₀ increases, the effect seems to fade, which also means that the cosine factor has a big influence on the origin intensity distribution when the beams tend to be Airy beams (χ₀ is small), while a bit effect when the beams tend to be Gaussian beams (χ₀ is large).

Fig. 1 Intensity distributions of the Airy-Gaussian beams (a1)–(a3) and cos-Airy-Gaussian beams (b1)–(b3) at the z=0 plane with different distribution factors χ₀.

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To investigate the intensity variation process of the cos-Airy-Gaussian beams, the intensity distributions on the x–y plane of the cos- and Airy-Gaussian LCP beams at different propagation distances with different distribution factors χ₀ are depicted in Fig. 2. From Figs. 2(b1)–2(b3) and compared with Figs. 2(a1)–2(a3), we can find that the intensity of the cos-Airy-Gaussian beams on the side lobe transfers to a certain lobe which is distinct from the central lobe of the Airy-Gaussian beams during the propagation. The main lobe of the cos-Airy-Gaussian beams in Fig. 2(b2) is under the central lobe of the Airy-Gaussian beams in Fig. 2(a2) in both x- and y-directions at 12Zr while that in Fig. 2(b3) is over the central lobe in Fig. 2(a3) in y-direction but still under in x-direction at 24Zr. These phenomena show that the Airy-Gaussian beams with the cosine factor have a special main lobe displacement along the z-axis which is unpredictable in analytical expression unlike the normal Airy-Gaussian beams. Nevertheless, the special feature becomes weaker and weaker with the distribution factors χ₀ increasing as shown in Figs. 2(d1)–2(d3) and Figs. 2(c1)–2(c3).

Fig. 2 Intensity distribution of the LCP beams of the Airy-Gaussian beams (a1)–(a3), (c1)–(c3) and cos-Airy-Gaussian beams (b1)–(b3), (d1)–(d3) at different propagation distances with different distribution factors χ₀. (Because the results from the RCP beams are similar, we only present the snapshots of the LCP beams)

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Figure 3 represents the intensity evolutions of the Airy-Gaussian beams and cos-Airy-Gaussian beams, and normalized maximum intensity distribution of the cos-Airy-Gaussian beams along the z-axis with different factors χ₀. Observing the four columns in Fig. 3, it is pronounced that the strong intensity position of the cos-Airy-Gaussian beams is different from that of the Airy-Gaussian beams. The intensity variation process of the cos-Airy-Gaussian beams can be exhibited as follows. At the very start, the intensity mainly distributes on the y-direction side lobe of the cos-Airy-Gaussian beams when χ₀ is small as shown in Fig. 2(b1). With the increase of propagation distance, the intensity on the y-direction side lobe gradually transfers to a certain lobe as depicted in Fig. 2(b2) which also can be seen in Fig. 3(b2) around 12Zr. Afterward, the intensity mostly distributes on the x-direction side lobe of the cos-Airy-Gaussian beams which is shown in Fig. 2(b3). The intensity variation process above is apparently consistent with the intensity evolution in Fig. 3(b2). The cos-Airy-Gaussian beams have an intensity transfer performance on a certain lobe during the propagation and the strong intensity position on the certain lobe can be adjusted by the distribution factor χ₀. The smaller the χ₀ is, the further the strong intensity position of the cos-Airy-Gaussian beams is, which is plotted clearly in Figs. 3(c1) and 3(c2).

Fig. 3 Intensity evolution along the z-axis of the Airy-Gaussian beams (a1)–(a4) and cos-Airy-Gaussian beams (b1)–(b4), and normalized maximum intensity evolution along the z-axis (c1)–(c2) of the cos-Airy-Gaussian beams with different distribution factors χ₀.

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To reveal the overall evolution properties of the Airy-Gaussian beams with cosine factor propagating in a chiral medium, we pay attention to investigate the interference intensity evolution and total intensity distribution along the z-axis of the cos-Airy-Gaussian beams with different distribution factors χ₀, which is manifested in Fig. 4. Combining Figs. 3(b1)–3(b4) with Figs. 4(a1)–4(a2) and 4(b1)–4(b2), it is intuitive that the interference intensity distribution relates to the propagation trajectories and the intensity distributions of the LCP and RCP beams deeply. Interestingly, with the increase of the distribution factor χ₀, the propagation trajectories of the LCP and RCP beams decline while the intensity distributions along the z-axis of those become closer as compared to the initial plane. It also means that the maximum interference intensity of the cos-Airy-Gaussian beams will become stronger and the corresponding position will become closer with the distribution factors χ₀ increasing, which are consistent with the curves in Fig. 4(c).

Fig. 4 Interference intensity (a1), (b1), total intensity (a2), (b2) and normalized interference intensity evolutions (c) along the z-axis of cos-Airy-Gaussian beams with different distribution factors χ₀.

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Based on the Eqs. (11) and (12), the central position and optical vortex position evolutions of the cos- or Airy-Gaussian-vortex beams along the z-axis with different distribution factors χ₀ are plotted in Fig. 5. From Figs. 5(a1) and 5(b1), it is evident that the central position of the LCP beams is always higher than that of the RCP beams, and they both increase with the distribution factors χ₀ decreasing. Compared Figs. 5(a1) and 5(b1) with 5(a2) and 5(b2), it is unambiguous that the positions of the vortex are always twice as high as their corresponding beams when we set the origin vortex position as (0,0), which also means that the velocity of the vortex is always twice as fast as its corresponding beams so that we can control the overlap position of the vortex and the beams by setting a certain origin vortex position. Also, we can adjust the overlap position by changing the distribution factor χ₀ because the propagation trajectories of the vortex and beams can be modulated by it.

Fig. 5 Central position (a1), (b1) and optical vortex position evolutions (a2), (b2) of the cos- or Airy-Gaussian-vortex beams along the z-axis with different distribution factors χ₀.

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Connecting the discussions from Fig. 5, we plot the diagrams depicting about the relations of the overlap position, the transverse origin vortex position, the distribution factor χ₀ and the chiral parameter γ in Figs. 6(b1) and 6(b2). It is intriguing that the overlap position of the LCP beams decreases with the increase of the chiral parameter γ while that of the RCP beams is opposite. Meanwhile, the overlap positions of the LCP and RCP beams both increase when increasing the distribution factor χ₀. Moreover, both LCP and RCP beams with a further transverse origin vortex position ( $- \frac{w_{1}}{16 χ_{0}^{4}} < x_{1} \leq 0$ ) have a further overlap position. According to Eqs. (13) and (14), the ultimate transverse displacements of the beams and the optical vortex are only influenced by the distribution factor χ₀ but not the chiral parameter or the types of beams (LCP or RCP), which is depicted in Figs. 6(a1) and 6(a2). The bigger the distribution factor χ₀ is, the smaller the ultimate transverse displacements of the beams and vortex are.

Fig. 6 Ultimate transverse displacement of the cos- or Airy-Gaussian beams and the optical vortex as a function of χ₀ (a1)–(a2), and overlap position (beams center and vortex are superimposed) as a function of the transverse origin vortex position with different chiral parameters γ and distribution factors χ₀ (b1)–(b2).

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In Fig. 7, we mainly investigate the effect of the origin vortex position on the LCP beams of the cos-Airy-Gaussian-vortex beams on account of the similar results from the RCP beams and the unpredictable main lobe position of the cos-Airy-Gaussian-vortex beams. Comparing Fig. 7(b2) with Fig. 7(c2), we find out that the effect of the origin vortex position on the origin intensity distribution become different. At the initial plane, the strong intensity domain moves down when the origin vortex position alters from −0.4mm to −0.8mm, but moves up from −0.8mm to −1.2mm. It is interesting that this kind of change can transmit to the after plane such as the z=12Zr shown in Figs. 7(b4) and 7(c4).

Fig. 7 Intensity and phase distributions of the LCP beams of the cos-Airy-Gaussian-vortex beams with different origin vortex positions. The black arrows in the third row indicate the locations of vortex. (0, −0.4) (a1)–(a4), (0, −0.8) (b1)–(b4), (0, −1.2) (c1)–(c4), (0, −1.6) (d1)–(d4).

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4. Conclusion

In this paper, we have investigated the effects of the cosine complex variable function on the Airy-Gaussian beams and Airy-Gaussian vortex beams in detail. The analytical expressions for the four types of beams, Airy-Gaussian beams, cos-Airy-Gaussian beams, Airy-Gaussian-vortex beams and cos-Airy-Gaussian-vortex beams propagating in a chiral medium are derived. It is shown that the cosine factor is capable to eliminate the central lobe and the x-direction side lobe of the origin intensity distribution when the Airy-Gaussian beams tend to be Airy beams, while this kind of effect fades when the Airy-Gaussian beams tend to be Gaussian beams. Meanwhile, we find out that the cos-Airy-Gaussian beams have an intensity transfer performance on a certain lobe during the propagation and the strong intensity position on the certain lobe can be adjusted by the distribution factor χ₀. The smaller the χ₀ is, the further the strong intensity position on the certain lobe of the cos-Airy-Gaussian beams is. Moreover, the maximum interference intensity of the cos-Airy-Gaussian beams becomes stronger and the corresponding position becomes closer with the distribution factors χ₀ increasing. In addition, it is intriguing that the Airy-Gaussian beams with the cosine factor have a special transverse displacement along the z-axis which is unpredictable in analytical expression unlike the normal Airy-Gaussian beams, but also fades when the beams tend to be Gaussian beams. More importantly, we find that there are always ultimate transverse displacements of the Airy-Gaussian beams and vortex because of the existence of the distribution factor χ₀. Overall, we can take full advantage of the special intensity transfer performance of the cos-Airy-Gaussian beams or the cos-Airy-Gaussian-vortex beams through controlling the distribution factor χ₀, the chiral parameter γ and even the original vortex position. And we believe our results may have potential applications in areas such as optical micromanipulation and optical sorting.

Appendix A: Derivation details of analytical expression of the cos-Airy-Gaussian beams propagating in a chiral medium

First, we obtain the origin electric expression of the cos-Airy-Gaussian beams as

E_{01} (x_{0}, y_{0}, 0) = A_{0} Ai (\frac{x_{0}}{w_{1}}) Ai (\frac{y_{0}}{w_{2}}) exp (\frac{a x_{0}}{w_{1}} + \frac{a y_{0}}{w_{2}}) exp (- \frac{x_{0}^{2} + y_{0}^{2}}{w_{0}^{2}}) cos (\frac{x_{0}}{w_{1}} + i \frac{y_{0}}{w_{2}}) .

Using the Euler’s formula, we get the exponential forms as

E_{(1)} (x_{0}, y_{0}, 0) = \frac{1}{2} A_{0} Ai (\frac{x_{0}}{w_{1}}) Ai (\frac{y_{0}}{w_{2}}) exp (\frac{a x_{0}}{w_{1}} + \frac{a y_{0}}{w_{2}}) exp (- \frac{x_{0}^{2} + y_{0}^{2}}{w_{0}^{2}}) exp (i \frac{x_{0}}{w_{1}} - \frac{y_{0}}{w_{2}}),

E_{(2)} (x_{0}, y_{0}, 0) = \frac{1}{2} A_{0} Ai (\frac{x_{0}}{w_{1}}) Ai (\frac{y_{0}}{w_{2}}) exp (\frac{a x_{0}}{w_{1}} + \frac{a y_{0}}{w_{2}}) exp (- \frac{x_{0}^{2} + y_{0}^{2}}{w_{0}^{2}}) exp (- i \frac{x_{0}}{w_{1}} + \frac{y_{0}}{w_{2}}) .

Substituting Eq. (17) into Eq. (2), we can write down expression as

\begin{array}{l} E_{(1)} (x, y, z) & = \frac{i k A_{0}}{4 π B} exp [- \frac{i k D}{2 B} (x^{2} + y^{2})] \\ \times \int_{- \infty}^{+ \infty} Ai (\frac{x_{0}}{w_{1}}) exp [- (\frac{1}{w_{0}^{2}} + \frac{i k A}{2 B}) x_{0}^{2} + (\frac{a + i}{w_{1}} + \frac{i k x}{B}) x_{0}] d x_{0} \\ \times \int_{- \infty}^{+ \infty} Ai (\frac{y_{0}}{w_{2}}) exp [- (\frac{1}{w_{0}^{2}} + \frac{i k A}{2 B}) y_{0}^{2} + (\frac{a - 1}{w_{w}} + \frac{i k y}{B}) y_{0}] d y_{0} . \end{array}

Set

M = \frac{1}{w_{0}^{2}} + \frac{i k A}{2 B}, N_{11} = \frac{a + i}{w_{1}} + \frac{i k x}{B}, N_{12} = \frac{a - 1}{w_{2}} + \frac{i k y}{B},

we can simplify Eq. (19) as

\begin{array}{l} E_{(1)} (x, y, z) & = \frac{i k A_{0}}{4 π B} exp [- \frac{i k D}{2 B} (x^{2} + y^{2})] \\ \times \int_{- \infty}^{+ \infty} Ai (\frac{x_{0}}{w_{1}}) exp (- M x_{0}^{2} + N_{11} x_{0}) d x_{0} \int_{- \infty}^{+ \infty} Ai (\frac{y_{0}}{w_{2}}) exp (- M y_{0}^{2} + N_{12} y_{0}) d y_{0} . \end{array}

Using the formula

\int_{- \infty}^{+ \infty} Ai (\frac{x}{a}) exp (- b x^{2} + c x) d x = \sqrt{\frac{π}{b}} exp (\frac{c^{2}}{4 b} + \frac{c}{8 a^{3} b^{2}} + \frac{1}{96 a^{6} b^{3}}) Ai (\frac{1}{16 a^{4} b^{2}} + \frac{c}{2 a b}),

we present the final expression for E₍₁₎(x, y, z) as

\begin{array}{l} E_{(1)} (x, y, z) & = \frac{i k A_{0}}{4 BM} exp [- \frac{i k D}{2 B} (x^{2} + y^{2}) + \frac{N_{11}^{2} + N_{12}^{2}}{4 M} + \frac{1}{8 M^{2}} (\frac{N_{11}}{w_{1}^{3}} + \frac{N_{12}}{w_{2}^{3}}) \\ + \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}})] Ai (\frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{11}}{2 w_{1} M}) Ai (\frac{1}{16 w_{2}^{4} M^{2}} + \frac{N_{12}}{2 w_{2} M}) . \end{array}

With the same method, we are able to derive the final expression for E₍₂₎(x, y, z) as

\begin{array}{l} E_{(2)} (x, y, z) & = \frac{i k A_{0}}{4 BM} exp [- \frac{i k D}{2 B} (x^{2} + y^{2}) + \frac{N_{13}^{2} + N_{14}^{2}}{4 M} + \frac{1}{8 M^{2}} (\frac{N_{13}}{w_{1}^{3}} + \frac{N_{14}}{w_{2}^{3}}) \\ + \frac{1}{96 M^{3}} (\frac{1}{w_{1}^{6}} + \frac{1}{w_{2}^{6}})] Ai (\frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{13}}{2 w_{1} M}) Ai (\frac{1}{16 w_{2}^{4} M^{2}} + \frac{N_{14}}{2 w_{2} M}), \end{array}

with

N_{13} = \frac{a - i}{w_{1}} + \frac{i k x}{B}, N_{14} = \frac{a + 1}{w_{2}} + \frac{i k y}{B} .

Finally, we obtain the analytical expression of the cos-Airy-Gaussian beams propagation in an optical ABCD system as

E_{01} (x, y, z) = E_{(1)} (x, y, z) + E_{(2)} (x, y, z) .

Appendix B: Derivation details of the evolution expressions of central lobe and vortex position along z-axis of these four kinds of beams

The central lobes of beams are defined from their corresponding analytical expressions. The evolution expressions of the central lobe in the x-direction along z-axis of the Airy-Gaussian beams and Airy-Gaussian-vortex beams can be obtained from the equation $f_{1} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{01}}{2 w_{1} M}$ . We can simplify the equation as

f_{1} (x) = \frac{i k}{2 {Bw}_{1} M} (x + \frac{B}{8 i k w_{1}^{3} M} + \frac{Ba}{i k w_{1}}) .

When the real part of the (

x + \frac{B}{8 i k w_{1}^{3} M} + \frac{B a}{i k w_{1}}

) term is zero, we can get the evolution expression of the central lobes in the x-direction along the z-axis of the Airy-Gaussian beams and Airy-Gaussian-vortex beams as [19]

x_{c} = \frac{A}{16 w_{1}^{3} M M^{*}} .

Besides, the evolution expression of the vortex along z-axis of the Airy-Gaussian-vortex beams can be got from the equation

K_{1} = [\frac{1}{w_{1}} (\frac{N_{01}}{2 M} + \frac{1}{8 w_{1}^{3} M^{2}} - x_{1}) + \frac{i}{w_{2}} (\frac{N_{02}}{2 M} + \frac{1}{8 w_{2}^{3} M^{2}} - y_{1})] L_{1}

. We can select the x-direction form from the equation and simplify it as

\frac{N_{01}}{2 M} + \frac{1}{8 w_{1}^{3} M^{2}} - x_{1} = \frac{i k}{2 BM} (x + \frac{Ba}{i k w_{1}} + \frac{B}{4 i k w_{1}^{3} M} - \frac{2 BM}{i k} x_{1}) .

When setting the real part of the (

x + \frac{Ba}{i k w_{1}} + \frac{B}{4 i k w_{1}^{3} M} - \frac{2 BM}{i k} x_{1}

) term to be zero, we can derive the evolution expression of the vortex in the x-direction along the z-axis of the Airy-Gaussian-vortex beams as [19]

x_{ov} = A x_{1} + \frac{A}{8 w_{1}^{3} M M^{*}} .

For the cos-Airy-Gaussian and cos-Airy-Gaussian-vortex beams, we need to consider two exponential forms from the cosine complex variable function. However, the cosine factor will not affect the central lobe and vortex position of the Airy-Gaussian and Airy-Gaussian-vortex beams, which can be seen from the following formulae.

f_{2} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{11}}{2 w_{1} M} = \frac{i k}{2 B w_{1} M} (x + \frac{B}{8 i k w_{1}^{3} M} + \frac{B (a + i)}{i k w_{1}}) .

When the real part of the (

x + \frac{B}{8 i k w_{1}^{3} M} + \frac{B (a + i)}{i k w_{1}}

) term is zero, we can get expression as

x_{c 1} = \frac{A}{16 w_{1}^{3} M M^{*}} - \frac{B}{k w_{1}} .

In the same way, we can get other similar expression as

f_{3} (x) = \frac{1}{16 w_{1}^{4} M^{2}} + \frac{N_{13}}{2 w_{1} M} = \frac{i k}{2 B w_{1} M} (x + \frac{B}{8 i k w_{1}^{3} M} + \frac{B (a - i)}{i k w_{1}}),

x_{c 2} = \frac{A}{16 w_{1}^{3} M M^{*}} + \frac{B}{k w_{1}} .

The evolution expression of the central lobe in the x-direction along z-axis of the cos-Airy-Gaussian and cos-Airy-Gaussian-vortex beams can be considered as

x_{c} = \frac{x_{c 1} + x_{c 2}}{2} = \frac{A}{16 w_{1}^{3} M M^{*}} .

So, cosine complex variable function will not affect the central lobe position. In addition, to the vortex position:

\begin{array}{l} P_{1} = [\frac{i}{w_{1}} (x_{1} - \frac{N_{11}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}}) - \frac{1}{w^{2}} (y_{1} - \frac{N_{12}}{2 M} - \frac{1}{8 w_{2}^{3} M^{2}})] L_{2}, \\ x_{1} - \frac{N_{11}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}} = - \frac{i k}{2 BM} (x + \frac{B (a + i)}{i k w_{1}} + \frac{B}{4 i k w_{1}^{3} M} - \frac{2 BM}{i k} x_{1}) . \end{array}

When setting the real part of the (

x + \frac{B (a + i)}{i k w_{1}} + \frac{B}{4 i k w_{1}^{3} M} - \frac{2 BM}{i k} x_{1}

) term to be zero, we can get expression as

x_{ov 1} = A x_{1} + \frac{A}{8 w_{1}^{3} M M^{*}} - \frac{B}{k w_{1}} .

In the same way, we can get other similar expressions as

\begin{array}{l} F_{1} = [\frac{i}{w_{1}} (x_{1} - \frac{N_{13}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}}) - \frac{1}{w^{2}} (y_{1} - \frac{N_{14}}{2 M} - \frac{1}{8 w_{2}^{3} M^{2}})] L_{3}, \\ x_{1} - \frac{N_{13}}{2 M} - \frac{1}{8 w_{1}^{3} M^{2}} = - \frac{i k}{2 BM} (x + \frac{B (a - i)}{i k w_{1}} + \frac{B}{4 i k w_{1}^{3} M} - \frac{2 BM}{i k} x_{1}), \end{array}

x_{ov 2} = A x_{1} + \frac{A}{8 w_{1}^{3} M M^{*}} - \frac{B}{k w_{1}} .

The evolution expression of the vortex position in the x-direction along z-axis of the cos-Airy-Gaussian-vortex beams can be regarded as

x_{ov} = \frac{x_{ov 1} + x_{ov 2}}{2} = A x_{1} + \frac{A}{8 w_{1}^{3} M M^{*}} .

When x_c = x_ov, we can obtain Eq. (15).

Funding

National Nature Science Foundation of China (NSFC) (11775083 and 11374108); National Training Program of Innovation and Entrepreneurship for Undergraduates (201810574177); Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation; The Extracurricular Scientific Program of School of Information and Optoelectronic Science and Engineering, South China Normal University (18GDGB01).

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Effects of the cosine complex variable function on the Airy-Gaussian and Airy-Gaussian-vortex beams in a chiral medium

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical simulation

4. Conclusion

Appendix A: Derivation details of analytical expression of the cos-Airy-Gaussian beams propagating in a chiral medium

Appendix B: Derivation details of the evolution expressions of central lobe and vortex position along z-axis of these four kinds of beams

Funding

References

Cited By

Figures (7)

Equations (53)

OSA Continuum