Abstract

We investigate the self-healing property of focused circular Airy beams (FCAB), and this property is associated with the transverse Poynting vector (energy flow) for a better interpretation. We both experimentally and numerically show the effect of the obstruction’s position, size and shape on the self-healing property of FCAB. It is found that FCAB will heal if the obstruction is placed at the area between the two foci of FCAB, and it has the least influence on the FCAB when the obstruction is placed near the lens’ rear focal plane, whereas FCAB cannot heal if the obstruction is out of the area between two foci. Our experimental results are in good agreement with numerical results.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Since Berry et al. theoretically found non-diffracting Airy waves in 1979 [1] and Siviloglou et al. theoretically introduced [2] and experimentally observed [3] finite-energy optical Airy beams in 2007, the study of propagation properties and applications of Airy beams is of great interest in optics [212], such as the Poynting vector and angular momentum [5]. Airy beams have several novel properties, such as self-healing [6], self-accelerating and self-similar propagating [2,3]. Self-healing property describes that the intensity distribution of Airy beam is stable and hard to be disturbed by small perturbation, and their intensity can recover within a short propagation distance. There are several optical beams exhibiting self-healing, including Bessel beams [1315], Pearcey beams [16], Helico-conical beams [17], cosh-Airy beams [18], and space-time light sheets [19]. The self-healing property makes Airy beams useful in optical imaging [20,21], atmospheric science [22] and optical manipulation [7,23]. The self-healing properties of Airy beams in scattering media, turbulent media and blocked by a rectangular block are extensively investigated. It is demonstrated that Airy beams have the high stability in scattering and turbulent media, compared with Gaussian beams [6]. Most of these investigations are about rectangular Airy beams (RABs).

Later, circular Airy beams (CABs) were proposed by Efremidis et al. [8], known as a kind of self-focusing beams. The self-focusing property makes CABs useful in optical manipulation [23]. As a kind of Janus waves [24], CABs can be divided into two self-focusing and self-defocusing components. The self-defocusing component can be refocused in the lens system, resulting that CABs will have two foci in the lens system. Especially, when the front focal plane of the lens is the initial plane of the CAB, its intensity distribution is symmetrical about the rear focal plane of the lens.

The self-focusing property of CABs and the self-accelerating property of RABs is essentially the same. However, the self-healing property of CABs is very distinct from that of RABs. Previous study demonstrated that the focus of CABs plays an important role in self-healing process [25]. Specifically, the self-healing process is slight before its focus and mainly takes place for a short distance after the focus. When CABs’ first few rings are covered or partially blocked, it can’t self-heal until it propagates after its focus. Thus, the self-healing distance for CABs is approximately equal to its self-focusing length, and sometimes it can be very long, whereas RABs can self-heal in a small propagation distance. However, the self-healing property of focused circular Airy beams (FCABs) has never been investigated. We found that the FCAB cannot self-heal when it is partially blocked at the initial plane, which indicates the self-healing property of the FCAB is distinct from the CAB. Therefore, the question still remains that whether the FCABs have the property of self-healing.

In this work, we investigate the self-healing property of FCABs both experimentally and numerically. We find that FCAB can self-heal only in a specific range between two foci, and observe the process of its self-healing during the propagation, which can be explained well by the transverse Poynting vector (energy flow). Then we quantify the self-healing property of FCAB and reveal the effect of the obstruction’s position, shape and size, and the propagation distance on the self-healing property of FCAB. Finally, it shows that the position of the obstruction will influence the process of self-healing, and there is a best obstruction’s position where the obstruction has the least effect on FCAB.

2. Propagation of FCABs with a block

The light field of the CAB at the initial plane can be expressed as [8]

$$E_{\mathrm{i}}(r,z=0)=\mathrm{Ai}\left(\frac{r_0-r}{w}\right)\exp\left[\frac{a\left(r_0-r\right)}{w}\right],$$
where $r$ is the radial coordinate, $\mathrm {Ai}(\cdot )$ denotes the Airy function, $r_0$ is the radius of the main ring, $w$ is the arbitrary scaling factor, and $a$ is the decay coefficient. The self-focusing length of the CAB in free space is given by [9] $f_{\mathrm {Ai}}=2k\sqrt {w^3(r_0+w)}$, where $k=2\pi /\lambda$ is the wave number. Now assume a sector-shaped block position at the initial plane, the light filed turns into
$$E_{\mathrm{b}}(r,\phi,z=0)= \begin{cases} 0, & \phi \in [-\Delta\theta/2,\Delta\theta/2]\\ E_\textrm{i}, & otherwise. \end{cases}$$
where $\Delta \theta$ is the angle spread of the sector-shaped block. For other shape blocks, one can do the similar calculation as follows, by replacig the block shape.

If a lens is inserted at a distance $z_{L}$ after the initial plane, the propagation evolution of the blocked CAB can be calculated by Collins formula [26,27]

$$\begin{aligned} E_{\mathrm{o}}(\rho ,\theta ,z)=& -\frac{ik}{2\pi B}\exp (ikz)\iint E_{ \mathrm{b}}(r,\phi ,z=0)\\ & \times \exp \left\{ \frac{ik}{2B}[Ar^{2}+D\rho ^{2}-2\rho r\cos (\phi -\theta )]\right\} rdrd\phi , \end{aligned}$$
where $A=1-(z-z_L)/f,B=z-z_{L}(z-z_L)/f,D=1-z_{L}/f$. The FCAB has two foci in the single lens system, and the positions of these foci can be expressed as [24]
$$z_{F,\pm }=z_L+\frac{f(f_{\mathrm{Ai}}\pm z_{L})}{f_{\mathrm{Ai}}\pm z_{L}\mp f},$$
where $f$ is the focal length of the lens. When $z_{L}=f$, i.e., the front focal plane of the lens is the initial plane of CAB, we have $z_{F,\pm }=2f\pm f^{2}/f_{\mathrm {Ai}}$. In this case, the two foci are symmetrical about the rear focal plane of lens. In our experiment, we use the following parameters: $r_{0}=1$ mm, $w=0.08$ mm, $a=0.05$, $\lambda =633$ nm, $f=125$ mm, thus we have $z_{F,-}=215$ mm and $z_{F,+}=285$ mm.

3. Experiment and results

For experimentally investigating the self-healing property of FCAB, we generate the CAB using a phase-only spatial light modulator (SLM, HoloeyePLUTO-2-VIS-056). We employ the technology developed in Ref. [28] to convert the amplitude information to the phase information, which is based on the grating diffraction to achieve amplitude encoding through phase-only SLM. The experimental setup is shown in Fig. 1. A linearly polarized He-Ne laser ($\lambda =633$ nm) first passes through the half-wave plate and polarized beam splitter, which both are combined to control the intensity of the incident light beam (the reflected light is blocked), then the expanded laser impinges onto the SLM. The computer-generated phase diagram is encoded on it and an aperture is inserted before the lens $L_{0}$ to select the first-order diffraction light. A CAB is generated at the rear focal plane of a $2-f$ lens system with the focal length $f_{0}=500$ mm, which is denoted as the initial plane of $z=0$. A lens $L$ with its focal length $f=125$ mm is placed at $z_{L}$, where the distance $z_{L}$ from the initial plane to the lens $L$ is $f$. Then a sector-shaped block is placed after the lens $L$, and the longitudinal distance between the block and the initial plane is $z_b$. Finally, the intensity distribution is recorded by a CCD camera and $z$ is the distance from the camera to the initial plane. The insert figure in Fig. 1 shows the section intensity distribution of the FCAB, which is symmetrical about the rear focal plane of the lens $L$ (i.e., the position of $z=250$ mm). The FCAB is hollow and its rings reduce rapidly before the first focus $z_{F,-}$ and expand rapidly after the second focus $z_{F,+}$. Between the area of two foci is Bessel-like solid beams.

 figure: Fig. 1.

Fig. 1. Experimental setup for measuring intensity distributions of FCAB, and a CAB is generated at the initial plane of $z=0$. The right-top insert figure shows the section intensity distribution of the FCAB after the lens $L$. Other notations are: HWP, half-wave plate; PBS, polarized beam splitter; SLM, spatial light modulator; AP, aperture; IP, initial plane at $z=0$.

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3.1 Case I: the block is placed at $z=0$

Figures 2(a) and 2(b) show the experimental intensity distributions of the blocked CAB and blocked FCAB at $z=600$ mm, respectively. In both of these situations, the sector-shaped block is placed at $z=0$. As in Ref. [25], it shows that the CAB does not self-heal until its own focus when its first few rings are blocked. We find that the angularly blocked CAB also not self-heal until its focus. After its focus, the CAB self-heals clearly and its intensity distribution is almost symmetrical. Note that the self-healing property of the CAB is not related to the position of the block, the blocked CAB can self-heal as long as it passes through its focus. However the blocked FCAB does not exhibit self-healing property, for the truncation is obvious and asymmetric (see Fig. 2(b)) . Our experimental results show that the blocked FCAB can’t self-heal when the block is placed before the lens $L$.

 figure: Fig. 2.

Fig. 2. Experimental intensity distributions for (a) the blocked CAB and (b) the blocked FCAB, at the plane of $z=600$ mm. In (b) the lens $L$ is placed at $z_{L}=f=125$ mm and the obstruction is placed at $z_b=0$.

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3.2 Case II: the block is placed between two foci

 figure: Fig. 3.

Fig. 3. Experimental ((a), (c) and (e)) and numerical ((b), (d) and (f)) results of the intensity distributions of the blocked FCAB under different propagation distance. The sector-shaped obstruction is placed at $z_b=$ 235 mm (a)-(b), 250 mm (c)-(d) and 265 mm (e)-(f). The numbers 1-5 (in (a)-(f)) are the cases of $z=350$ mm, 400 mm, 450 mm, 500 mm and 600 mm, respectively. Here the sector-shaped obstruction has the angle spread $\Delta \theta =\pi /3$.

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Now we consider the self-healing property of the FCAB when the block is placed after its first focus (i.e., $z_b>z_{F,-}=215$ mm as we have calculated in the above). Figure 3 shows the experimental and numerical results of the intensity distributions of the blocked FCAB under different propagation distance $z$. It is found that the self-healing process of the FCAB is dependent on the block position $z_b$. In the case of $z_b=235$ mm and 250 mm, as the propagation distance $z$ increases, the blocked FCAB recovers or self-heals better, see Figs. 3(a)–3(d). However, in the case of $z_b=265$ mm, the blocked FCAB recovers or self-heals better initially (see Figs. 3(e1)–3(e2) and 3(f1)–3(f2)), and then it becomes inhomogeneous slightly (see Figs. 3(e3)–3(e5) and 3(f3)–3(f5)), which indicates that there is an optimal self-healing propagation distance. When the propagation distance is short, the blocked FCAB self-heals better in the case of $z_b=265$ mm, comparing to the cases of $z_b=235$ mm and 250 mm, in Figs. 3(a)–3(d). From Fig. 3, one can also find that the experimental measurements are in good agreement with the theoretical numerical results.

In order to show the effect of the block position on the self-healing property of the FCAB much clearly, here we fix the observation plane of the CCD at $z=600$ mm and change the position of the sector-shaped block between the two foci of the FCAB. Figure 4 shows the effect of the block position $z_b$ on the self-healing property of the FCAB both experimentally and numerically. When the block position is before or at the first focus, i.e., $z_b\leq z_{F,-}=215$ mm, the observed truncation is very strong, see Figs. 4(a1) and 4(b1). When the block is placed between the two foci, i.e., $z_{F,-}<z_b<z_{F,+}$, the observed truncation gradually becomes weak and the first few rings of the FCAB almost heal although the intensity is still inhomogeneous. As $z_b$ is close to the value of $(z_{F,-}+z_{F,+})/2$, the self-healing effect of observed FCAB becomes more significant, see Figs. 4(a2)–4(a4) and 4(b2)–4(b4). When $z_b$ equals to the value of $(z_{F,-}+z_{F,+})/2$, there is the maximal optimal effect of the self-healing property for such FCABs. When the block position is after or at the second focus, i.e., $z\geq z_{F,+}=285$ mm, the observed truncation becomes very strong again, see Figs. 4(a5) and 4(b5). Note that in the focusing region, i.e., the range between the two foci, the blocked FCAB will self-heal in a short propagation distance, which is not dependent on the block position $z_b$. Moreover, due to the highly focusing of the FCAB, most of the energy of the FCAB is located at the center and it is unnecessary to discuss the self-healing property in the focusing region. However, beyond the second focus, i.e., $z\ge z_{F,+}$, the self-healing property of the FCAB is distinct and strongly depends on the block position $z_b$ as stated above.

 figure: Fig. 4.

Fig. 4. Experimental (top panel) and numerical (bottom panel) results of the intensity distributions of a blocked FCAB at $z=600$ mm under different obstruction’s position $z_b$. (a1)-(a5) These figures are the cases of $z_b=195$ mm, $235$ mm, $250$ mm, $265$ mm and $305$ mm, respectively, and (b1)-(b5) are the corresponding numerical results. Here the sector-shaped obstruction has the angle spread $\Delta \theta =\pi /3$.

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 figure: Fig. 5.

Fig. 5. The transverse Poynting vector of the blocked FCAB at the planes of (a) $z=500$ mm and (b) $z=600$ mm, under the different obstruction’s position. The index numbers 1-3 in subfigures are the cases of $z_b=195$ mm, 250 mm and 305 mm. The white arrows represent the direction and relative strength of $\vec {S}_{//}$.

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In most cases, the self-healing properties of a specific beam, like [6,17,29] are explained by the transverse Poynting vector. In order to explain such self-healing process of the blocked FCAB, we calculate the transverse Poynting vector. For a linear-polarized light, the transverse Poynting vector can be expressed as [6]

$$\vec{S}_{//}=\frac{i}{4k}\sqrt{\frac{\epsilon _{0}}{\mu _{0}}}\left[ E_{\mathrm{b}}\nabla E_{\mathrm{b}}^{\ast }-E_{\mathrm{b}}^{\ast }\nabla E_{\mathrm{b}}\right],$$
where $\nabla =\hat {x}\partial /\partial x+\hat {y}\partial /\partial y$, $\epsilon _{0}$ and $\mu _{0}$ are the dielectric constant and permeability of vacuum, respectively; $i$ is the imaginary unit, and “$\ast$” refers to the complex conjugate. Here we calculate the transverse Poynting vector at the planes of $z=500$ mm and $600$ mm, and change the position of the sector-shaped obstruction after the lens $L$. Figure 5 shows the effect of the obstruction’s position $z_b$ on the transverse Poynting vector $\vec {S}_{//}$. From Fig. 5, one can find that the arrows show the direction of the transverse energy flow or the Poynting vector. It clearly delineates the construction of the annular distribution of light in the blocked region by the flow of energy from the side unblocked portion as the propagating distance increases. It is the energy flowing from the unblocked region that reconstructs the blocked region. When the obstruction is out of the region between two foci (see the subfigures with index numbers 1 and 3), the transverse energy flow pointing to the blocked area is insignificant, which results that the blocked FCAB cannot self-heal. When the obstruction is placed between two foci (the optimal position located at the middle of two foci), there exists the clear energy flowing from the unblocked region to the blocked portion.

3.3 Effect of the size and shape of the block on the self-healing property of the FCABs

 figure: Fig. 6.

Fig. 6. Experimental results of the intensity distributions of the blocked FCABs under different obstruction’s size and shape, (a) the cases of the sector-shaped obstruction with $\Delta \theta =\pi /6$, (b) the cases of the rectangular obstruction crossing half of the FCAB and (c) the cases of the rectangular obstruction crossing over the whole of the FCAB. The index numbers 1-5 in (a)-(c) are the places of $z=350$ mm, 400 mm, 450 mm, 500 mm, 600 mm, respectively. Note that all the obstructions are placed at $z_b=250$ mm.

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The effect of the shape and size of the block on the self-healing property of the FCAB is also investigated, and here we consider another case of the sector-shaped block with $\Delta \theta =\pi /6$ and the cases of the rectangular block, see Fig. 6. In this situation, all the blocks are placed at $z_b=250$ mm, which corresponds to the optimal position for the self-healing process. Figure 6(a) shows the intensity distributions of the blocked FCAB at different propagation distance when the angle spread of the sector-shaped block is $\Delta \theta =\pi /6$. Compared with the cases of Fig. 3(c), it is found that as $\Delta \theta$ decreases, the self-healing ability of the FCAB becomes better. In Fig. 6(b), it shows the self-healing process when the rectangular block with a width $0.08$ mm is used and it only crosses the half of the FCAB. It demonstrates clearly that the blocked FCAB can heal very well as it propagates. However, when the rectangular block crosses over the whole of the FCAB, see Figs. 6(c1)–6(c5), although the self-healing effect still exists, the self-healing ability becomes worse since the rectangular object blocks too large area within the transverse region. In Figs. 6(c1)–6(c5), it is seen that, even though the first few rings can heal, their intensity is weak. Note that the width of the rectangular block also influences the self-healing property of the FCAB. The FCAB cannot self-heal when the width of the rectangular block is too large.

 figure: Fig. 7.

Fig. 7. The similarity $F$ between the FCAB and the blocked FCAB, (a) as a function of the propagation distance $z$ under different $z_b$ for the sector-shaped obstruction with $\Delta \theta =\pi /3$, (b) as a function of the angle spread $\Delta \theta$ for the sector-shaped obstructions under the fixed $z_b$ and $z$, and (c) as a function of the obstruction’s position $z_b$ with a fixed value of $z=3000$ mm. The discrete dots and the error bars in (b) are the experimental data from the experimental intensity distributions according to Eq. (5).

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Next we consider the effect of the block position $z_b$, the angle spread $\Delta \theta$ of the sector-shaped block and propagation distance $z$ on the self-healing ability of the FCAB quantitatively. The self-healing ability of FCAB can be quantified by comparing the similarity between FCAB and blocked FCAB, which is defined as [30]

$$F(z_b,z)=\frac{\iint I(z)I_{b}(z_b,z)dxdy}{\sqrt{\iint I^{2}(z)dxdy\cdot \iint I_{b}^{2}(z_b,z)dxdy}},$$
where $I(z)$ is the intensity of the FCAB without the block and $I_{b}(z_b,z)$ is the intensity of the blocked FCAB in the presence of the block. Figure 7(a) shows the effect of the propagation distance $z$ on the self-healing ability of FCAB. When the block is placed between the first focus and the rear focal plane of the lens $L$, i.e., $z_{F,-}<z_b<(z_{F,-}+z_{F,+})/2$, the blocked FCAB gradually reconstructs with propagating and finally stabilizes, see the red, bule and green curves of Fig. 7(a) . As $z_b$ is close to the value of $(z_{F,-}+z_{F,+})/2$, the ability of self-healing is more significant. However, the blocked FCAB reconstructs faster at the beginning as $z_b$ is close to the value of $z_{F,+}$. When the block is placed between the rear focal plane of the lens $L$ and the second focus, i.e., $(z_{F,-}+z_{F,+})/2<z_b<z_{F,+}$, it is found that the curves of $F$ first reach to the maximum and then decrease slightly, and such behavior tells us that there is an optimal position to obtain the self-healing effect in these situations, also see the intensity distributions in Figs. 3(e1)–3(e5) or 3(f1)–3(f5). For example, the blocked FCAB can be reconstructed better around $z=400$ mm for the case of $z_b=265$ mm, see the black dashed curve in Fig. 7(a). When $z$ further increases, the value of $F$ decreases slightly and finally becomes stable.

Figure 7(b) shows the effect of the angle spread $\Delta \theta$ of the sector-shaped block on the self-healing property of FCAB. As stated above, the sector-shaped block has less influence on the FCAB when the angle spread $\Delta \theta$ decreases. Quantitively, the similarity $F$ linearly depends on the angle spread $\Delta \theta$, and the experimental data are in good agreement with the theoretical prediction.

Figure 7(c) shows the effect of the block position $z_b$ on the self-healing ability of FCAB at $z=3000$ mm, where the self-healing process of the blocked FCAB is almost completed (from the information in Fig. 7(a)). When the block is placed before the first focus, i.e., $z_b<z_{F,-}=215$ mm, the value of $F$ is a relatively low value. When the block is placed after the first focus, the value of $F$ increases and reaches a maximum at $z_b=(z_{F,-}+z_{F,+})/2$. Then the value of $F$ decreases until the block is placed after the second focus $z_{F,+}$. These properties are matched very well with the results in Figs. 3 and 4.

Therefore, when the block is placed out of the range between two foci, the blocked FCAB does not exhibit the self-healing property. Only in the specific range between two foci, the blocked FCAB exhibits the self-healing property. As the block approaches to the plane of $z_b=(z_{F,-}+z_{F,+})/2$, the self-healing property is more significant. There is the optimal effect of the self-healing property for the blocked FCAB when $s$ equals to the value of $(z_{F,-}+z_{F,+})/2$.

4. Conclusion

We have investigated the effect of the block's position, size, and shape on the self-healing property of the FCAB and have quantified their self-healing properties by comparing the similarity of the FCAB and the blocked FCAB. It shows that the blocked FCAB can self-heal when the block is placed in the specific range between the two foci of the FCAB, where the transverse Poynting vector (energy flow) pointing to the truncation area is significant. The self-healing ability linearly depends on the angle spread of the sector-shaped block. In the distant field where the self-healing process completes, as the block approaches the rear focal plane of the lens, the effect of the block on FCAB decreases gradually and there is the optimal effect of the self-healing property when the block is placed at the rear focal plane of the lens. Moreover, during the process of self-healing, the blocked FCAB gradually self-heals as the propagation distance increases, when the obstruction is placed in the range between the first focus and the rear focal plane of the lens. Also when the block is placed in the range between the rear focal plane and the second focus, there exists an optimal self-healing distance. The experimental results are in good agreement with numerical results. As self-healing Airy beams are applied wildely in optical-imaging, such as light-sheet microscopy [20] and label-free imaging through turbid media [20,21], our results may also provide the potential applications in optical-imaging and beam-propagation technologies.

Funding

National Natural Science Foundation of China (11674284, 11974309); Natural Science Foundation of Zhejiang Province (LD18A040001); National Key Research and Development Program of China (2017YFA0304202).

Disclosures

The authors declare no conflicts of interest.

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30. X. X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014). [CrossRef]  

References

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  • |

  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
  4. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019).
    [Crossref]
  5. H. Sztul and R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [Crossref]
  6. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref]
  7. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  8. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [Crossref]
  9. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
    [Crossref]
  10. P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
    [Crossref]
  11. A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
    [Crossref]
  12. H. E. Kondakci and A. F. Abouraddy, “Airy wave packets accelerating in space-time,” Phys. Rev. Lett. 120(16), 163901 (2018).
    [Crossref]
  13. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
    [Crossref]
  14. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
    [Crossref]
  15. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004).
    [Crossref]
  16. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
    [Crossref]
  17. N. Hermosa, C. Rosales-Guzman, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. 38(3), 383–385 (2013).
    [Crossref]
  18. G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
    [Crossref]
  19. H. Esat Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43(16), 3830–3833 (2018).
    [Crossref]
  20. T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
    [Crossref]
  21. H. Nagar, E. Dekel, D. Kasimov, and Y. Roichman, “Non-diffracting beams for label-free imaging through turbid media,” Opt. Lett. 43(2), 190–193 (2018).
    [Crossref]
  22. Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
    [Crossref]
  23. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, “Trapping and guiding microparticles with morphing autofocusing airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
    [Crossref]
  24. D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41(20), 4656–4659 (2016).
    [Crossref]
  25. N. Li, Y. F. Jiang, K. K. Huang, and X. H. Lu, “Abruptly autofocusing property of blocked circular airy beams,” Opt. Express 22(19), 22847–22853 (2014).
    [Crossref]
  26. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [Crossref]
  27. S. Wang and D. Zhao, Matrix optics, (CHEP, 2000).
  28. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38(23), 5004–5013 (1999).
    [Crossref]
  29. P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011).
    [Crossref]
  30. X. X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014).
    [Crossref]

2019 (2)

N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019).
[Crossref]

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

2018 (3)

2017 (1)

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

2016 (1)

2014 (3)

2013 (2)

N. Hermosa, C. Rosales-Guzman, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. 38(3), 383–385 (2013).
[Crossref]

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

2012 (1)

2011 (3)

2010 (2)

2008 (3)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

2004 (1)

1999 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

1996 (1)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1970 (1)

Abouraddy, A. F.

H. E. Kondakci and A. F. Abouraddy, “Airy wave packets accelerating in space-time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref]

H. Esat Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43(16), 3830–3833 (2018).
[Crossref]

Alfano, R.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Boothroyd, S. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Campos, J.

Chen, R. P.

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

Chen, Z.

Chen, Z. G.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Christodoulides, D. N.

Chrostowski, J.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Chu, X. X.

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

X. X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014).
[Crossref]

Cizmar, T.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Collins, S. A.

Coll-Llado, C.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Cottrell, D. M.

Couairon, A.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

Dalgarno, H. I.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Davis, J. A.

Dekel, E.

Dennis, M. R.

Dholakia, K.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Efremidis, N. K.

Esat Kondakci, H.

Fedorov, V. Y.

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41(20), 4656–4659 (2016).
[Crossref]

Ferrier, D. E.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Gbur, G.

Gu, Y.

Gunn-Moore, F. J.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Hermosa, N.

Huang, K. K.

Jiang, Y. F.

Kasimov, D.

Kondakci, H. E.

H. E. Kondakci and A. F. Abouraddy, “Airy wave packets accelerating in space-time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref]

Koulouklidis, A. D.

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

Li, N.

Lindberg, J.

Lu, X. H.

MacDonald, R. P.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Mazilu, M.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Mills, M. S.

Moreno, I.

Mourka, A.

Nagar, H.

Nylk, J.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Okamoto, T.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Panagiotopoulos, P.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

Papazoglou, D. G.

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41(20), 4656–4659 (2016).
[Crossref]

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

Prakash, J.

Ring, J. D.

Roichman, Y.

Rosales-Guzman, C.

Segev, M.

Singh, R. P.

Siviloglou, G. A.

Syreet, B. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Sztul, H.

Tao, S. H.

Torres, J. P.

Tzortzakis, S.

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41(20), 4656–4659 (2016).
[Crossref]

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

Vaity, P.

Vettenburg, T.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Wang, S.

S. Wang and D. Zhao, Matrix optics, (CHEP, 2000).

Wen, W.

Yuan, X.

Yzuel, M. J.

Zhang, P.

Zhang, Z.

Zhao, D.

S. Wang and D. Zhao, Matrix optics, (CHEP, 2000).

Zhou, G. Q.

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

Zhou, Y. M.

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Laser Phys. (1)

G. Q. Zhou, X. X. Chu, R. P. Chen, and Y. M. Zhou, “Self-healing properties of cosh-Airy beams,” Laser Phys. 29(2), 025001 (2019).
[Crossref]

Nat. Commun. (1)

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4(1), 2622 (2013).
[Crossref]

Nat. Methods (1)

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Llado, D. E. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Nat. Photonics (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Opt. Commun. (2)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syreet, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122(4-6), 169–177 (1996).
[Crossref]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Opt. Express (5)

Opt. Lett. (10)

H. Esat Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43(16), 3830–3833 (2018).
[Crossref]

P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

N. Hermosa, C. Rosales-Guzman, and J. P. Torres, “Helico-conical optical beams self-heal,” Opt. Lett. 38(3), 383–385 (2013).
[Crossref]

H. Nagar, E. Dekel, D. Kasimov, and Y. Roichman, “Non-diffracting beams for label-free imaging through turbid media,” Opt. Lett. 43(2), 190–193 (2018).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, “Trapping and guiding microparticles with morphing autofocusing airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref]

D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Janus waves,” Opt. Lett. 41(20), 4656–4659 (2016).
[Crossref]

Optica (1)

Phys. Rev. Lett. (3)

A. D. Koulouklidis, D. G. Papazoglou, V. Y. Fedorov, and S. Tzortzakis, “Phase memory preserving harmonics from abruptly autofocusing beams,” Phys. Rev. Lett. 119(22), 223901 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Airy wave packets accelerating in space-time,” Phys. Rev. Lett. 120(16), 163901 (2018).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Other (1)

S. Wang and D. Zhao, Matrix optics, (CHEP, 2000).

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Figures (7)

Fig. 1.
Fig. 1. Experimental setup for measuring intensity distributions of FCAB, and a CAB is generated at the initial plane of $z=0$ . The right-top insert figure shows the section intensity distribution of the FCAB after the lens $L$ . Other notations are: HWP, half-wave plate; PBS, polarized beam splitter; SLM, spatial light modulator; AP, aperture; IP, initial plane at $z=0$ .
Fig. 2.
Fig. 2. Experimental intensity distributions for (a) the blocked CAB and (b) the blocked FCAB, at the plane of $z=600$ mm. In (b) the lens $L$ is placed at $z_{L}=f=125$ mm and the obstruction is placed at $z_b=0$ .
Fig. 3.
Fig. 3. Experimental ((a), (c) and (e)) and numerical ((b), (d) and (f)) results of the intensity distributions of the blocked FCAB under different propagation distance. The sector-shaped obstruction is placed at $z_b=$ 235 mm (a)-(b), 250 mm (c)-(d) and 265 mm (e)-(f). The numbers 1-5 (in (a)-(f)) are the cases of $z=350$ mm, 400 mm, 450 mm, 500 mm and 600 mm, respectively. Here the sector-shaped obstruction has the angle spread $\Delta \theta =\pi /3$ .
Fig. 4.
Fig. 4. Experimental (top panel) and numerical (bottom panel) results of the intensity distributions of a blocked FCAB at $z=600$ mm under different obstruction’s position $z_b$ . (a1)-(a5) These figures are the cases of $z_b=195$ mm, $235$ mm, $250$ mm, $265$ mm and $305$ mm, respectively, and (b1)-(b5) are the corresponding numerical results. Here the sector-shaped obstruction has the angle spread $\Delta \theta =\pi /3$ .
Fig. 5.
Fig. 5. The transverse Poynting vector of the blocked FCAB at the planes of (a) $z=500$ mm and (b) $z=600$ mm, under the different obstruction’s position. The index numbers 1-3 in subfigures are the cases of $z_b=195$ mm, 250 mm and 305 mm. The white arrows represent the direction and relative strength of $\vec {S}_{//}$ .
Fig. 6.
Fig. 6. Experimental results of the intensity distributions of the blocked FCABs under different obstruction’s size and shape, (a) the cases of the sector-shaped obstruction with $\Delta \theta =\pi /6$ , (b) the cases of the rectangular obstruction crossing half of the FCAB and (c) the cases of the rectangular obstruction crossing over the whole of the FCAB. The index numbers 1-5 in (a)-(c) are the places of $z=350$ mm, 400 mm, 450 mm, 500 mm, 600 mm, respectively. Note that all the obstructions are placed at $z_b=250$ mm.
Fig. 7.
Fig. 7. The similarity $F$ between the FCAB and the blocked FCAB, (a) as a function of the propagation distance $z$ under different $z_b$ for the sector-shaped obstruction with $\Delta \theta =\pi /3$ , (b) as a function of the angle spread $\Delta \theta$ for the sector-shaped obstructions under the fixed $z_b$ and $z$ , and (c) as a function of the obstruction’s position $z_b$ with a fixed value of $z=3000$ mm. The discrete dots and the error bars in (b) are the experimental data from the experimental intensity distributions according to Eq. (5).

Equations (6)

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E i ( r , z = 0 ) = A i ( r 0 r w ) exp [ a ( r 0 r ) w ] ,
E b ( r , ϕ , z = 0 ) = { 0 , ϕ [ Δ θ / 2 , Δ θ / 2 ] E i , o t h e r w i s e .
E o ( ρ , θ , z ) = i k 2 π B exp ( i k z ) E b ( r , ϕ , z = 0 ) × exp { i k 2 B [ A r 2 + D ρ 2 2 ρ r cos ( ϕ θ ) ] } r d r d ϕ ,
z F , ± = z L + f ( f A i ± z L ) f A i ± z L f ,
S / / = i 4 k ϵ 0 μ 0 [ E b E b E b E b ] ,
F ( z b , z ) = I ( z ) I b ( z b , z ) d x d y I 2 ( z ) d x d y I b 2 ( z b , z ) d x d y ,

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