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Self-healing of the bored helico-conical beam

Jinfu Zeng, Shubo Cheng, Shuo Liu, Geng Zhang, Shaohua Tao, and Wenxing Yang

Open Access Open Access

Abstract

We investigate the dynamic evolution and self-healing properties of the bored helico-conical beams with different filter parameter S in this paper. The relative error coefficient, D, is utilized to judge the self-healing effect of the bored helico-conical beam. The result demonstrates that the self-healing effect of the bored beams will be perfect when D approaches to 0. We also discuss the influence of the filter parameter S on the effective self-healing distance theoretically and experimentally. The result demonstrates that the effective self-healing distance decreases exponentially with the increasing filter parameter S. Moreover, the corresponding transverse energy flows of the bored beams are analyzed. The experimental results of the dynamic evolution for the bored helico-conical beams agree with the simulation ones well.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Structured light involves converting an initial beam (such as Gaussian mode of a laser beam) into a desired structure (i.e., tailoring or shaping of light) [1,2]. Over the past few decades, different structured light have been demonstrated and implemented in different fields such as super-resolution microscopy, optical communication and optical manipulation [3–5]. For example, Zhao et al. demonstrated that the spherical vortex beams can trap large particles [6]. Suarez et al. developed a holographic optical tweezers system with arrayed Airy beams carrying different initial launch angles, for trapping microparticles more stably and efficiently [7]. Xin et al. applied the structured light to biological applications, and created a thermoplasmonics combined optical trapping technique for optical manipulation [8–11]. Different from the conventional structured light, such as Laguerre–Gaussian beams [4,6], Hermite–Gaussian beams [12], Bessel beams [13], Airy beams [14,15], an optical beam with a non-separable radial and azimuthal phase, i.e., helico-conical beam, generated with a computer-generated hologram, possess the spiral intensity distributions and obey a simple scaling relationship between the observed spiral geometry and initial phase distributions [16].

Recently, the generation, propagation and application of the helico-conical beams have been reported [16–18]. Based on the interference phase measurement technique, N. Hermosa et al. used a Mach–Zehnder interferometer to determine the phase structures of the helico-conical beams [18]. The orbital angular momentum (OAM) properties of the helico-conical beams, detected by a common path interferometer, have been also analyzed [19]. The superposition of multiple helico-conical beams at the focal plane was presented and will be promising in optical manipulation [20]. Furthermore, Cheng et al. proposed a new kind of modified helico-conical optical beam [21–23], which can generate spiral intensity distributions with adjustable openings. The propagation characteristics of the polarized helico-conical Lorentz-Gauss beams, radially polarized helico-conical Airy beams and Bessel–Gaussian beam mixing helico–conical phase wavefront have been also proposed and analyzed in detail [24–26]. Moreover, the blocked helico-conical beam proved to possess the self-healing property [27,28], which is similar to that of the beams such as Bessel beams [13], Airy beams [14] and Pearcey beams [29]. In Refs. [30–32], a modified helical phase obtained by eliminating the on-axis screw-dislocation has been proposed to generate a new kind of intensity patterns. In this paper we bore the helico-conical phase to generate the controllable intensity distributions. The generation method of the bored helico-conical beams will be described in detail. The self-healing property of the bored helico-conical beam is analyzed theoretically and experimentally. The bored helico-conical beams can be customized and may find potential applications in various fields, including optical trapping and optical guiding.

2. Bored helico-conical beam

The noncanonical phase function of the helico-conical optical beam can be expressed by

$$\psi ({r,\theta } )= l\theta ({K - r/{r_0}} )$$
where (r, θ) denotes the polar coordinates, K is a constant with a value of either 1 or 0, r0 is the normalization factor of the radial coordinate r, and l represents the topological charge [16]. The sign of the topological charge l can change the spiral direction of the beam. For simplicity, l takes a positive value and K is equal to 1 in this paper. The bored helico-conical beam can be generated with the customized helico-conical phase. The phase can be expressed by a matrix in the simulations. The customized helico-conical phase can be expressed with the equation ${\psi _{_c}} = \psi ({r,\theta } )\cdot T$, where T is a filter. The filter T can be given by Eq. (2).
$$T({r,\theta } )= \left\{ {\begin{array}{c} {1,\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 0 \le l\theta ({K - r/{r_0}} )\le 2\pi S\; \; \; \; \; \; \; \; }\\ {0,\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \textrm{others}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; } \end{array}} \right.$$
where S = 0, 1, …, l-1, l. As an example, Fig. 1(a) shows the whole helico-conical phase with K = 1, l = 15. The designed filter T with K = 1, l = 15, and S = 5 is shown in Fig. 1(b). The customized helico-conical phase ${\psi _{_c}}$ is shown in Fig. 1(c).

 figure: Fig. 1.

Fig. 1. (a) The whole helico-conical phase with K = 1, l = 15. (b) The designed filter T with K = 1, l = 15, and S = 5. (c) The customized helico-conical phase with K = 1, l = 15, and S = 5.

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The light field of the bored helico-conical beam in the source plane can be defined as

$${u_0}({x,y,z = 0} )= \exp \left( { - \frac{{{x^2} + {y^2}}}{{{w^2}}}} \right)\exp ({i{\psi_c}} )$$
where x and y are the dimensionless coordinates in Cartesian coordinates, z is the propagation distance, w is the beam waist of the Gaussian beam.

The propagation evolution of the bored helico-conical beam can be simulated by the angular plane wave spectrum theory, which is expressed as

$$u({x,y,z} )= iFT\{{FT[{{u_0}({x,y,z = 0} )} ]H} \}$$
where $u({x,y,z} )$ denotes the complex amplitudes at the exit plane, $FT$ and $iFT$ are the Fourier transform and inverse Fourier transform, respectively, and H is the transfer function defined in [23]
$$H = \exp \left[ {i2\pi z\sqrt {\frac{1}{{{\lambda^2}}} - k_x^2 - k_y^2} } \right]$$
where λ is the wavelength of light, which is set as 532 nm in the following simulations. kx and ky denote the wave vector in the x and y directions, respectively, which can be obtained from the following equation [6]
$$FT\{{u({x,y,z = 0} )} \}= \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {u({x,y,z = 0} )} } \exp [{ - i({k_x}x + {k_y}y)} ]dxdy$$

Thus, the intensity distribution at the exit plane can be calculated,

$$I({x,y,z} )= \overline u ({x,y,z} )u({x,y,z} )$$
where the overbar represents the conjugate.

3. Numerical simulations and experimental results

Here, the bored helico-conical beam can be generated by using a spatial light modulator (SLM). Figure 2(a) shows the experimental setup. The expanded beam (λ = 532 nm) is passed through a polarizer onto the SLM (Holoeye, Pluto-VIS-096, 1920×1080 pixels, 8µm×8µm/pixel, reflective type) encoded with the customized helico-conical phase. A blazed grating is added to separate the beam, as shown in Fig. 2(b). The beam reflected from the SLM is passed through the filter, and the intensity pattern can be captured by a charge coupled device (CCD).

 figure: Fig. 2.

Fig. 2. (a) The experimental setup. (Focal length, L1 = 30 mm, L2 = 200 mm). (b) The customized helico-conical phase hologram with a blazed grating.

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Figures 3(a)–3(d) show the phase holograms used to generate the whole helico-conical beam with K = 1, l = 20, 30, 40, 50, respectively. With the topological charge l increased at the propagation distance z = 0.3m (from SLM to CCD), the openings of the helico-conical beams get wider in Figs. 3(e)–3(h). The experimental results shown in Figs. 3(i)–3(l) agree with the theoretical ones in Figs. 3(e)–3(h). The intensity distributions of the bored helico-conical beams are also analyzed in this paper. As an example, the customized helico-conical phases with K = 1, l = 50, and S = 10, 15, 25, 30 are shown in Figs. 4(a)–4(d), respectively. When the helico-conical phase is customized by using Eq. (2), the corresponding helico-conical beam will be bored. The intensity patterns of the bored helico-conical beams at the propagation distance z = 0.4m are correspondingly shown in Figs. 4(e)–4(h). From Figs. 4(c)–4(d) and Figs. 4(g)–4(h) it can be seen that the small bored regions of the helico-conical phases have little effect on the wholeness of the helico-conical beam. Moreover, analyzing Figs. 4(a)-(d) and Figs. 4(e)–4(h), we can find that the bored regions of the helico-conical beams get larger with the parameter S decreased. The corresponding experimental results shown in Figs. 4(i)–4(l) are consistent with those shown in Figs. 4(e)–4(h).

 figure: Fig. 3.

Fig. 3. (a)- (d) The helico-conical phase holograms of l = 20, l = 30, l = 40, l = 50 for K = 1, respectively; (e)-(h) The corresponding simulated intensity distributions of (a)-(d) at the propagation distance z = 0.3 m; (i)-(l) The corresponding CCD-captured intensity distributions of (e)-(h).

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 figure: Fig. 4.

Fig. 4. (a)-(d) The bored helico-conical phase holograms with K = 1, l = 50, and S = 10, 15, 25, 30, respectively; (e)-(h) The corresponding simulated intensity distributions of (a)-(d) at the propagation distance z = 0.4 m; (i)-(l) The corresponding CCD-captured intensity distributions of (e)-(h).

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In order to analyze the self-healing properties of the bored helico-conical beams, we compare the intensity distributions of the whole helico-conical beam with K = 1, l = 50 and those of the bored helico-conical beam with K = 1, l = 50, S = 20 at the different propagation distances. Figures 5(a)–5(e) show the intensity distributions of the whole helico-conical beam with K = 1, l = 50 at the propagation distance z = 0.1m, 0.2m, 0.3m, 0.4m, 0.5m, respectively. The corresponding experimental intensity patterns are shown in Figs. 5(k)–5(o), respectively. For comparison, Figs. 5(f)–5(j) show the intensity distributions of the bored helico-conical beam with K = 1, l = 50, S = 20 at the propagation distance z = 0.1m, 0.2m, 0.3m, 0.4m, 0.5m, respectively. And the corresponding experimental intensity patterns are shown in Figs. 5(p)–5(t), respectively. The tails of the beams in Figs. 5(f)–5(j) are extremely similar as those in Figs. 5(a)–5(e). Moreover, it can be seen from Figs. 5(f)–5(j) that the background beams gradually converge toward to the center area and the bored region (the head of the spiral intensity) of the helico-conical beam can be gradually reconstructed. In other words, the bored helico-conical beam can evolve into a whole one when the bored helico-conical beam with K = 1, l = 50, S = 20 propagates along the axial direction over 0.5m. The experimental results shown in Figs. 5(k)–5(t) are consistent with the simulation ones shown in Figs. 5(a)–5(j) well.

 figure: Fig. 5.

Fig. 5. The intensity distributions of the whole helico-conical beam with K = 1 and l = 50 at the propagation distance of (a) z = 0.1 m, (b) z = 0.2 m, (c) z = 0.3 m, (d) z = 0.4 m, (e) z = 0.5 m; (k)-(o) The corresponding CCD-captured intensity distributions of (a)-(e); The intensity distributions of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the propagation distance of (f) z = 0.1 m, (g) z = 0.2 m, (h) z = 0.3 m, (i) z = 0.4 m, (j) z = 0.5 m; (p)-(t) The corresponding CCD-captured intensity distributions of (f)-(j).

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We also analyze the self-healing properties of the bored helico-conical beams with the Poynting vector [33]. The Poynting vector of the helico-conical beam can be written as $\vec{S} = {\varepsilon _0}\vec{E} \times \vec{B}$, where ${\varepsilon _0} = c/4\pi $ and c is the speed of light. In the Lorenz gauge, the time-averaged Poynting vector can be calculated and written as

$$\begin{array}{l} \left\langle {\overrightarrow S } \right\rangle = {\varepsilon _0}\left\langle {\overrightarrow E \times \overrightarrow B } \right\rangle = \frac{{{\varepsilon _0}}}{2}[{({{E^\ast } \times B} )+ ({E \times {B^\ast }} )} ]\\ \mathop {}\limits^{} {\mathop {}\limits^{} _{}} = \left[ {\frac{{i\omega c}}{{8\pi }}\left( {u\frac{{\partial u^{\ast} }}{{\partial x}} - u^{\ast} \frac{{\partial u}}{{\partial x}}} \right)} \right]\overrightarrow x + \left[ {\frac{{i\omega c}}{{8\pi }}\left( {u\frac{{\partial u^{\ast} }}{{\partial y}} - u^{\ast} \frac{{\partial u}}{{\partial y}}} \right)} \right]\overrightarrow y + \frac{{\omega kc}}{{4\pi }}{|u |^2}\overrightarrow z \end{array}$$
where, $\vec{x}$, $\vec{y}$ and $\vec{z}$ denote the unit vectors along the x, y and z directions, respectively. The sign * represents the complex conjugate. $\omega $ and k are the angular frequency and the wave number of the beam, respectively. The $\vec{x}$ and $\vec{y}$ terms contribute the non-zero transverse components (x-y plane) of the Poynting vector. The $\vec{z}$ term is the longitudinal component of the Poynting vector, which denotes the energy flow in the z-direction. In the discussion, we are interested in the $\vec{x}$ and $\vec{y}$ contribution to the energy flow, which can exhibit the healing of the transverse intensity distribution.

The dynamic evolution of the Poynting vector can be used to explain the self-healing phenomenon of the bored helico-conical beam during the propagation. The magnitude and direction of the white arrows demonstrate the counterpart of the energy flow in the transverse planes. The transverse energy flows of the helico-conical beams in Figs. 5(a)–5(e) are shown in Figs. 6(a)–6(e), with the cross-section intensity of the beams in the backplane of each frame. Meanwhile, Figs. 6(f)–6(j) also show the transverse energy flows of the bored helico-conical beams in Figs. 5(f)–5(j). Compared with those in Figs. 6(a)–6(e), as the propagation distance z increases in Figs. 6(f)–6(j), the transverse energy flows from the surrounding to the central area and gradually recovers in the missing part of the beam. Especially, in Fig. 6(h) the energy flow towards the bored head of the helico-conical beam explicitly. The energy around the head exhibits a stronger and larger density distribution. The energy flow of the bored beam changes dynamically during the propagation and exhibits the healing of the missing intensity distribution. Generally, the bored energy flow at the origin head of the helico-conical beam was recovered basically after the propagation distance z = 0.3m [see Figs. 6(h) and 6(i)]. The self-healing phenomenon of the beam is apparent. It means that the transverse energy flows are the source of the bored helico-conical beams self-healing.

 figure: Fig. 6.

Fig. 6. (a)-(e) The transverse energy flows of the helico-conical beam with K = 1 and l = 50 at the distances of z = 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, respectively; (f)-(j) The transverse energy flows of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the same prorogation distances of z = 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, respectively. [Background intensity distributions are the versions of Figs. 5(a)–5(j).]

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In this paper we introduce an intensity similarity factor D, i.e., the relative error coefficient, in order to analyze the self-healing effect of the bored helico-conical beam. The relative error theory is widely used in mathematics and physics. The calculation formula can be written as

$$D = \frac{{M - N}}{M}$$
where, M and N represent the intensities of the whole helico-conical beam and the bored helico-conical beam, respectively. The maximum value of the relative error coefficient D is 1. The bored helico-conical beam evolves into a whole one when D is equal to 0.

The head of the helico-conical beams will be bored gradually with the parameter S decreased. In Figs. 7(a)–7(e), the heads of the whole helico-conical beams are marked with the white circles at the same location. For comparison, the corresponding regions of the bored helico-conical beams in Figs. 7(f)–7(j) are also marked with the white circles at the same location. The corresponding local intensity distributions in the circle are shown in Figs. 7(k)–7(t). Comparing the local intensity distributions in Figs. 7(k)–7(o) with those in Figs. 7(p)–7(t), we find that the local intensity distributions shown in Fig. 7(o) and 7(t) have a perfect match. It means that the self-healing effect of the bored helico-conical beam with l = 50, K = 1 and S = 20 is perfect at prorogation distances of z = 0.58m. The plot of the similarity factor D versus the propagation distance z (from 0.1m to 0.9m) is shown in Fig. 7(u). It can be seen that the relative error coefficient D approaches to 1 before the prorogation distance of z = 0.3m, which means that there is an obvious difference between the whole helico-conical beam and the bored helico-conical beam. When z increases from 0.3m to 0.5m, the value D drops rapidly, indicating that the similarity gap between the whole helico-conical beam and the bored helico-conical beam gradually decreases. Then the value D gradually approaches to 0 with the propagation distance z increased, and the similarity gap also gets smaller.

The plots of the similarity factor D versus the propagation distance z for the different filter parameter S = 15, 20, 25 are shown in Fig. 8(a). The red dotted line represents the threshold line of D = 0.1. The curves have the three crossing positions with the threshold line, and the propagation distance corresponding to the position is called the effective self-healing distance ze. Thus, it means that the bored helico-conical beam carrying different filter parameter S has different effective self-healing distance ze. The plot of the effective self-healing distance ze versus the filter parameter S is shown in Fig. 8(b). The exponential curve fitting can be implemented. The result shows that the effective self-healing distance ze gradually decreases exponentially with the filter parameter S increased. In other words, the helico-conical beam with a large bored region will be reconstructed effectively over a long distance.

 figure: Fig. 7.

Fig. 7. (a)-(e) The intensity distributions of the whole helico-conical beam with l = 50 and K = 1 at the propagation distance z = 0.18 m, 0.28 m, 0.38 m, 0.48 m, 0.58 m, respectively; (k)-(o) the corresponding local intensity distributions marked with the white circles in (a)-(e); (f)-(j) the intensity distributions of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the propagation distance z = 0.18 m, 0.28 m, 0.38 m, 0.48 m, 0.58 m, respectively; (p)-(t) the corresponding local intensity distributions at the same locations in (f)-(j). (u) Plot of the similarity factor D versus the propagation distance z.

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 figure: Fig. 8.

Fig. 8. (a) Plots of the similarity factor D versus the propagation distance z (from 0.1 m to 0.9 m) for the different filter parameter S = 15, 20, 25. (b) Plots of the effective self-healing distance ze versus the filter parameter S.

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4. Conclusion

In this paper the bored helico-conical beam is proposed. The self-healing diffraction characteristics of the beams are verified theoretically and experimentally. The relative error coefficient D is introduced to analyze the self-healing effect of the bored helico-conical beam. When the value D greatly approaches to 0, the self-healing helico-conical beam is in good agreement with the whole one. Moreover, the self-healing distance ze gradually decreases exponentially with the filter parameter S increased. It is expected that the bored helico-conical beams can be promising in the research fields of multi-plane optical manipulation.

Funding

National Natural Science Foundation of China (11904032, 11674401, 11774054, 11775190, 12075036).

Disclosures

There are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. (a) The whole helico-conical phase with K = 1, l = 15. (b) The designed filter T with K = 1, l = 15, and S = 5. (c) The customized helico-conical phase with K = 1, l = 15, and S = 5.

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Fig. 2.
Fig. 2. (a) The experimental setup. (Focal length, L1 = 30 mm, L2 = 200 mm). (b) The customized helico-conical phase hologram with a blazed grating.

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Fig. 3.
Fig. 3. (a)- (d) The helico-conical phase holograms of l = 20, l = 30, l = 40, l = 50 for K = 1, respectively; (e)-(h) The corresponding simulated intensity distributions of (a)-(d) at the propagation distance z = 0.3 m; (i)-(l) The corresponding CCD-captured intensity distributions of (e)-(h).

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Fig. 4.
Fig. 4. (a)-(d) The bored helico-conical phase holograms with K = 1, l = 50, and S = 10, 15, 25, 30, respectively; (e)-(h) The corresponding simulated intensity distributions of (a)-(d) at the propagation distance z = 0.4 m; (i)-(l) The corresponding CCD-captured intensity distributions of (e)-(h).

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Fig. 5.
Fig. 5. The intensity distributions of the whole helico-conical beam with K = 1 and l = 50 at the propagation distance of (a) z = 0.1 m, (b) z = 0.2 m, (c) z = 0.3 m, (d) z = 0.4 m, (e) z = 0.5 m; (k)-(o) The corresponding CCD-captured intensity distributions of (a)-(e); The intensity distributions of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the propagation distance of (f) z = 0.1 m, (g) z = 0.2 m, (h) z = 0.3 m, (i) z = 0.4 m, (j) z = 0.5 m; (p)-(t) The corresponding CCD-captured intensity distributions of (f)-(j).

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Fig. 6.
Fig. 6. (a)-(e) The transverse energy flows of the helico-conical beam with K = 1 and l = 50 at the distances of z = 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, respectively; (f)-(j) The transverse energy flows of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the same prorogation distances of z = 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, respectively. [Background intensity distributions are the versions of Figs. 5(a)–5(j).]

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Fig. 7.
Fig. 7. (a)-(e) The intensity distributions of the whole helico-conical beam with l = 50 and K = 1 at the propagation distance z = 0.18 m, 0.28 m, 0.38 m, 0.48 m, 0.58 m, respectively; (k)-(o) the corresponding local intensity distributions marked with the white circles in (a)-(e); (f)-(j) the intensity distributions of the bored helico-conical beam with K = 1, l = 50, and S = 20 at the propagation distance z = 0.18 m, 0.28 m, 0.38 m, 0.48 m, 0.58 m, respectively; (p)-(t) the corresponding local intensity distributions at the same locations in (f)-(j). (u) Plot of the similarity factor D versus the propagation distance z.

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Fig. 8.
Fig. 8. (a) Plots of the similarity factor D versus the propagation distance z (from 0.1 m to 0.9 m) for the different filter parameter S = 15, 20, 25. (b) Plots of the effective self-healing distance ze versus the filter parameter S.

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Equations (9)

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ψ ( r , θ ) = l θ ( K − r / r 0 )
T ( r , θ ) = { 1 , 0 ≤ l θ ( K − r / r 0 ) ≤ 2 π S 0 , others
u 0 ( x , y , z = 0 ) = exp ⁡ ( − x 2 + y 2 w 2 ) exp ⁡ ( i ψ c )
u ( x , y , z ) = i F T { F T [ u 0 ( x , y , z = 0 ) ] H }
H = exp ⁡ [ i 2 π z 1 λ 2 − k x 2 − k y 2 ]
F T { u ( x , y , z = 0 ) } = ∫ − ∞ + ∞ ∫ − ∞ + ∞ u ( x , y , z = 0 ) exp ⁡ [ − i ( k x x + k y y ) ] d x d y
I ( x , y , z ) = u ¯ ( x , y , z ) u ( x , y , z )
⟨ S → ⟩ = ε 0 ⟨ E → × B → ⟩ = ε 0 2 [ ( E ∗ × B ) + ( E × B ∗ ) ] ⁡ = [ i ω c 8 π ( u ∂ u ∗ ∂ x − u ∗ ∂ u ∂ x ) ] x → + [ i ω c 8 π ( u ∂ u ∗ ∂ y − u ∗ ∂ u ∂ y ) ] y → + ω k c 4 π | u | 2 z →
D = M − N M
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