Propagation dynamics of Janus vortex waves

Wenlei Yu; Shuofeng Zhao; Peipei Jiang; Yunfeng Jiang

doi:10.1364/OE.27.034484

1. Introduction

The Janus waves were proposed in 2016 [1]. These light waves could generate two distinct foci under the action of a focusing lens, thus have potential great application in many fields. The circular Airy beam (CAB) is one of the most common members in the family of Janus waves [1]. It’s well known that CAB could abruptly autofocus at a real focus in + z direction [2], but in fact, it still has a virtual focus in -z direction. So, under the action of the lens, two new real images could be generated by objects located at “real focus” and “virtual focus”. This property is called dual focus property in this paper. The abruptly autofocusing property of CAB or other similar beams are widely investigated in these years [3–15]. But there are few studies about the dual focus property of CAB [16,17]. Recently, Papazoglou’s group demonstrate that this property could be applied in strong field science for phase memory preserving [18].

Optical vortex (OV) has attracted intensive interest in last decades. Many interesting phenomena and important applications about OV have been investigated, such as optical rotation, confinement of cold atoms, orbital angular momentum of photons and so on [19–25]. The OV imposed in different kinds of special light beams, such as Bessel beams [26], Airy beams [27–31], partially coherent beams [32–34], perfect light ring [35–37], circle Pearcey beams [14], have been widely investigated. These studies have greatly expanded the application of these special light beams; and they are also helpful for us to understand the physical behavior of OV. To our best knowledge, the OV imposed in Janus waves, and its effect on the double foci, has not been investigated yet. It is natural to ask could the beam keep its special dual focus property and how to control this property by different parameters, when OVs are imposed in. On the other hand, it is well-known that the off-axis vortex could rotate in single focus light beam; and a pair of opposite OVs would collide and annihilate exactly at the focus [5,38,39]. So, it is also interesting to investigate what will happen when this pair of OVs are imposed in dual focus light beams. Will they annihilate at the first or the second focus, or at the lens focus?

In this paper, we have investigated the propagation characteristics of Janus vortex waves in detail, based upon the expression of circular Airy vortex beams (CAVB). The propagation dynamics of CAVB through ABCD paraxial optical system are calculated by numerical Hankel transform method; the influences of different parameters on dual focus property are investigated. After that, the rotation of an off-axis OV in Janus vortex wave is studied, some interesting results are found. Finally, we investigate the case when two same or opposite OVs are imposed in CAB, the repel and attraction of two OVs in dual focus light beam are analyzed.

2. Theoretical simulation of propagation of Janus vortex wave

The CAB is demonstrated as a type of Janus wave [1]. The CAB with an OV located at (r_k, φ_k) in polar coordinates could be expressed as [2,5],

(1)$${u_0}({r,\varphi ,z = 0} )= G \cdot Ai\left( {\frac{{{r_0} - r}}{w}} \right)\exp \left( {a\frac{{{r_0} - r}}{w}} \right) \cdot {({r{e^{i\varphi }} - {r_k}{e^{i{\varphi_k}}}} )^n},$$

where r₀ is a parameter related with the initial radius, w is a scaling parameter, a is a decaying parameter, n is the topological charge, G is a constant related with the incident power. The propagation of CAVB through any ABCD paraxial optical system could be calculated by Collins formula [40],

u({x,y,z} )= \frac{{{e^{ikz}}}}{{i\lambda B}}\int\!\!\!\int {{u_0}({{x_0},{y_0}} )\exp \left\{ {\frac{{ikA}}{{2B}}\left[ {({x_0^2 + y_0^2} )+ \frac{D}{A}({{x^2} + {y^2}} )- \frac{2}{A}({x{x_0} + y{y_0}} )} \right]} \right\}d{x_0}d{y_0}}

(2)$$\begin{array}{l} \mbox{ = }\frac{{{e^{ikz}}}}{{i\lambda B}}\exp \left[ {\frac{{ik}}{{2B}}\left( {D - \frac{1}{A}} \right)({{x^2} + {y^2}} )} \right]\\ \mbox{ } \times \int\!\!\!\int {{u_0}({{x_0},{y_0}} )\exp \left\{ {\frac{{ik}}{{2AB}}[{{{({A{x_0} - x} )}^2} + {{({A{y_0} - y} )}^2}} ]} \right\}d{x_0}d{y_0}} \end{array}$$

\begin{array}{l} \mbox{ = }\frac{{{e^{ikz}}}}{{i\lambda B}}\exp \left[ {\frac{{ik}}{{2B}}\left( {D - \frac{1}{A}} \right)({{x^2} + {y^2}} )} \right]\\ \mbox{ } \times \int\!\!\!\int {\frac{1}{{{A^2}}}{u_0}\left( {\frac{{{x_t}}}{A},\frac{{{y_t}}}{A}} \right)\exp \left\{ {\frac{{ik}}{{2AB}}[{{{({{x_t} - x} )}^2} + {{({{y_t} - y} )}^2}} ]} \right\}d{x_t}d{y_t}} , \end{array}

where x_t=Ax₀; and (x,y) and (x₀,y₀) are Cartesian coordinates of the output plane and input plane, respectively. By using the convolution theory of Fourier transform, Eq. (2) could be written as,

(3)$$u(x,y,z) = \mbox{exp}\left[ {\frac{{ikC}}{{2A}}({{x^2} + {y^2}} )} \right] \times F{T^{ - 1}}[{{{\tilde{u}}_0}(A{f_x},A{f_y}) \cdot prop} ],$$

where k is the wave number, f_x and f_y are spatial frequency, ${\tilde{u}_0} = FT({u_0})$ is the Fourier transform of light fields in the initial plane, FT⁻¹ denotes the inverse Fourier transform; prop is the famous propagating factor, whose formula in ABCD system in Eq. (3) should be,

(4)$$prop = FT\left( {\frac{{{e^{ikz}}}}{{i\lambda B}}\exp \left[ {\frac{{ik}}{{2AB}}({{x^2} + {y^2}} )} \right]} \right) = \frac{{{e^{ikz}}}}{A}\exp [{ - iBA\lambda \pi f_r^2} ],$$

where ${f_r} = \sqrt {f_x^2 + f_y^2} $ is the radial spatial frequency. When A = 1, B = z, we can get the propagating factor in the free space.

Usually, the split-step Fourier transform method can be applied to numerically calculate Eq. (3) [41]. However, this method is quite time-consuming and requires a large computer memory, especially when some lenses are included in the optical system; sometimes the result would be inaccurate. Fortunately, the input light fields in our study is radially symmetric (or could be expanded to the sum of radially symmetric fields), so the 2D Fourier transform in Eq. (3) can be simplified to Hankel transform. Thus, a simple and accurate quasi-discrete Hankel transform method could be applied here [42,43]. However, we should extend this method to ABCD optical system first.

According to the binomial theorem, the initial electric fields u₀ could be expanded to the sum of radially symmetric fields ${u_0} = \sum\limits_{l = 0}^n {{P_l}{u_{0l}}} $, where,

(5)$${P_l} = G{({ - 1} )^{n - l}}\left( {\begin{array}{{c}} n\\ l \end{array}} \right){r_k}^{n - l}{e^{i({n - l} ){\varphi _k}}},$$

(6)$${u_{0l}}({r,\varphi } )= Ai\left( {\frac{{{r_0} - r}}{w}} \right)\exp \left( {a\frac{{{r_0} - r}}{w}} \right){r^l}{e^{il\varphi }}.$$

For a radially symmetric light field, u_0l=g(r)exp(ilφ), Eq. (3) could be expressed as [43,44],

(7)$$\tilde{g}({{f_r}} )= 2\pi \int\limits_0^\infty {g\left( {\frac{r}{{|A |}}} \right){J_l}({i2\pi {f_r}r} )rdr} ,$$

(8)$${\tilde{u}_{\mbox{0}l}}(A{f_x},A{f_y}) = \tilde{g}({{f_r}} )\exp \left[ {il\left( {\varphi + \frac{{1 - sign(A )}}{2}\pi } \right)} \right],$$

(9)$${u_l}({r,\varphi ,z} )= 2\pi \exp \left[ {il\left( {\varphi + \frac{{1 - sign(A )}}{2}\pi } \right)} \right]\int\limits_0^\infty {\tilde{g}({{f_r}} ){J_l}({i2\pi {f_r}r} )\cdot prop \cdot d{f_r}} .$$

Note that Eqs. (7) and (9) could not be applied when A = 0. When A is 0, the output light beam is just the Fourier transform of the initial light field, and could be easily obtained from Eq. (7). Finally, the light field in the output plane could be calculated by,

(10)$$u({r,\varphi ,z} )= \sum\limits_{l = 0}^n {{P_l}{u_l}({r,\varphi ,z} )} .$$

And the output intensity in this paper is I=|u|².

The thin lens system is shown in Fig. 1, its optical matrix is,

(11)$$\left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right] = \left[ {\begin{array}{{cc}} {1 - \frac{z}{f}}&{ - \frac{{zs}}{f} + s + z}\\ { - \frac{1}{f}}&{1 - \frac{s}{f}} \end{array}} \right],$$

where, f is the focal length of the lens, z is the distance between the output plane and the lens, s is the distance between the input plane and the lens.

Fig. 1. The CAVB with + 1 OV in the input plane (a) and the lens system (b). The focus of the lens is f, f₁ and f₂ denote the two foci of focused CAVB.

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Substituting Eq. (1) and Eq. (11) into Eqs. (7)–(10), we can obtain the propagation dynamics of Janus vortex waves, and analyze its dual focus property. In our simulation, we choose r₀=1 mm, w = 0.2 mm, a = 0.1, f = 0.3 m, s = 0 unless stated otherwise; the wavelength is 632.8 nm, and the incident power is 10 mW.

The dual focus property of common CAB under the action of a thin lens is shown in Fig. 2(a). The position where light intensity reaches a peak value and the radius of beam size reaches a minimum value is defined as focus position. It is obvious that, in Fig. 2(a), two foci could be found at z = f₁=0.26 m and z = f₂=0.35 m, respectively. When an on-axis OV with different topological charge n is imposed in, the CAVB will keep the dual focus property [Figs. 2(d) and 2(g)]. As shown in Figs. 2(d)–2(i), two dark foci could be observed. In each dark focal plane, the intensity increases to a maximum value and the radius of the light ring decreases to a minimum value. Thus, a perfect light hollow bottle could be formed, which might be useful in optical micromanipulation [45,46].

Fig. 2. Propagation dynamics of focused CAVB (the first column), the intensity pattern at the first focus (the second column) and the second focus (the third column). (a)–(c) n = 0; (d)–(f) n = 1; (g)–(i) n = 2.

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The CAVBs with different n almost focus at two same regions as CAB, if other parameters are not changed. But in fact, as shown in Fig. 3(a), the accurate focus positions change with n. As n increases, the distance between two foci |f₂-f₁| would be slightly shortened: f₁ becomes larger, and f₂ becomes smaller. And the focal intensities at two foci also decrease with n. The initial radius r₀ can also influence the dual focus property. As shown in Fig. 3(b), as r₀ increases, the distance between two dark foci would also be shortened, but unlike the influence of n mentioned before, here, the focal intensity would increase as |f₂-f₁| decreases.

Fig. 3. Changes of the maximum intensity along z axis with different parameters: (a) with different n, when r₀=1 mm, s = 0; (b) with different r₀, when n = 1, s = 0; (c) with different s, when n = 1, r₀=1 mm.

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From the imaging theory of a thin lens, it is known that positions of two foci of Janus wave could be controlled by varying the distance between the input plane and the lens s. As s increases, the virtual (or real) object distance would decrease (or increase), leading to the decrease of the real image distance f₁ (or f₂). The same conclusion could be found in [1] and Fig. 3(c). It is also worth noting that the relative intensity at two foci could be effectively controlled by varying s. Usually, when s = 0, the intensity at first focus is much larger than the second [Figs. 3(a)–3(c)]; but as s increases to 0.3 m, the two focal intensities are nearly the same; when s is 0.6 m, the focal intensity at the second focus exceeds the first one [Fig. 3(c)]. This is of great use for the application of Janus vortex waves in the dual-trap optical tweezes [47,48]. By changing s, we can easily control the trapping force of each trap.

3. Rotation of an off-axis OV in Janus vortex wave

It is demonstrated that an off-axis OV would be forced to move to the center and rotated by some angle in the single-focusing process [5]. It is interesting to investigate the behavior of off-axis OV in the dual-focusing process. Figure 4 shows that the CAVB could keep its dual focus property when an off-axis OV is located at r_k=0.5mm: the radial coordinate of OV reaches minimum values in two focus planes [Figs. 4(c) and 4(g)]; and intensity reaches maximum values in two focal planes (Fig. 5). Similar with the case of on-axis OV, the two foci for the beam with off-axis OV would also become closer with a larger topological charge n. In the two focal regions, the positive OV would rotate anticlockwise around the beam axis. Similarly, the negative OV would rotate clockwise. In other positions, the off-axis OV would keep nearly at the same angle position, only the radial position is changed.

Fig. 4. Intensity patterns [(a)–(h)] and phase patterns [(a1)–(h1)] of focused CAVB at different z position. (a) z = 0; (b) z = 0.24 m; (c) first focus plane, z = 0.26 m; (d) z = 0.28 m; (e) lens focus plane, z = 0.3 m; (f) z = 0.32 m; (g) second focus plane, z = 0.35 m; (h) z = 0.4 m. The inset arrows in the third column denote the rotation direction of OV.

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Fig. 5. Changes of maximum intensity along z axis for the focused CAVB with an off-axis OV.

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The positions of the off-axis OV could be accurately detected from phase patterns at different output planes. The trajectory of the off-axis OV in focused CAVB is shown in Fig. 6, and is compared with the single focus light beam (i.e., focused Gaussian beams). For Gaussian beam (GB), the rotation of OV is greatly accelerated in the focal region of the lens (z = f), but the OV in CAVB almost keeps still in this region. The OV in CAVB rotates rapidly in its two own focal regions (z = f₁ and z = f₂). Moreover, for GB, the OV could not finish even one full rotation, it slowly rotates to the angle position of π after the focus plane. But for CAVB, in our case, the OV would finish 4 complete rotation at each focal plane. At the first focal region, the OV rotates from 0 to a final position of π/2; at the second focal region, from π/2 to a final position of π. So, the OV for both beams would finally reach a same angle position when the propagation distance is large enough.

Fig. 6. Comparison of angle position of OV in dual focus light beams (i.e., focused CAVB) and in single focus light beams (i.e., focused GB).

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4. Propagation dynamics of Janus vortex wave with two OVs

For the single-focus beam, it is proved that the two OV could repel or attract each other, when they are very close to each other in the focal region [5,38]. Similar conclusions could also be found in dual-focus light beams. We assume the two OVs are located symmetrically in x-axis, and we only consider n = 1 here for simplicity,

(12)$${u_0}({r,\varphi ,z = 0} )= G \cdot Ai\left( {\frac{{{r_0} - r}}{w}} \right)\exp \left( {a\frac{{{r_0} - r}}{w}} \right)({r{e^{i\varphi }} - {r_k}} )({r{e^{ {\pm} i\varphi }}\mbox{ + }{r_k}} ).$$

Similar with the calculation for the beam with an off-axis OV, by substituting the expansion of Eq. (12) into Eqs. (7)–(10), we can get the intensity patterns and phase patterns at different positions. Here we use (+1, +1) to denote two same OVs with charge + 1; and (+1, −1) for two opposite OVs.

As shown in Fig. 7, when two same OVs (+1, +1) are imposed on, the beam would keep hollow in the whole propagation process; and two OVs would rotate around each other in two focal regions. The two same OVs would move closer in two focal regions, but they cannot completely overlap each other, thus cannot combine into a charge-2 OV under paraxial condition. The same conclusion could also be found in Fig. 8, where r_OV denotes the distance between + 1 OV and beam axis. Because of the dual focusing effect, r_OV reaches minimum values in two focal regions, and becomes larger in lens focal region and other positions. Comparing with the case of single off-axis OV, the repelling effect of two same OV in focused CAVB is obvious, especially in lens focal region. As shown in Fig. 8, for the single + 1 OV, the distance between OV and beam center is 28µm in lens focal plane z = f; but for the case of two same OVs, this distance would increase to 38µm because of the repelling effect. So, this pair of OVs do not just rotate rigidly as nested in GB [38].

Fig. 7. Intensity profiles [(a)–(f)] and phase patterns [(a1)–(f1)] when two same OVs (+1, +1) are imposed in focused CAVB. (a) z = 0; (b) z = 0.2 m; (c) first focus plane, z = 0.26 m; (d) lens focus plane, z = 0.3 m; (e) second focus plane, z = 0.35 m; (f) z = 0.4 m.

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Fig. 8. Changes of the distance between OV and beam axis r_OV with z in dual focusing process, r_k=0.5 mm for both cases.

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Figure 9 shows the intensity patterns and phase patterns of focused CAVB with two opposite OVs (+1, −1) in different output planes. As we can see in this figure, the two opposite OVs would rotate in different direction, attract each other and annihilate exactly in the first focal plane z = f₁ [Fig. 9(c1)]. Because of the disappearance of OV, the focal intensity profile would become Gaussian-like [Fig. 9(c)]. The beam could keep the nonhollow profile in the lens focus plane [Fig. 9(d)] and the second focus plane [Fig. 9(e)]. After the second focal plane, the beam would become hollow and unsymmetrical again [Fig. 9(f)], but the two opposite OVs would never reappear [Fig. 9(f1)]. Because of the collision of two opposite OVs, the beam with (+1, −1) OVs shows similar dual focus property as CAB without OV [ Fig. 10(a)]. However, because of the introduction of OVs, the beam with opposite OVs shows some interesting differences. As shown in Fig. 10(b), when two opposite OVs are imposed in, the axial focal widths would be broadened, and the intensity in lens focal region would be greatly reduced.

Fig. 9. Intensity profiles [(a)–(f)] and phase patterns [(a1)–(f1)] when two opposite OVs (−1, +1) are imposed in focused CAVB. (a) z = 0; (b) z = 0.2 m; (c) first focus plane, z = 0.26 m; (d) lens focus plane, z = 0.3 m; (e) second focus plane z = 0.35 m; (f) z = 0.4 m.

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Fig. 10. Propagation dynamics of focused CAVB with (+1, −1) OVs in y-z plane (a) and the comparison of intensity distribution along beam axis with focused CAB without OV (b).

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5. Conclusions

The dual focus property of Janus vortex waves is investigated in this paper based upon the expression of focused CAVB. A convenient and accurate numerical method based upon Hankel transform is presented to simulate the propagation dynamics of light beams through ABCD optical system. By using this method, the propagation characteristics of focused CAVB with an on-axis OV, an off-axis OV and a pair of OVs are investigated in detail.

When an on-axis OV is imposed in Janus vortex waves, two dark foci could be generated, thus a perfect light hollow bottle could be formed. This hollow bottle might be useful in optical micromanipulation. As the topological charge n or the initial radius r₀ increases, the two dark foci would become closer. The focal intensities at two foci would decrease with n, but increase with r₀. By varying the distance between the input plane and the thin lens s, the focal positions would be changed. And most importantly, the relative intensity between the two focal intensities could be controlled by s. When s is small, the intensity at first focus is larger; but as s increases, the second focal intensity would increase and even exceeds the first focal intensity. These modulation methods might be of great help for the application of Janus vortex wave in dual-trap optical tweezers or other fields.

The off-axis OV would rotate rapidly in two focal regions of Janus vortex waves, but keeps nearly still in the lens focal region. The off-axis OV in Janus vortex wave shows interesting behaviors when compared with the common GB. The off-axis OV in GB could not finish even one complete rotation, it slowly reaches to the π position after the focus. But the OV in CAVB experiences several complete rotations in two focal regions; and in each focusing process, the angular displacement of OV is nearly π/2. Two same OVs would rotate around each other and exhibit repelling effect in focusing process. Two opposite OVs would annihilate exactly at the first focus, and the beam would become nonhollow. After the second focus, the beam would become hollow again, but the OV would never reappear.

We believe that the Janus vortex wave proposed in this paper is helpful to understand the interesting behavior of OV; and could also be used in optical micromanipulation, optics communication and other fields..

Funding

Natural Science Foundation of Zhejiang Province (LY20A040006); National Natural Science Foundation of China (11504274).

Disclosures

The authors declare no conflicts of interest.

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Propagation dynamics of Janus vortex waves

Abstract

1. Introduction

2. Theoretical simulation of propagation of Janus vortex wave

3. Rotation of an off-axis OV in Janus vortex wave

4. Propagation dynamics of Janus vortex wave with two OVs

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (10)

Equations (14)

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