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Abstract
Orbital angular momentum (OAM) multiplexing technology has been developed in the optical information encryption fields. Here, the modulated chiro-optical OAM (MC-OAM) holography is proposed to further improve information security capacity, which integrates the OAM multiplexing technology with the chiro phase modulation. The orthogonality of the axicon parameter, chiro coefficient and rotation angle modulating the chiro phase distributions are analyzed, respectively, which demonstrate their potential usages as extra degrees of freedom besides the topological charge (TC). Those three parameters combining TC serve as four optical keys, which provides a four-dimensional spatial multiplexing method for information security. Furthermore, we have demonstrated that TC minimum interval of the fractional MC-OAM reaches 0.01. The experimental and simulation results exhibit the essential properties in selectivity and multiplexing of MC-OAM holography. This method can significantly increase the holographic information capacity and safety and inspire widespread applications, such as display, information security and communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the angular momentum (OAM) was introduced into the optical beams by Allen [1] in 1992, the OAM properties arouse tremendous applications. The light carrying OAM formulated by a phase factor $\exp (il\varphi )$ is known as the optical vortices (OV) [2], where $l$ indicates the topological charge (TC) and $\varphi$ is the azimuthal angle. During the last three decades [3], the studies of optical vortices have experienced from the fundamental studying to the versatile applications. The advent of new OV modulation techniques successively promotes the applications. For example, the airy vortex beam [4,5] and Pearcy vortex beam [6,7] present the non-diffraction property, which have been used in light propagation through atmospheric turbulence, in optical tweezers to bypass obstacles to deliver cell bodies or drug particles to the target area. The inherent orthogonality of OAM provides the additional degree of freedom to enhance the information capacity for broad prospects in optical communication, optical entropy, optical image, etc.
Recently, Fang et al. [8] has implemented the OAM as an independent information channel in holography to achieve the OAM-preserved, OAM-selective and OAM-multiplexing hologram. Moreover, they revealed the OAM selectivity and showed up to 10-bit OAM-dependent encoded hologram for high-security encryption. Whereafter, the OAM holography has been achieved rapid development [9–11]. When OAM beams are incorporated with modulated technology of the light beam, many methods have been proposed to increase the dimensions of the optical OAM multiplexing hologram and further increase the data storage ability, the security and fidelity in different holographic systems. For instance, several target images can be encrypted both in different ellipticity and OAM channels [12]. The reconstruction processes need the incident elliptic optical vortex (EOV) with an inverse helical mode index under a specific ellipticity. The ultra-dense perfect integer and fractional OAM multiplexed holography [13] are introduced to extend the traditional one-dimensional OAM holography to two-dimensional spatial division multiplexed holography. By radially and angularly modulating the spiral phase and the axicon phase of the fractional OAM holography, the resolution of OAM holographic encoding method can be up to 0.01. In the angular multiplexing OAM holography [14], the angular space is put forward as a new degree of freedom for OAM holography for encoding. It is demonstrated that the partial phase modes which are divided from one OAM phase mode can be used to encode several target images. The decoding light carrying the correct section distribution can decrypt the corresponding target image. As the coming of the information explosion era, more degrees of coding freedom are required to meet the challenges in high-security and high-capacity.
Spiral-optical beams [15–20] have been intensively studied by researchers in the last few years, which carry independent radial and azimuthal phase distributions. Owing to their superiority in modulating the spatial distribution, the spiral-optical beams have been used in optical communication [21] and chiral isotropic material fabrication [22]. In this paper, we introduce a phase structure of chiro-optical beam [23] into multi-dimensional OAM holography. The intensity and phase distribution of the chiral beam can be modulated by changing the chiro coefficient, axicon parameter, rotation angle, and the topological charge. Based on these modulation parameters, the controllable intensity twist lobes and rotated direction could be used to encode optical signal. The orthogonality of the chiro-optical beam has been theoretically analyzed according to axicon parameter $\beta$, the chiro coefficient $A$, and the rotation angle $\alpha$ besides TC, which provide new degrees of freedom. The simulations and experiments are carried out to verify the feasibility of the proposed method. The remainder of this paper is organized as follows. In Sec. 2, we theoretically derived the orthogonality of the parameter ‘$A$’, ‘TC’, ‘$\alpha$', ‘$\beta$‘, and further illustrated the helix-shaped OAM selective holography schematic. The details of the experimental configuration and the experimental/simulation results of the different- dimensional multiplexed holography schemes are discussed in Sec. 3. Finally, the study is summarized in Sec. 4.
2. Method
To achieve OAM holography encryption with multi-channel, the phase of the chiro-optical beam can be implemented as information carrier. In the cylindrical coordinate, the complex amplitude of the chiro-optical beam can be expressed as [23]:
Fig. 1. (a1) -(a4) The HPPs with $A = 0,0.2,0.4,0.6$, $\beta = 4.8m{m^{ - 1}}$, $l = 2$, $\alpha = 0$. (b1)-(b4) the simulation of the intensity of the chiro-optical beam. (C) Peak intensity of the chiro-optical beam as a function of $A$ range. (d) The overlap [see Eq. (7)] between two chiro-optical beams with different parameter A, the two beams are orthogonal when ${A_1} \ne {A_2}$. (e) The overlap [see Eq. (2)] between a non-rotation state and the state rotated $\alpha$, the two states are orthogonal when $\alpha = \frac{\pi }{2},\frac{{3\pi }}{2}$. (f) Schematic illustration of the helix-shaped OAM selective hologram capable of encoding a letter 3. (g) The target optical field “3” is reconstructed by a chiro beam with inverse phase mode, namely, $A = 0.2$, $l ={-} 2$, $\beta ={-} 4.8m{m^{ - 1}}$, $\alpha = 0$.
Their inner product in the angular direction can be obtained
Since the general result of Eq. (3) is difficult to be obtained, we discuss the exceptional cases instead. Firstly, let’s consider two chiro-optical beams with the same A, rotation angle $\alpha$ and axicon parameters but different TCs.
In the formula:
It can be known that the chiro-optical beam with different TCs is orthogonal. In the second step, we discuss the overlap between two chiro-optical beams with the same TC, rotation angle $\alpha$ and A but different axicon parameters. By integrating the inner product of the ${E_1}$ and ${E_2}$ in the radial and the angular direction and with help of $\delta (ax) = \frac{1}{{|a |}}\delta (x)$, Eq. (3) can be written as
When ${\beta _2} \ne {\beta _1}$, the axicon parameters of the chiro-optical beam are of orthogonality. In the third step, we prove whether two light beams with the same TC, rotation angle $\alpha$ and $\beta$ but different A values are orthogonal.
From this expression of the left-hand side of Eq. (6), it is possible to calculate the modulus of the overlap
This integral result is plotted in Fig. 1(d) for various values of A. It illustrates that ${A_2} \ne {A_1}$, the integral result equals to zero, namely, the chiro-optical beam with different parameters in coaxial transmission can be separated from each other. That is, the chiro-optical beams carrying different A are orthogonal.
We numerically calculate $\int\limits_0^{2\pi } {\int\limits_{ - \infty }^\infty {{E_1}^\ast } {E_2}drd\theta }$ to testify the orthogonality of the rotation angle $\alpha$. We now consider make one chiro-optical beam oriented at rotation angle $\alpha = 0$, and the other with orientation $\alpha \ne 0$. We calculate the overlap between two chiro-optical beams with the same A, TC and axicon parameters but different $\alpha$, The integral result is plotted in Fig. 1(e), we can clearly see that, when the state is rotated over $\alpha = {\pi / 2},{{3\pi } / 2}$ by rotating the HPP, the state is orthogonal to the nonrotated state.
By reason of the foregoing, the feasibility of A, TC, $\beta$, $\alpha$ of the chiro-optical beams as independent information channels are verified. We bring four parameters of the HPP into OAM holography for increasing the degrees of freedom in it. We design the OAM preserved-hologram by adopting the Gerchberg–Saxton (GS) iteration algorithm [25,26] to encode the sampled target image. The sampling function is a two-dimensional comb array, the interval of which is calculated from the diameter of intensity profile of the modulated chiro phase in spatial frequency domain. Since the distribution of the chiro phase distribution varying with the chiro coefficient, chiro coefficient, axicon coefficient and rotation angle, the sampling interval is unique. After combining the preserved-hologram and a HPP with different coding degrees of freedom l, $\beta$ and $A$, we can obtain a helix-shaped-selective hologram. The multiplexed selective hologram is formed by adding multiple selective holograms. The principle is illustrated in the Fig. 1. (f). For a chiro-optical beam carrying with a specific inverse l, $\beta$ together with the same $A$, the target images can be reconstructed at the imaging plane, as shown in Fig. 1. (g).
3. Simulation and experimental results
3.1 Principle of $\textrm{A}$-encrypted MC-OAM multiplexing holography
A schematic illustration of the MC-OAM holography with the chiro coefficient A key is shown in Fig. 2, the feasibility of which has been verified through simulation and experiment. The numbers ‘2’, ‘3’, ‘4’, ‘5’ can be respectively encrypted into four preserved holograms by using four HPPs with $A$ of 0.2, 0.4, 0.6, 0.8, topological charge of 2 and axicon parameter of $4.8\textrm{m}{\textrm{m}^{ - 1}}$, respectively. The four preserved holograms are encoded with four different phase modes based the chiro coefficient $A$ resulting to four selective holograms, respectively. And the four selective holograms can be further superposed to generate a MC-OAM multiplexing hologram with the A key. The encrypted letters are encoded into four preserved holograms and combined them to generate a MC-OAM multiplexing hologram with the A key. When a corresponding chiro-optical beam is used to illuminate the MC-OAM multiplexing hologram, the target number can be respectively and accurately decoded in series of strong-contrast Gaussian spots. As a comparison, the unwanted targets will appear in weak spiral-shape spots. When we illuminate the hologram using a planar beam or chiro-optical beam with error parameter [Fig. 2(f) and (g)], four image channels will interfere with each other, leading to an indiscernible image pattern in the reconstructed image plane. Furthermore, we use the receive operator characteristic (ROC) curve to calculate the area under ROC curve (AUC) to measure the error rate of experiment compared with simulation. We can clearly see in Fig. 2 (b1)–(e2) that the values of AUC get close to 0.99, which proved the experiment results agree well as the simulation.
Fig. 2. (a). Schematic diagram for calculating MC-OAM multiplexing holograms with the A key. (b-e). Simulation reconstruction of A-dependent holographic images based on incident beams with $A = 0.2,0.4,0.6,0.8$, $\beta ={-} 4.8m{m^{ - 1}}$, $l ={-} 2$, $\alpha = 0$. (b1)–(e1) Experimental results. (f) Reconstruction image by a planar wave. (g) Reconstruction image by chiro-optical beam with error parameter $\beta$.
In addition, we let the enciphered message ‘NJNU’ in QR-code, namely the abbreviation of Nanjing Normal University, divide into four image parts. Each section of QR-code is sampled and encoded into a hologram, respectively, and further multiplies the HPP with $l ={-} 2$, $\beta = 4.8m{m^{ - 1}}$ and $A = 0.2,0.4,0.6,0.8$. Finally, we combined the four holograms into a MC-OAM multiplexing hologram with the A key. The sketch of the encoding process is shown in the Fig. 3. In the decryption process, four sections of one QR-code image can be reconstructed by multiplexed incident chiro-optical decoding beams with four different chiro coefficients ‘$A$’, namely, ${A_{de}}$ of $0.2,0.4,0.6,0.8$ respectively. Only the ciphertext of $A$-encrypted MC-OAM multiplexing hologram is simultaneously illuminated by the four correct chiro-optical beams carrying with the customized $A$ key. We can successfully extract the complete encrypted QR information in the optical system. Therefore, it is demonstrated that multiple keys enable optical holographic encryption with a higher level of security.
Fig. 3. Illustration of the holographic decryption results. (a-d) Simulation reconstruction separately of the four parts of QR code, based on incident chiro-optical beams with $A = 0.2,0.4,0.6,0.8$ (a1-d1) The corresponding experimental results. (e) The decryption result of the complete QR based on multiply the four-decoding chiro-optical beams. (e1) The corresponding experimental results to e.
We carried out an experiment setup to verify the proposed method, which is arranged as shown in Fig. 4. The solid-state laser with a wavelength of 532.8 nm is used as the light source in the system. After being collimated and expanded, a plane wave incident on SLM1 to obtain the desired decrypted light. The specially designed phase pattern loaded on SLM1 is a modulated HPP, the distribution of which is determined by four parameters namely A, TC, $\beta$ and rotation angle. The modulated beam reflected by the SLM1 was focused by a lens L1, and the corresponding chiro-optical beam is generated. The coded information is uploaded on the SLM2. The decrypted image is captured by a digital camera with help of a lens L2.
3.2 $\textrm{A}$-$\textrm{l}$ encrypted MC-OAM multiplexing holography
The continuous change value of parameter ‘$A$’ exhibits enormous potential in increasing the safety performance of OAM holography. A prefix is added before the term ‘OAM holography’ to denote the encoded parameters in the following part. Therefore, Al-OAM means we use both TC and parameter A as encode channels. With the method introduced above, $A$-$l$ encrypted MC-OAM multiplexing hologram can be generated by superposition of different images that be encoded into different $A$ and $l$ channels. The two-channel multiplexing results are demonstrated in Fig. 5(a-e), where the hologram ‘I, m, a, g, e’ is attached with $\beta ={-} 4.8m{m^{ - 1}}$, different TC (from $l = 2$ to $l = 10$, with the step of 2), different A from 0 to 0.8 with the step with 0.2. The encrypted letters can be reconstructed when intentional chiro-optical beams with specified topological charges and appropriate parameter ‘$A$’ are used, as shown in Fig. 5.
Fig. 5. The results of the MC-OAM holography with the chiro coefficient $A$ and topological charge $l$ key. (a)–(e) Simulated decrypt information. (a1)–(e1) Experimental results.
The fractional vortex beam has gained increasing practical applications in the optical communication, microparticle manipulation and optical information coding. Due to the fact that a beam with unlimited integer or fractional value helix mode index, it provides high degrees of freedom for multiplexing information, which boosts the transmitted capacity in optical communication processes. Up to now, the resolution of a receiver to precisely recognize OAM modes is 0.01 reported by the Ref. [27,28]. To further verify the availability of fractional HPP as the information carrier, we design a $A$-$l$ encrypted fractional MC-OAM hologram multiplexing scheme. The iterative GS transform under chiro-optical beam is used to encode the images of ‘6’, ‘7’, ‘8’ ‘9’. Each encrypted image corresponds to superimposed a HPP with different ‘$A$’ values from 0.2 to 0.8 with the step of 0.2 and the fractional TC, where the TC values ranging from 0.01 to 0.04 with the step of 0.01. Therefore, we obtained four MC-OAM holograms with the A and fractional $l$ key, which can be combined into a phase-only phase hologram, as shown in Fig. 6. Both TC number and parameter ‘$A$’ of HPP used in the iterative operation of phase recovery algorithm are the keys needed in the decryption process. When the chiro phase mode used in the decryption process are inverse mode to the hologram, the decryption image can be obtained on the output plane of the system.
Figure 7 shows the encoding process by modulating the rotation angle and parameter A. Four modulating phase modes $\varphi (l,A,\beta ,\alpha )$, namely, $\varphi (5,0.2,4.8,0)$, $\varphi (5,0.2,4.8,{\pi / 2})$, $\varphi (5,0.2,4.8,{{3\pi } / 2})$, $\varphi (5,0.6,4.8,0)$ are used to multiplex four coded image information ‘C’, ‘O’, ‘D’, ‘E’. We can obtain four $A - \alpha$ MC-OAM holograms, multiply these four holograms to superimpose into one $A - \alpha$ MC-OAM multiplexing hologram, as shown in Fig. 7(a0). When the hologram is illuminated respectively with different chiro-optical beams carrying with different value of $A - \alpha$, namely, ${\varphi _{de}}( - 5,0.2, - 4.8,0)$, ${\varphi _{de}}( - 5,0.2, - 4.8,{\pi / 2})$, ${\varphi _{de}}( - 5,0.2, - 4.8,{{3\pi } / 2})$, ${\varphi _{de}}( - 5,0.6, - 4.8,0)$, four target images can be respectively captured by CCD. The simulation and experimental results are shown in Fig. 7(a)–(d), Fig. 7(a1)–(d1), respectively. Thus, chiro-optical beams with different $A - \alpha$ can be used to demultiplex holographic images from the $A - \alpha$ MC-OAM multiplexing hologram in high contrast Gaussian spots form.
Fig. 6. The simulation and experimental results of the A-encrypted fractional MC-OAM multiplexing holography. (b)–(e) Simulated decrypt information. (b1)–(e1) Experimental results.
We design the $l - \alpha$ MC-OAM multiplexing hologram, in which two different topological charges (l = −5,5) and three different rotation angles ($\alpha = 0,{\pi / 2},{{3\pi } / 2}$) are combined to encrypt four characters. Four modulating phase modes $\varphi ( - 5,0.2,4.8,0)$, $\varphi ( - 5,0.2,4.8,{\pi / 2})$, $\varphi (5,0.2,4.8,0)$ and $\varphi (5,0.2,4.8,{\pi / 2})$ are respectively used to multiplex four characters ‘$A$’, ‘$\beta$’, ‘$\alpha$’, ‘$l$’. TC and rotation angle jointly determine the decryption results. The reconstructed image information is presented in Fig. 8, and each is decoded by unique modulating phase mode $\varphi (l,A,\beta ,\alpha )$. The $l - \alpha$ encrypted the MC-OAM multiplexing holography experiment has been implemented (Fig. 8) to verify the feasibility of the rotation angle modulation. Furthermore, we further implement the fractional MC-OAM multiplexing holography with the $l$-$\alpha$ key to verify the feasibility of the rotation angle modulation to achieve the topological charge resolution of 0.01. Three coded images ‘$y$’, ‘$e$’, ‘$s$’ and corresponding three modulating phase modes ${\varphi _{in}}(5.01,0.2,4.8,0)$, ${\varphi _{in}}(5.02,0.2,4.8,{\pi / 2})$, ${\varphi _{in}}(5.03,0.2,4.8,{{3\pi } / 2})$ are multiplexed into a hologram. Figure 9(a) to (c 1) exhibit the decoding results of the $\alpha$ encrypted the fractional MC-OAM multiplexing holography, where the decoding phase mode of the incident beams are, ${\varphi _{de}}( - 5.01,0.2, - 4.8,0)$, ${\varphi _{de}}( - 5.02,0.2, - 4.8,{\pi / 2})$, ${\varphi _{de}}( - 5.03,0.2, - 4.8,{{3\pi } / 2})$. It can be understood that when $\exp(i{\varphi _{de}}) = \exp( - i{\varphi _{in}})$ the phase of the desired target image will become zero. The selected letter pattern can be displayed clearly.
Fig. 8. The simulation and experimental results of the MC-OAM multiplexing holography with the $l - \alpha$ key.
Fig. 9. The simulation and experimental results of the $l$- $\alpha$ fractional MC-OAM multiplexing holography.
3.3 Designed $\mathrm{A\ -\ l\ -\ \alpha }$ encrypted MC-OAM multiplexing holography
The proposed concept of three-dimensional multiplexing of the chiro-optical of light is illustrated in Fig. 10(A). We encode different images hidden in hologram, then let the 9 preserved holograms respectively multiply different nine HPPs with axicon parameter equal to $4.8m{m^{ - 1}}$ (TC values from -5 to 5 with the step of 5, A values from $0.1$ to $0.9$ with the step of $0.4$, $\alpha$ values 0, ${\pi / 2}$, $3{\pi / 2}$). Those parameters generate nine conceivable combinations. As mentioned above, the encryption image decoded by chiro-optical beams with specific combinations of TC, rotation angle and parameter ‘$A$’ under a specific axicon parameter. The simulation results and experimental results are shown in Fig. 10(B) and Fig. 11, respectively.
Fig. 10. The simulation results of the $A - l - \alpha$ encrypted MC-OAM multiplexing holography. Each image ‘w’, ‘o’, ‘n’, ‘d’, ‘e’, ‘r’, ‘f’, ‘u’, ‘l’ is coded as hologram and superposed by a HPP with corresponding topological charge, parameter A, rotation angle $\alpha$.
3.4 $\mathrm{A\ -\ l\ -\ \alpha -\ \beta }$ encrypted MC-OAM multiplexing holography
Moreover, according to analyze results in section 2, the four parameters in the chiro-optical beam can be used as independent channels and have the feasibility of the synchronous modulation. And theoretically unlimited helical mode index of the modulated chiro-optical makes information capacity theoretically infinite. Based on the orthogonality of four parameters in theoretically, a maximal OAM-multiplexing channel number could be achieved for OAM holography by satisfying the Nyquist Shannon sampling theorem. Therefore, the information channels enormously increased through the four parameters for faithfully encoding and decoding information. We verify the effectiveness of the four- dimensional multiplexed holography by demonstrating the decryption results of 20 images through experiments and simulation. The different topological charges ($l ={-} 12, - 6,0,6,12, - 12, - 6,0,6,12$), two different axicon parameter (${\beta _1},{\beta _2}$) and a rotation angle ${\alpha _1}$, and a parameter ${A_1}$ are adopt to encode ten numbers from 0 to 9, which are in ‘Platinum Beat BTN’ font . The other group number ‘0,1,2,3,4,5,6,7,8,9’, which is used the other font ‘Isoline Light’ to distinguish. The other ten numbers are encrypted by ten HPPs with different topological charges ($l ={-} 12, - 6,0,6,12, - 12, - 6,0,6,12$), two different axicon parameter (${\beta _3},{\beta _4}$) and a rotation angle ${\alpha _2}$, and a parameter ${A_2}$. Superposing those twenty selective holograms generates a MC-OAM multiplexing hologram based on $A - l - \alpha - \beta$ of the modulated chiro-optical phase. With various four parameters modulated incident chiro-optical beam, only the incident beam with the corresponding invers TC and axicon coefficient, the same rotation angle and parameter A could accurately recover the target number into a series of Gaussian spots. The simulation results and experimental results are shown in Fig. 12 and Fig. 13, respectively.
Fig. 12. The decoding result of the twenty images from different incident modulated chiro-optical beam. The encrypted number are encoded and superposed of different multiplex $\alpha$, $\beta$, $A$ and $l$ channels $(l ={-} 12, - 6,0,6,12,A = 0.2,0.6,\,\,\alpha = {\pi / 2},3{\pi / 2},\,\,\beta ={-} 14.8m{m^{ - 1}}, - 4.8m{m^{ - 1}},4.8m{m^{ - 1}},14.8m{m^{ - 1}})$.
Fig. 13. The experimental results of the $A - l - \alpha - \beta$ encrypted MC-OAM multiplexing holography.
4. Conclusion
In summary, an innovative encrypted and decrypted scheme is proposed by using the phase of modulated chiro-optical beam. Four parameters modulated the chiro-optical beam, namely A, TC, the axicon coefficient, and the rotation angle can be used as independent degree of freedom. Since TC, the chiro coefficient and the axicon coefficient are continuous variables, the information capability multiply towards infinity in theoretically. In the simulations and experiments, we can choose any one, two, three or four parameters to encrypt and decrypt information. Furthermore, we have demonstrated the fractional MC-OAM multiplexing-based parameter $A$ or based rotation angle $\alpha$ encrypted holography with the topological charge interval of 0.01. Furthermore, as a result, MC-OAM holography based on modulated chiro-optical beam will significantly increase the holographic information capacity and safety, inspire widespread applications, such as display, security, and communication.
Funding
National Natural Science Foundation of China (62175112, 61975081); Graduate Research and Innovation Projects of Jiangsu Province (No. KYCX22_1540).
Acknowledgments
This study was supported by the National Natural Science Foundation of China (NSFC)(62175112, 61975081); Postgraduate Research &Practice Innovation Program of Jiangsu Province (No. KYCX22_1540).
Disclosures
The authors declare that 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.
References
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
2. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989). [CrossRef]
3. Y. Shen, X. Wang, Z. Xie, et al., “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]
4. B. Y. Wei, S. Liu, P. Chen, S. X. Qi, Y. Zhang, W. Hu, Y. Q. Lu, and J. L. Zhao, “Vortex Airy beams directly generated via liquid crystal q-Airy-plates,” Appl. Phys. Lett. 112(12), 121101 (2018). [CrossRef]
5. N. K. Efremidis, Z. G. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019). [CrossRef]
6. L. P. Zhang, D. M. Deng, X. B. Yang, G. H. Wang, and H. Z. Liu, “Effects of the modulated vortex and second-order chirp on the propagation dynamics of ring Pearcey Gaussian beams,” Opt. Lett. 44(19), 4654–4657 (2019). [CrossRef]
7. L. Y. Zhu, Y. Chen, Z. X. Fang, W. P. Ding, and R. D. Lu, “Experimental demonstration and investigation of vortex circular Pearcey beams in a dynamically shaped fashion,” Opt. Express 29(13), 19819–19830 (2021). [CrossRef]
8. X. Y. Fang, H. R. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020). [CrossRef]
9. X. O. Yang, Y. Xu, M. C. Xian, Z. W. Feng, L. W. Zhu, Y. Y. Cao, S. Lan, B. O. Guan, C. W. Qiu, M. Gu, and X. P. Li, “Synthetic helical dichroism for six-dimensional optical orbital angular momentum multiplexing,” Nat. Photonics 15(12), 901–907 (2021). [CrossRef]
10. H. Q. Zhou, B. Sain, Y. T. Wang, C. Schlickriede, R. Z. Zhao, X. Zhang, Q. S. Wei, X. W. Li, L. L. Huang, and T. Zentgraf, “Polarization-Encrypted Orbital Angular Momentum Multiplexed Metasurface Holography,” ACS Nano 14(5), 5553–5559 (2020). [CrossRef]
11. Q. Xiao, Q. Ma, T. Yan, L. W. Wu, C. Liu, Z. X. Wang, X. Wan, Q. Cheng, and T. J. Cui, “Orbital-Angular-Momentum-Encrypted Holography Based on Coding Information Metasurface,” Adv. Optical Mater. 9(11), 2002155 (2021). [CrossRef]
12. P. Y. Cheng, S. J. Huang, and C. Yan, “Ellipticity-encrypted orbital angular momentum multiplexed holography,” J. Opt. Soc. Am. A 38(12), 1875–1883 (2021). [CrossRef]
13. G. X. Zhu, Z. Y. Bai, J. Y. Chen, C. R. Huang, L. P. Wu, Fu Cailing, and Wang Yiping, “Ultra-dense perfect optical orbital angular momentum multiplexed holography,” Opt. Express 29(18), 28452–28460 (2021). [CrossRef]
14. F. L. Wang, X. C. Zhang, R. Xiong, X. Y. Ma, and X. Q. Jiang, “Angular multiplexation of partial helical phase modes in orbital angular momentum holography,” Opt. Express 30(7), 11110–11119 (2022). [CrossRef]
15. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: A product of helical and conical phase fronts,” Opt. Express 13(5), 1749–1760 (2005). [CrossRef]
16. J. F. Zeng, S. B. Cheng, S. Liu, G. Zhang, S. H. Tao, and W. X. Yang, “Self-healing of the bored helico-conical beam,” Opt. Express 30(6), 9924–9933 (2022). [CrossRef]
17. J. Lin, X. C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, “Deterministic approach to the generation of modified helical beams for optical manipulation,” Opt. Express 13(10), 3862–3867 (2005). [CrossRef]
18. S. B. Cheng, T. Xia, M. S. Liu, Y. Y. Jin, G. Zhang, Y. Xiong, and S. H. Tao, “Power-exponent helico-conical optical beams,” Opt. Laser Technol. 117, 288–292 (2019). [CrossRef]
19. D. H. Yang, Y. Li, D. Deng, Q. M. Chen, Y. Zhang, Y. Liu, J. M. Gao, and M. J. Sun, “Chiral optical field generated by an annular subzone vortex phase plate,” Opt. Lett. 43(19), 4594–4597 (2018). [CrossRef]
20. J. K. Zhou, H. X. Ma, Y. Q. Zhang, S. S. Zhang, C. J. Min, and X. C. Yuan, “Energy flow inversion in an intensity-invariant focusing field,” Opt. Lett. 47(6), 1494–1497 (2022). [CrossRef]
21. N. P. Hermosa II and C. O. Manaois, “Phase structure of helico-conical optical beams,” Opt. Commun. 271(1), 178–183 (2007). [CrossRef]
22. J. C. Ni, C. W. Wang, C. C. Zhang, Y. L. Hu, L. Yang, Z. X. Lao, B. Xu, J. W. Li, D. Wu, and J. R. Chu, “Three-dimensional chiral microstructures fabricated by structured optical vortices in isotropic material,” Light: Sci. Appl. 6(7), e17011 (2017). [CrossRef]
23. R. Liu, Y. Li, D. Deng, Y. Liu, L. T. Hou, and Y. H. Han, “Chiro-optical fields with asymmetric orbital angular momentum and polarization,” Opt. Express 30(5), 7467–7476 (2022). [CrossRef]
24. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
25. P. W. M. Tsang, “Computer-generated Phase-only Holograms for 3D Display,” Cambridge University (2021).
26. F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5(7), 1058–1065 (1988). [CrossRef]
27. Z. Liu, S. Yan, H. Liu, and X. Chen, “Superhigh-Resolution Recognition of Optical Vortex Modes Assisted by a Deep-Learning Method,” Phys. Rev. Lett. 123(18), 183902 (2019). [CrossRef]
28. S. M. A. Hosseini-Saber, E. A. Akhlaghi, and A. Saber, “Diffractometry-based vortex beams fractional topological charge measurement,” Opt. Lett. 45(13), 3478–3481 (2020). [CrossRef]
