Open Access
Add to BibSonomy
Abstract
An effective approach is proposed for obtaining a long-distance THz diffraction-free beam with meter-scale length. Multiple 3D-printed lens-axicon doublets are cascaded to form the generation system. In order to manifest the physical mechanism behind the generation process of this long-distance diffraction-free beam, we make a detailed comparative analysis of three beams: the ideal Bessel beam, the quasi-Bessel beam generated by single axicon, and the diffraction-free beam generated by the lens-axicon doublets. Theoretical results show that the zero-radial-spatial-frequency component plays a key role during the generation process of the third beam. Moreover, the intensities of this component are enhanced with the increase in the number of lens-axicon doublets, making the diffraction-free length longer. An experiment containing three lens-axicon doublets is performed to demonstrate the feasibility of our design. A 0.1-THz beam with one-meter diffraction-free length was successfully generated. Further experiments indicate that this THz diffraction-free beam also has a self-healing property. We believe that such long-distance diffraction-free beams can be used in practical THz remote sensing or imaging.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the zeroth-order Bessel beam was firstly introduced by Durnin in 1987 [1,2], so-called “diffraction-free beam (DFB)” has been a research hotspot in optics until now [3–6]. According to Sommerfeld's definition, diffraction is the deviation of optical waves during their rectilinear propagation, not caused by reflection or refraction [3]. Mathematically, the light field $E(x,y,z)$ of a DFB can be generally written as $E(x,y,z) = U(x,y)V(z)$, i.e. the transverse part $U(x,y)$ does not depend on the longitudinal coordinate. For a monochromatic wave, there are four exact solutions of Helmholtz equation that can be separable into a transverse and a longitudinal part in different coordinate systems: cosine function in Cartesian coordinates, Bessel function in circular cylindrical coordinates, Mathieu function in elliptic coordinates, and parabolic cylinder functions in parabolic coordinates. Physically, these solutions correspond to four ideal DFBs existing in the free space: plane wave, Bessel beam, Mathieu beam and parabolic beam respectively [6]. Besides of these DFBs, there is another well-known DFB called Airy beam in free space, which mathematically corresponds to an Airy wave packet solution of the Schrödinger equation [7–9].
By using different optical elements or device, the aforementioned DFBs have been generated in the past three decades [10–14]. However, it is worth noting that these generated DFBs are not beams in the conventional sense, but the quasi-DFBs because the ideal DFBs occupy the entire space and need infinite energy. In other words, the realization of ideal DFBs is impossible in practice. Although these beams are not real DFBs, they have been widely used in various optical application due to their unique features, e.g. microscopy [15], laser micro drilling [16], optical trapping or manipulating [17,18], optical communication [19,20] and imaging with extended depth of fields (DOF) [21,22].
Using a simple optical element, i.e. glass axicon made by polishing technique, one can easily obtain a common and widely used optical quasi-DFB: quasi-Bessel beam (QBB) or Bessel-Gaussian beam [23–25]. Similar to the optical QBB, THz QBB has also attracted much research interest in the past decade [26–31]. In order to get an axicon operating in THz domain, the researchers firstly used the milling or injection molding techniques. However, these conventional technologies consume a lot of time and money. Thanks to the advent of 3D-printing technology, either traditional refractive or complex diffractive axicons for THz waves can be easily and quickly manufactured nowadays [32]. It has been demonstrated recently that a 0.3-THz QBB with over 100mm diffraction-free region could be realized by using 3D-printed diffractive axicon, leading to a THz reflection imaging system with an extended DOF and high anti-interference capability [30]. However, this QBB with such diffraction-free length is still not enough for practical THz remote sensing or imaging.
In principle, an axicon transforms the incident Gaussian beam into a converging conical wave within the interfering area, resulting to a QBB. For this generation scheme, there are three key parameters determining characteristics of the QBB: the diffraction-free length ${z_{\textrm{m}} }$, the radius of central main lobe ${r_{\textrm{m}} }$, and the power of central main lobe ${P_{\textrm{m}} }$, while these three parameters are constraint with each other. For example, one can use the axicon with large apex angle to obtain a very small ${r_{\textrm{m}} }$ and large ${P_{\textrm{m}} }$, but the diffraction-free length will become short. As we know, longer ${z_{\textrm{m}} }$ will benefit the practical application of THz QBB, especially the remote imaging for nondestructive testing. In practice, there is a simple method that can elongate ${z_{\textrm{m}} }$, i.e. increasing the size of input Gaussian beam. However, the radius of central main lobe is simultaneously enlarged, leading to a reduced ${P_{\textrm{m}} }$.
In order to balance these parameters, a lens-axicon doublet has been proposed to obtain a long-distance quasi-DFB with small ${r_{\textrm{m}} }$[33–35]. In addition, manipulating the separation between the lens and axicon can also reduce spherical aberrations of the beam profile [35]. Thus, this generation scheme offers us a powerful approach to get long-distance THz quasi-DFB. It is worth noting that the previous researches mainly focused on the spatial profile of this long-distance quasi-DFB, lacking a comprehensive analysis of the physical mechanism behind the generation process, e.g. its characteristics in spatial frequency domain. In this paper, we perform a detailed comparative analysis of three beams: the ideal Bessel beam (IBB), the QBB generated by a single axicon and the quasi-DFB generated by the lens-axicon doublets. Our theoretical results show that it is the existence of zero-radial-spatial-frequency (ZRSF) component in the third beam that results in a very long diffraction-free length, while the short-distance QBB generated by single axicon doesn’t have this component. Thus we believe that the long-distance beam generated by the lens-axicon doublets is a new class of quasi-DFB. Based on our theoretical analysis, a 0.1-THz quasi-DFB with one-meter diffraction-free length has been successfully generated by using the three cascaded lens-axicon doublets. Moreover, longer quasi-DFB can be achieved with the increasing of the number of lens-axicon doublet. In addition, further experiments indicate that this THz diffraction-free beam also has the self-healing property.
2. Theory analysis and design
2.1 Ideal Bessel beam
An IBB can be written as:
Fig. 1. The IBB: the intensities distribution in (a) $x\textrm{ - }y$ and (b) $x\textrm{ - }z$ plane; the angular spectrum distribution in (c) ${f_x}\textrm{ - }{f_y}$ and (d) ${f_x}\textrm{ - }z$ plane.
Performing 2D-Fourier transform of Eq. (1),
2.2 Quasi-Bessel beam generated by single axicon
As mentioned above, using the axicon is a common approach to get the QBB in laboratory. Next we will analyze this generation scheme based on the well-developed angular spectrum theory (AST) [36]. The beam impinged on the axicon is a 0.1-THz ($\lambda = \textrm{3 mm}$) Gaussian beam with the beam width radius ${w_0} = \textrm{15 mm}$. The electric field at the entrance surface of the axicon can be written as ${E_{in}}(\rho ) = {E_0}\textrm{exp}({{ - {\rho ^2}} / {w_0^2}})$. The apex angle of used axicon is $\gamma = {15^\textrm{o}}$ and the radius of its aperture is ${R_1} = \textrm{50}\textrm{.8 mm}$. All selected parameters are corresponding to the following experiments.
Assuming the axicon is a pure phase-shifting device, the absorption of materials can be neglected. Thus, the phase-shifting of the axicon can be written as:
Based on AST and Eqs. (3)–(4), one can easily get all information of the output beam. Figure 2(a) illustrates the intensities spatial distribution of output beam in $x\textrm{ - }y$ and $x\textrm{ - }z$ plane, respectively. At the exit plane $z = 0$, the spatial profile of output beam is like the original Gaussian beam. Subsequently, the beam is rapidly focused to be a QBB, keeping its transverse intensities distribution within a near $\textrm{200 mm}$ propagation distance. The axial evolution of central-peak intensities [the white curve in Fig. 2(a)] also demonstrates this conclusion, i.e. ${z_\textrm{m}} \approx \textrm{200 mm}$. Note that the QBB rapidly spread to two side lobes after $z = \textrm{230 mm}$, resulting in disappear of the main lobe.
Fig. 2. The quasi-Bessel beam generated by single axicon: (a) the intensities distribution in $x\textrm{ - }y$ and $x\textrm{ - }z$ plane; (b) the angular spectrum distribution in ${f_x}\textrm{ - }{f_y}$ and ${f_x}\textrm{ - }z$ plane.
The calculated angular spectrum shows an annular distribution [see Fig. 2(b)], which is similar to the ring shape of IBB. This is the reason why Durnin et al. call it Bessel beam [1,2]. Note that there is no ZRSF component, i.e. ${f_\rho } = 0$ component, in the angular spectrum for the IBB and QBB generated by single axicon, while this component is very important in the following analysis.
2.3 Design of multiple cascaded lens-axicon doublets
Next we propose an approach to generate a long distance quasi-DFB by using multiple cascaded lens-axicon doublets, as shown in Fig. 3. In this system, every lens-axicon doublet contains of a lens ${L_\textrm{n}}$ with $\textrm{100 mm}$ focal length and an axicon ${A_\textrm{n}}$ with ${\gamma _\textrm{n}} = {15^\textrm{o}}$ apex angle, respectively. The function of the first lens ${L_0}$ is to transform an expanded incoming Gaussian beam emitted from 0.1-THz transmitter to a collimated beam with width radius ${w_0} = \textrm{15 mm}$. On the other hand, the first axicon ${A_\textrm{0}}$ generates a QBB with a diffraction-free length ${z_\textrm{m}} = \textrm{200 mm}$, as shown in Fig. 2(a).
Fig. 3. The proposed multiple cascaded lens-axicon doublets and the intensities distribution in $x\textrm{ - }y$ plane of the input beam at the entrance surface of each elements.
Using Eqs. (3)–(4) and AST again, one can easily calculate the E-field of the output beams behind every element in the Fig. 3. Without loss of generality, we will take three lens-axicon doublets (${L_0}\textrm{ - }{A_0}\textrm{ - }{L_1}\textrm{ - }{A_1}\textrm{ - }{L_2}\textrm{ - }{A_2}$) as an example. The second lens ${L_1}$ is placed within the diffraction-free region of the QBB generated by ${A_0}$ ($\textrm{100 mm}$ behind ${A_0}$, i.e. ${A_0}$ is placed in the front focal plane of ${L_1}$). Since a lens performs the Fourier transform on the light E-field [36], the ${L_1}$ will map the annular angular spectrum distribution in spatial frequency domain of a QBB onto the space domain, i.e. obtaining an annular beam in free space (see Fig. 3). Subsequently, this annular beam is impinged onto ${A_1}$.
Figure 4(a) depicts the intensities of the output beam behind the second axicon ${A_1}$ in $x\textrm{ - }z$ plane (without ${L_2}$ and ${A_2}$). Based on this intensity distribution, the output beam is somewhat similar to the QBB generated by single axicon, i.e. a quasi-DFB with a strong central main lobe and a series of side lobe. According to the calculated full widths at half-maximum (FWHM) of the main lobe [the white curve in Fig. 4(a)], it can be seen that the diffraction-free length of this quasi-DFB is increased to about $\textrm{500 mm}$. Although the beam begins to obviously spread after $z = \textrm{500 mm}$, the main lobe still exists. Such characteristic is significantly different from the QBB generated by one axicon [see Fig. 2(a)]. Further calculation gives the angular spectrum of this quasi-DFB [Fig. 4(c)], which is also not similar with the annular angular spectrum of QBB. The main difference is that the intensities of ${f_\rho } = 0$ component in this angular spectrum is not only nonzero but also very strong, while the intensities of ${f_\rho } \ne 0$ components are relatively weak. Note that although the quasi-Bessel beam generated by ${A_0}$ has no ${f_\rho } = 0$ component, it has plenty of ${f_\rho } \ne 0$ spatial component, see Fig. 2(b). Due to the focusing property of lens ${L_1}$ and axicon ${A_1}$, some of ${f_\rho } \ne 0$ components are transformed into ${f_\rho } = 0$ component, leading to the angular spectrum shown in Fig. 4(c).
Fig. 4. The theoretical results: the calculated intensity distribution in $x\textrm{ - }z$ plane of the output quasi-DFBs behind (a):${A_1}$ and (b):${A_2}$, the white curves are the calculated FWHMs of main lobes ; the calculated angular spectrum in ${f_\rho }\textrm{ - }z$ plane behind (c):${A_1}$ and (d):${A_2}$
For the third lens-axicon doublet, the lens ${L_2}$ is also placed within the diffraction-free region of the quasi-DFB generated by ${A_1}$. The distance between ${A_1}$ and ${L_2}$ is $\textrm{150 mm}$, which means that ${A_1}$ is not at the front focal plane of lens again. So the ${L_2}$ doesn’t map the angular spectrum distribution of the quasi-DFB generated by ${A_1}$ onto the space domain. While calculated results show that the beam impinged on ${A_2}$ is still annular, see Fig. 3. Figure 4(b) depicts the final output beam from the ${L_0}\textrm{ - }{A_0}\textrm{ - }{L_1}\textrm{ - }{A_1}\textrm{ - }{L_2}\textrm{ - }{A_2}$ system. The calculated FWHMs of the main lobe [the white curve in Fig. 4(b)] show that the diffraction-free length of the final quasi-DFB is increased to approximate $\textrm{1000 mm}$. Compared to the output beam behind ${A_1}$, the intensities of ${f_\rho } = 0$ component in the angular spectrum of the final quasi-DFB get stronger, see Fig. 4(d).
3. Detailed analysis and experimental demonstration
In this section, we will comprehensively analyze our designed multiple cascaded lens-axicon doublets and experimentally demonstrate their feasibility. Figure 5 plots the calculated angular spectrum distribution with respect to the spatial frequency ${f_\rho }$ of output quasi-DFBs from the different axicons in our designed structure. The two blue arrows in Fig. 5 are the angular spectrum of an IBB, i.e. two Dirac impulses.
Fig. 5. The calculated angular spectrum distribution with respect to the spatial frequency of output quasi-DFBs behind ${A_0}$, ${A_1}$ and ${A_2}$, respectively. The two blue arrows are the angular spectrum of an IBB.
It can be clearly seen that the ${f_\rho } = 0$ component is nonzero for the beams behind ${A_1}$ and ${A_2}$. While for the IBB and the QBB behind ${A_0}$, ${f_\rho } = 0$ component is vanished or has a very small quantity, respectively. According to the conservation law of energy $k_\rho ^2 + k_z^2 = k_0^2$, i.e. $f_\rho ^2 + f_z^2 = f_0^2 = {({1 / \lambda })^2}$, the ${f_\rho } = 0$ component corresponds to an plane wave propagating along the $z\textrm{ - }$ direction with the spatial frequency $f_z^{} = f_0^{}$. For the QBB generated by single axicon, the overlap of ${f_\rho } \ne 0$ components [see Fig. 2(b)] forms its diffraction-free region. Since the QBB has no or very small ${f_\rho } = 0$ component, it will rapidly spread and lose the main lobe behind the diffraction-free region. On the contrary, there is always a spatial frequency $f_z^{} = f_0^{}$ component that propagates along the $z\textrm{ - }$ direction in the outgoing beams from ${A_1}$ and ${A_2}$, see Fig. 5. It is this nonzero ${f_\rho } = 0$ component can guarantee the exist of the quasi-DFB’s main lobe over a long distance, as shown in Figs. 4(a) and 4(b). Further calculation shows that the ${f_\rho } = 0$ component and its adjacent components get more energy with the increasing of the number of lens-axicon doublet, as shown in Fig. 5. This property makes stronger ${f_\rho } = 0$ component propagating $z\textrm{ - }$ direction, leading to a longer diffraction-free length. So we believe that this long-distance beam generated by the multiple cascaded lens-axicon doublets is a new class of quasi-DFB, but not a QBB.
To experimentally demonstrate the feasibility of our approach, we still choose three lens-axicon doublets as an example. The 0.1-THz continuous wave source (GKa-100, SPACEK LABS) is an InP Gunn diode with the power of $\textrm{25 mW}$. All the lenses and axicons are fabricated via a 3D-printer (Objet30, Stratasys Ltd.). The transverse resolution of this machine is 600 dpi (42 µm) and longitudinal resolution is 900 dpi (28 µm). The absorption coefficient $\alpha$ and refractive index $n$ of 3D-printing materials are $1.2\textrm{ c}{\textrm{m}^{\textrm{ - 1}}}$ and 1.645, respectively [32]. A broadband high sensitivity detector with zero-bias Schottky diode is connected with the SR830 lock-in amplifier to obtain the 0.1-THz signal. This detector is mounted on a motorized three-axis translation stage, in which the distance of x, y and z-axis is $\textrm{200 mm}$, $\textrm{200 mm}$ and $\textrm{800 mm}$, respectively.
Figure 6(a) illustrates the measured intensities in $x\textrm{ - }z$ plane of the final output beam behind ${A_2}$, in which seven insets are the intensity distribution in $x\textrm{ - }y$ plane at different $z$ coordinates. The exit surface of ${A_2}$ is at $z = 0$. Figure 6(b) plots the measured axial evolution of central-peak intensities of the output beams behind each axicon. Note that the measured data along z-axis are spliced because the distance of our translation stage is only $\textrm{800 mm}$. As can be seen in Fig. 6(a), the experimental results are well matched with the previous calculation, i.e. Fig. 4(b). Conservatively speaking, the diffraction-free length of the generated quasi-DFB is about $\textrm{1000 mm}$ based on the FWHM [the white curve in Fig. 6(a)]. Correspondingly, the curves of central-peak intensities become flatter with the increasing of lens-axicon doublet, see Fig. 6(b). Such experimental results can be explained by the aforementioned theoretical conclusion, i.e. the intensities of ${f_\rho } = 0$ component and its adjacent components will be enhanced with the increasing of the number of lens-axicon doublet, leading to a longer diffraction-free length.
Fig. 6. The experimental results: (a) the measured intensity distribution in $x\textrm{ - }z$ and $x\textrm{ - }y$ plane of the THz quasi-DFB behind ${A_2}$; (b) the measured axial evolution of central-peak intensities of the output beams behind ${A_0}$, ${A_1}$ and ${A_2}$, the blue line is that of an IBB.
In addition, the FWHM of generated THz quasi-DFB at $z = \textrm{200 mm}$ has the least value $\textrm{9}\textrm{.9 mm}$. Between $z = \textrm{400 mm}$ and $z = \textrm{1400 mm}$, the FWHM is increased from $\textrm{24}\textrm{.3 mm}$ to $\textrm{47 mm}$. If we assume a focusing Gaussian beam with $\textrm{9}\textrm{.9 mm}$ beam waist, calculated results show that it will expand to $\textrm{160 mm}$ beam-width after propagating $\textrm{1000 mm}$. This comparison indicate the beam behind ${A_2}$ is diffraction-free.
It should be addressed here that the energy efficiency of the proposed approach is key issue for its practicality. Based on the experimental data, the transmission efficiency of the three lens-axicon doublets setup is only 2.7%. There are two possible major losses in the lens-axicon doublets, i.e. absorption loss of materials and reflection loss of the flat entrance surface of each element in Fig. 3. Based on aforementioned absorption coefficient and refractive index of 3D-printing materials, the calculated transmission efficiency of each lens and axicon are 65.8% and 54.9%, respectively. If only considering the absorption of materials, we get 74.4% and 62.1%, respectively. In addition, the used lens and axicon are both refractive elements. Thus we think that the main loss of a lens-axicon doublet is the absorption of materials. If we only consider the absorption of materials, further calculations show that transmission efficiency of three lens-axicon doublets setup is about 4.7%. Note that the incident beam used in calculation is a plane wave for simplicity, but not the real THz Gaussian beam. In order to increase the energy efficiency, it is believed that using diffractive elements [30] is a potential method.
Self-healing is a unique property of the quasi-DFB. Figure 7 illustrates an experiment about this property of the generated THz quasi-DFB. A metal rod as the obstacle is placed behind the last axicon ${A_2}$ with $\textrm{200 mm}$ distance. At $z = \textrm{210 mm}$ (this z-axis is same with that in Fig. 6), the image of the rod can be clear seen, which means that the generated quasi-DFB is destroyed. At $z = 6\textrm{10 mm}$, the beam begins to reconstruct and recovers to the quasi-DFB finally, as shown in the lower panel of Fig. 7.
Fig. 7. Self-healing experiments: the measured intensity distribution in $x\textrm{ - }y$ plane of the output beam behind the obstacle (a metal rod).
4. Conclusion
In summary, we have demonstrated the effectiveness of an experiment scheme, i.e. multiple cascaded lens-axicon doublets, for generating long-distance THz diffraction-free beam. All used lenses and axicons can be quickly fabricated by 3D-printing technology. The measured diffraction-free length is over one-meter. Comprehensive theoretical analysis show that the nonzero ${f_\rho } = 0$ component of the beam generated by lens-axicon doublets guarantees the exist of its main lobe over a long distance. Moreover, such component will get more energy with the increasing of the number of lens-axicon doublet, leading to a longer diffraction-free length. In addition, experimental results show that this long-distance THz quasi-DFB also has the self-healing property. We believe this generated diffraction-free beam has potential applications for THz remote sensing or imaging.
Appendix: Derivation of Eq. (3)
As shown in Fig. 8, the phase-shifting between the input and output electric field for an axicon comes from two parts: the propagation inside the axicon with distance ${l_1}$ (which exists only if $\rho < {R_1}$), and the propagation in air with distance ${l_2}$, where
Thus, the phase-shifting is obtained as
Funding
Fundamental Research Funds for the Central Universities (2017KFYXJJ029); National Natural Science Foundation of China (61905232); National Defense Pre-Research Foundation of China (61422160107).
Disclosures
The authors declare no conflicts of interest.
References
1. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]
2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
3. Z. Bouchal, “Nondiffracting Optical Beams: Physical Properties, Experiments, and Applications,” Czech. J. Phys. 53(7), 537–578 (2003). [CrossRef]
4. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2009). [CrossRef]
5. H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, “Non-diffracting Waves”, (Wiley, Weinheim, 2014).
6. U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Prog. Opt. 61, 237–281 (2016). [CrossRef]
7. C. W. Wong, “Nonspreading Wave Packets,” Am. J. Phys. 64(6), 792–799 (1996). [CrossRef]
8. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
9. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]
10. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]
11. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef]
12. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef]
13. P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012). [CrossRef]
14. C. E. Garcia-Ortiz, V. Coello, Z. H. Han, and S. I. Bozhevolnyi, “Generation of diffraction-free plasmonic beams with one-dimensional Bessel profiles,” Opt. Lett. 38(6), 905–907 (2013). [CrossRef]
15. F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nature Commun. 3(1), 632–638 (2012). [CrossRef]
16. M. K. Bhuyan, P. K. Velpula, J. P. Colombier, T. Olivier, N. Faure, and R. Stoian, “Single-shot high aspect ratio bulk nanostructuring of fused silica using chirp-controlled ultrafast laser bessel beams,” Appl. Phys. Lett. 104(2), 021107 (2014). [CrossRef]
17. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
18. K. Jongki, J. Yoonseob, L. Sejin, H. Woosung, S. Jeon-Soo, and O. Kyunghwan, “Fourier optics along a hybrid optical fiber for Bessel-like beam generation and its applications in multiple-particle trapping,” Opt. Lett. 37(4), 623–625 (2012). [CrossRef]
19. Y. P. Xie, T. Lei, C. W. Yang, L. P. Du, and X. C. Yuan, “Beam wander relieved optical switch using Bessel beams in turbulent atmosphere,” Chin. Opt. Lett. 17(9), 090602 (2019). [CrossRef]
20. N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, R. Botha, and A. Forbes, “Free-space optical communication link with shape-invariant orbital angular momentum Bessel beams,” Appl. Opt. 58(16), 4258–4264 (2019). [CrossRef]
21. T. Breen, N. Basque-Giroux, U. Fuchs, and I. Golub, “Tuning the resolution and depth of field of a lens using an adjustable ring beam illumination,” Appl. Opt. 59(15), 4744–4749 (2020). [CrossRef]
22. Y. C. Hu, Z. J. Chen, L. Z. Xiang, and D. Xing, “Extended depth-of-field all-optical photoacoustic microscopy with a dual non-diffracting Bessel beam,” Opt. Lett. 44(7), 1634–1637 (2019). [CrossRef]
23. O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16(17), 12688–12700 (2008). [CrossRef]
24. G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of Bessel beams from semiconductor lasers,” Quantum Electron. 43(5), 423–427 (2013). [CrossRef]
25. A. J. Asuncion and R. A. Guerrero, “Generating superimposed Bessel beams with a volume holographic axicon,” Appl. Opt. 56(14), 4206–4212 (2017). [CrossRef]
26. X. L. Wei, C. M. Liu, L. T. Niu, Z. Q. Zhang, K. J. Wang, Z. G. Yang, and J. S. Liu, “Generation of arbitrary order Bessel beams via 3D printed axicons at the terahertz frequency range,” Appl. Opt. 54(36), 10641–10649 (2015). [CrossRef]
27. Z. Wu, X. K. Wang, W. F. Sun, S. F. Feng, P. Han, J. S. Ye, Y. Yu, and Y. Zhang, “Vectorial diffraction properties of THz vortex Bessel beams,” Opt. Express 26(2), 1506–1520 (2018). [CrossRef]
28. A. Bitman, I. Moshe, and Z. Zalevsky, “Improving depth-of field in broadband THz beams using nondiffractive Bessel beams,” Opt. Lett. 37(19), 4164–4166 (2012). [CrossRef]
29. A. Bitman, S. Goldring, I. Moshe, and Z. Zalevsky, “Computed tomography using broadband Bessel THz beams and phase contrast,” Opt. Lett. 39(7), 1925–1928 (2014). [CrossRef]
30. L. T. Niu, K. J. Wang, Y. Q. Yang, Q. Wu, X. Ye, Z. G. Yang, J. S. Liu, and H. Y. Yu, “Diffractive elements for zero-order Bessel beam generation with application in the terahertz reflection imaging,” IEEE Photonics J. 11(1), 5900212 (2019). [CrossRef]
31. D. M. Mittleman, “Twenty years of terahertz imaging,” Opt. Express 26(8), 9417–9431 (2018). [CrossRef]
32. Z. Q. Zhang, X. L. Wei, C. M. Liu, K. J. Wang, J. S. Liu, and Z. G. Yang, “Rapid fabrication of terahertz lens via three-dimensional printing technology,” Chin. Opt. Lett. 13(2), 022201 (2015). [CrossRef]
33. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010). [CrossRef]
34. W. T. Zheng, F. T. Wu, Q. A. Zhang, and Z. M. Cheng, “A new technique for generating non-diffracting beam with long propagation distance using two axicons,” Acta Phys. Sin. 61(14), 144201 (2012). [CrossRef]
35. I. S. Chowdhury, R. P. Roberts, G. Molina-Terriza, and X. Vidal, “Lens-axicon separation to tailor aberration free focused Bessel-Gaussian beams in the paraxial regime,” Opt. Express 27(8), 11160–11173 (2019). [CrossRef]
36. J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W.H. Freeman, 2017).
