Abstract

Perfect vortex beam (PVB), whose ring radius is independent of its topological charge, play an important role in optical trapping and optical communication. Here, we experimentally demonstrate the reconfigurable double-ring PVB (DR-PVB) generation with independent manipulations of the amplitude, the radius, the width, and the topological charge for each ring. Based on complex amplitude modulation (CAM) with a phase-only spatial light modulator (SLM), we successfully verify the proposed DR-PVB generation scheme via the computer-generated hologram. Furthermore, we carry out a quantitative characterization for the generated DR-PVB, in terms of both the generation quality and the generation efficiency. The correlation coefficients of various reconfigurable DR-PVBs are above 0.8, together with the highest generation efficiency of 44%. We believe that, the proposed generation scheme of reconfigurable DR-PVB is desired for applications in both optical tweezers and orbital angular momentum (OAM) multiplexing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Vortex beams associated with the helical wavefront are able to carry orbital angular momentum (OAM) of $l\hbar$ per photon, where l is the topological charge and $\hbar$ is the reduced Planck constant [1]. Those vortex beams can transfer their OAM to mesoscopic particles [24] and provide a new degree of freedom for multiplexing [57]. Typical vortex beams such as Bessel beam and Laguerre-Gaussian beam with ring intensity profile at the transverse cross-section have been explored for several emerging applications in manipulating cold atoms [810], optical tweezers [11,12], and optical communications [13,14]. However, the ring parameter of those vortex beams strongly depends on its value of topological charge, leading to a constraint for various applications. For instance, it results in a low coupling efficiency and mode profile distortion, when the vortex beam is coupled into a ring-core fiber with fixed refractive index (RI) profile [6]. Moreover, a large value of topological charge and a small ring diameter is in an urgent need for the optical trapping. In order to overcome those issues for traditional vortex beams, Ostrovsky et al. firstly proposed the concept of perfect vortex beam (PVB) whose radius is independent of its topological charge [15]. Such unique property has stimulated extensive research interests over the world [1622]. The PVB can be obtained by either the Fourier transformation of the Bessel-Gaussian beam [16] or the energy redistribution with the help of an optimal phase element [17]. Meanwhile, a ring laser beam can be generated by the Gaussian beam illumination of an axicon, then the PVB can be obtained at the first diffraction order of the spatial light modulator (SLM) [18,19]. However, previous investigations were only focused on the PVB with a single bright ring, which prevents the trapping of low-RI particles. Alternatively, the double-ring PVB (DR-PVB) is a potential solution, where two bright rings can trap and rotate the high-RI particles, while low-RI particles can be captured by the dark ring. Meanwhile, the DR-PVB is able to improve the OAM multiplexing capacity, the super-resolution imaging technique [23], and the laser structuring [2426]. The DR-PVB with two close rings can be obtained by the Fourier transform of azimuthally polarized Bessel beams [27]. However, the intensity at the dark region between two bright rings increases with the variation of topological charge, leading to a sharpness reduction of the dark ring. The double-ring beam can be also observed at the focal region of the binary axicon, when the polarized laser beam is introduced [28]. However, the detailed characterization of DR-PVB is not provided. A narrow double-ring shaped radially polarized spiral laser beam with variable amplitudes and ring widths can be obtained by a combination of a binary axicon with a polarization interference. The width of each ring is determined by the diameter of the input beam and the amplitude distribution can be controlled by the energy redistribution among different diffraction orders [29]. Meanwhile, the double-ring beam can be generated by the use of circular diffractive grating [30]. However, since double rings of the generated beam are very close to each other, it is impossible to vary the distance between them. Although both the Fourier transform of superposition of two coaxial Bessel beams and the binary-phase diffractive optical element (DOE) can be referred for the DR-PVB generation [31,32], the verification results of the perfect characteristic are not provided. Therefore, a reconfigurable DR-PVB generation with precise manipulation of transverse cross-section profile, where its amplitude, ring radius, ring width and topological charge at each ring can be arbitrarily set, is highly desired for trapping and rotating particles with both the size and the RI variable. Generally, beam shaping technique is indispensable for the PVB generation. Digital micromirror device (DMD) has been reported to realize the PVB generation [20,21], but the maximum theoretical modulation efficiency of DMD is only 10% [33]. Alternatively, the SLM provides another option with high efficiency. Since most SLM can only realize the modulation of spatial phase, we need to implement the complex amplitude modulation (CAM) technique for the purpose of simultaneous amplitude and phase modulation [34]. When the computer-generated hologram is loaded into the SLM, the input beam is diffracted into different orders. In particular, the diffraction efficiency can be optimized, and the desired output beam can be obtained after the spatial filtering [35].

In current submission, we focus on the reconfigurable generation of DR-PVB with the help of the CAM enabled by the phase-only SLM. After providing a definition of flexible manipulation for the DR-PVB, we explain the operation principle of corresponding computer-generated hologram. Then, by loading the holograms into the SLM, we successfully demonstrate the DR-PVB generation with flexible manipulation of its amplitude, radius, ring width, and topological charge. Finally, we experimentally characterize the generated DR-PVB by the presence of the optical vortices, when the generated DR-PVB and the Gaussian reference beam are interfered. Meanwhile, the correlation coefficients of the generated DR-PVBs after flexible manipulation are above 0.8, indicating of the excellent generation quality. Meanwhile, we explore the generation efficiency of the proposed DR-PVB generation scheme.

2. Theoretical investigations

2.1 Model of reconfigurable DR-PVB

The transverse distribution of single-ring PVB with a topological charge of $l$ can be analytically described as

$$E(\rho ,\theta ) = \delta (\rho - {w_0})\textrm{exp} (\textrm{i}l\theta )$$
where $({\rho , \theta )} $ is the position vector under the polar coordinate, ${w_0}$ is the radius of the PVB, and $\delta$ is the Dirac function. However, it is difficult to realize it in practice, so that a ring field profile with a Gaussian approximation is commonly-used. Thus, Eq. (1) can be rewritten as [19]
$$E(\rho ,\theta ) = \textrm{exp} \left[ { - \frac{{{{({\rho - {w_0}} )}^2}}}{{{\Delta ^2}}}} \right]\textrm{exp} (\textrm{i}l\theta )$$
where $\Delta $ is the ring width of the PVB. In our investigation, we treat the DR-PVB as a superposition of two single-ring PVBs
$$E(\rho ,\theta ) = {A_1}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_1}} )}^2}}}{{\Delta _1^2}}} \right]\textrm{exp} (\textrm{i}{l_1}\theta ) - {A_2}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_2}} )}^2}}}{{\Delta _2^2}}} \right]\textrm{exp} (\textrm{i}{l_2}\theta )$$
where ${A_1}$ and ${A_2}$ are the maximum amplitude of two rings, ${w_1}$ and ${w_2}$ are the radius of two rings, ${\Delta _1}$ and ${\Delta _2}$ are the width of two rings, respectively. Using the sign of – for the superposition of two rings intends to maintain the sharpness of the dark ring. Next, Eq. (3) can be expressed as
$$E(\rho ,\theta ) = ({{a_1} + {b_1}i} )- ({{a_2} + {b_2}i} )= ({{a_1} - {a_2}} )+ ({{b_1} - {b_2}} )i$$
where,
$${a_1} = {A_1}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_1}} )}^2}}}{{\Delta _1^2}}} \right]\cos ({l_1}\theta )$$
$${b_1} = {A_1}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_1}} )}^2}}}{{\Delta _1^2}}} \right]\sin ({l_1}\theta )$$
$${a_2} = {A_2}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_2}} )}^2}}}{{\Delta _2^2}}} \right]\cos ({l_2}\theta )$$
$${b_2} = {A_2}\textrm{exp} \left[ { - \frac{{{{({\rho - {w_2}} )}^2}}}{{\Delta _2^2}}} \right]\sin ({l_2}\theta )$$

Thus, the amplitude of Eq. (3) is

$$A(\rho ,\theta ) = |{E(\rho ,\theta )} |= \sqrt {{{({{a_1} - {a_2}} )}^2} + {{({{b_1} - {b_2}} )}^2}}$$
and the phase of Eq. (3) is
$$\phi (\rho ,\theta ) = \textrm{angle} ({E(\rho ,\theta )} )= \left\{ {\begin{array}{{ll}} {\arctan \frac{{{b_1} - {b_2}}}{{{a_1} - {a_2}}}}&{\textrm{ when }{a_1} - {a_2} > 0}\\ {\pi + \arctan \frac{{{b_1} - {b_2}}}{{{a_1} - {a_2}}}}&{\textrm{ when }{a_1} - {a_2} < 0}\\ {\frac{\pi }{2}}&{\textrm{ when }{a_1} = {a_2}\textrm{ and }\;{b_1} > {b_2}}\\ { - \frac{\pi }{2}}&{\textrm{ when }{a_1} = {a_2}\textrm{ and }\;{b_1} < {b_2}} \end{array}} \right.$$

By adjusting the parameters of Eq. (3) properly, the transverse field with double-ring intensity profile is schematically presented in Fig. 1(a) and (b). l1 and l2 are the topological charges of the outer and inner rings, respectively. The phase factors $\textrm{exp} (\textrm{i}{l_1}\theta )$ and $\textrm{exp} (\textrm{i}{l_2}\theta )$ indicate that the DR-PVB exhibits a spiral phase, as shown in the Fig. 1(c). The phase of each ring changes continuously from -π to π around the center. Meanwhile, there occur several phase singularities at the center and the position where the phase of two rings contacts. Since the topological charges of DR-PVB can be flexibly manipulated, the intensity profile with different topological charges is shown in Fig. 1(d), the inset of Fig. 1(d) indicates that the radial profile is independent of topological charges. Here, we fix the inner ring values of ${A_2}$, ${w_2}$ and ${\Delta _2}$, and assume that the above-mentioned variables satisfy the conditions of $k = {{{A_1}} / {{A_2}}}$, $p = {{{w_1}} / {{w_2}}}$ and $q = {{{\Delta _1}} / {{\Delta _2}}}$. Therefore, the DR-PVB is denoted as ‘$k - p - q$’, and the transverse profile of the DR-PVB can be flexibly manipulated by changing the values of k, $p$ and $q$.

2.2 Hologram generation using the CAM

The key of CAM technique is to encode the arbitrary amplitude and phase information simultaneously into a phase hologram [36,37]. Consequently, the input beam can be transformed to the desired beam by loading the hologram into the SLM. It is well-known that an electrical field with amplitude $A(x,y)$ and phase $\phi (x,y)$ can be expressed as

$$E(x,y) = A(x,y)\textrm{exp} ({i\phi (x,y)} )$$
where $A(x,y)$ and $\phi (x,y)$ represent the spatial distribution of amplitude and phase, whose value is constrained within the range of [0,1] and $[ - \pi ,\pi ]$, respectively. One way to realize the CAM technique is
$$h(x,y) = \textrm{exp} \{ i\Phi [A(x,y),\phi (x,y)]\}$$

 figure: Fig. 1.

Fig. 1. Schematic DR-PVB: (a) transverse intensity profile, (b)cross-section intensity profile, (c) isophase surface profile under the condition of l1= l2 = 2, and (d) cross-sectional intensity profile with different topological charges.

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As a result, the desired field can be achieved as ${E_{\textrm{des}}}(x,y) = {E_{\textrm{in}}}(x,y) \times h({x,y} )$. The key point is to identify the function $\Phi [A(x,y),\phi (x,y)]$, which is the hologram to be loaded into the SLM. In our research, we choose Eq. (13) which is able to realize the CAM with a better beam quality, in comparison with other methods [35,38,39],

$$\Phi [A(x,y),\phi (x,y)] = f(A(x,y))\phi (x,y)$$
where $f(A(x,y))$ is given by
$$f(A(x,y)) = 1 - {\textrm{sinc} ^{ - 1}}({A({x,y} )} )$$

The amplitude and phase can be easily obtained from the Eq. (3). Furthermore, a linear phase grating is added to the hologram, in order to successfully separate the different diffraction orders. The transfer function of the phase grating has a form [40]

$${t_g}(x,y) = \textrm{exp} [i2\pi (ux + vy)]$$
where u and v are the spatial frequencies of the grating along with the horizontal and vertical direction, respectively. The spatial coordinate (U, V) of the first diffraction order at the far field is determined by the grating frequencies. In this work, we set the horizontal and vertical frequencies as 8000 and 0, respectively. Finally, the hologram with the phase grating is
$$\Phi = \bmod \{{[1 - {{\textrm{sinc} }^{ - 1}}(A(x,y))][\phi (x,y) + 2\pi (ux + vy)]} \}$$
where mod $\{{\cdot} \}$ represents the modulus function that wraps the phase around $2\pi$.The first part represent $f(A(x,y))$, and the second part contains the phase of the target field and the phase grating. A sample hologram of DR-PVB generated by Eq. (16) is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. A sample hologram of DR-PVB

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3. Experimental setup

The experimental setup is shown in Fig. 3. The Gaussian beam emitted from a 1550nm distributed feedback (DFB) laser (Yenista Optics OSICS-DFB, YO11490139) is collimated through a collimator, after its state of polarization (SOP) is managed by a single-mode fiber based polarization controller (SMF-PC). Then, the Gaussian beam is expanded by a 5× beam expander (BE), in order to increase the beam waist from 2.1mm to 10.5mm and fit the active area of SLM. A reflective phase-only SLM(HOLOEYE-PLUTO) with a resolution of 1920 × 1080 pixels and a pixel pitch of 8 µm is used to implement the CAM, after the Gaussian beam is free-space transmitted through a beam splitter (BS1) and reflected by a mirror (M1). The loaded hologram leads to an optical diffraction, after the beam shaping is implemented with the SLM. For the ease of characterizing the generated DR-PVB, we build a 4f system including two lenses with a focal length of 200mm (f1=200mm) and 100mm (f2=100mm), respectively, in order to delay the image plane of the SLM to the CCD (OPHIR Spiricon SP928-1550) with 1600 × 1200 pixels and a pixel pitch of 4.4 µm. An adjustable diaphragm is placed at the Fourier plane of L1 to only obtain the first diffraction order. Meanwhile, the Mach–Zehnder interference setup is developed to characterize the generated DR-PVB. The Gaussian beam from the same semiconductor laser reflected by the BS1 is used as a reference beam, and consequently interferes with the generated DR-PVB. During the characterization of intensity profile for the DR-PVB, BS1 and BS2 are removed for reducing the power loss.

 figure: Fig. 3.

Fig. 3. Experimental setup to generate DR-PVB.

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4. Characterization results and discussions

When the hologram generated by the CAM technique is successfully loaded into the SLM, we experimentally generate the DR-PVB and the corresponding intensity profile is recorded by the CCD. Without the use of BS1 and BS2, we first characterize the intensity profile of the generated DR-PVB with variable amplitudes, radius and beam widths. The quality of the generated DR-PVB is quantitatively evaluated by a correlation coefficient between the intensity profiles of the either numerically or experimentally generated DR-PVB and that of targeted DR-PVB, which is defined as [41],

$$C = \frac{{\sum\nolimits_m {\sum\nolimits_n {({{A_{mn}} - \bar{A}} )({{B_{mn}} - \bar{B}} )} } }}{{\sqrt {\left( {{{\sum\nolimits_m {\sum\nolimits_n {({{A_{mn}} - \bar{A}} )} } }^2}} \right)\left( {{{\sum\nolimits_m {\sum\nolimits_n {({{B_{mn}} - \bar{B}} )} } }^2}} \right)} }}$$
where m and n are the pixels of the length and width of the transverse intensity profile. Amn and Bmn are the intensity of each pixel of the generated DR-PVB and targeted DR-PVB, and $\bar{A}$ and $\bar{B}$ are the mean intensity values of the generated DR-PVB and targeted DR-PVB, respectively. The value of C is at a range from 0 to 1, where 1 is a perfect matching, indicating the realization of the identical DR-PVB generation. Higher C indicates the stronger similarity between the generated DR-PVB and targeted DR-PVB. Generally, the bench-mark of good quality is 0.8, when the damaged pixels of the CCD are removed [42]. Figure 4 shows three sets of DR-PVB intensity profiles with different amplitudes, radius, and widths, indicating that the generated intensity profile has excellent agreement with the targeted intensity profile. Since corresponding correlation coefficients denoted at the bottom of Fig. 4(a-c) are around 0.9, the generated DR-PVB with flexible manipulation has good beam quality. Furthermore, we experimentally capture a group of DR-PVB intensity profiles and calculate the simulation and experiment correlation coefficients, as shown in Fig. 5. All simulation correlation coefficients are beyond 0.98, indicating of a perfect similarity. The experimental correlation coefficients are slightly lower than that of the simulation, because the background noise appears during the DR-PVB generation, but still higher than 0.8. The similarity becomes weak with the reduction of intensity, because the captured figure is sensitive to the noise under the condition of low signal-to-noise ratio (SNR). Meanwhile, the experimental correlation coefficients decrease, when the radius of outer ring is larger. We infer that it is due to the reduced SNR by the Gaussian envelope limitation of the input field. Since the intensity of the generated DR-PVB is mainly determined by the generation efficiency, here we define the term as the total power of the generated beam over that of the incident beam on the SLM. Next, we calculate the simulation and experimental efficiencies of those generated DR-PVBs, as shown in Fig. 6. It is worth noting that the intensity values are normalized for the ease of performance comparison. The highest generation efficiency of our proposed generation method is up to 44%, and the lowest generation efficiency is more than 13%. In general, the trend of experimental generation efficiencies is in consistent with that of simulation results. The experimental generation efficiency is higher than the simulation one, when k and q are small, because the radius of fixed spatial filter is relatively large for the small beam size, resulting in an interference from other diffraction orders or the stray light. Thus, the experimental generation efficiency is higher than that from the numerical simulation. As for the case of ‘k-2-1’, the peak intensity of outer ring is weaker under the condition of k <1, leading to a low generation efficiency. When k>1 is satisfied, since the stray light has a trivial impact on the total power, the experimental generation efficiency is lower than the simulation one. Meanwhile, the peak intensity of the outer ring remains stable while that of the inner ring decreases at the limitation of the Gaussian envelope, leading to the reduction of generation efficiency with the growing k. In Fig. 6(b), p starts with 2 in order to keep the outer ring away from the inner ring, in order to avoid the enhancement of the minimal intensity for the dark ring. When p<2.7 is satisfied, the generation efficiency increases with the growing radius of the outer ring. We infer that, more pixels are included for the outer ring with the growing radius. Under the condition of p>2.7, the generation efficiency decreases at the limitation of the Gaussian envelope, when the radius of the outer ring increases. Meanwhile, owing to more pixels appeared in the outer ring, the generation efficiency becomes higher with the increase of outer ring width, as shown in Fig. 6(c). Finally, we record the intensity profile of the same value of k, p and q with different topological charges. The same as the definition of DR-PVB, the ring radius of the transverse intensity profiles remains constant for all topological charges, as shown in Fig. 7.

 figure: Fig. 4.

Fig. 4. Correlation coefficient between experimental and theoretical intensity profile of DR-PVB, (a) variable amplitude, (b) variable radius, and (c) variable beam width.

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 figure: Fig. 5.

Fig. 5. Correlation coefficient variation of DR-PVB versus (a) amplitude, (b) radius, and (c) beam width.

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 figure: Fig. 6.

Fig. 6. Efficiencies variation of DR-PVB intensity profile versus its (a) amplitude, (b) radius, and (c) beam width.

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 figure: Fig. 7.

Fig. 7. Cross-sectional intensity profile of generated DR-PVB with different topological charges.

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Next, we carry out experimental verification of spatial phase distribution for the generated DR-PVB, based on the optical interference pattern between the generated DR-PVB and the Gaussian reference beam. The optical interference patterns with the same topological charges arising in two rings are presented in Fig. 8. Our generated DR-PVB can carry different topological charges within two rings at the same time. Figure 9 shows the optical interference patterns of two topological charges with different values. Those optical interference patterns confirm the presence of the optical vortex. The rotation direction of the tailing is counterclockwise when the sign of topological charge is positive, while it rotates clockwise when the sign is negative. The tailing phenomena become more obvious, when the ring radius is larger. The experimental results agree well with the simulation results for all cases. The experimental interference patterns are slightly uneven, because the two beams cannot be perfectly coaxial due to the environmental perturbation, meanwhile, it may be due to the non-centric illumination of the hologram. Since the topological charges of two rings can be independently set, it can be used for controlling the rotation of optically trapped microscopic particles [43]. Moreover, since it can take different information in two rings at the same time, our proposed DR-PVB generation is a potential option for enhancing the channel capacity based on the OAM multiplexing.

 figure: Fig. 8.

Fig. 8. Optical interference pattern under conditions of the same topological charges arising in two rings.

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 figure: Fig. 9.

Fig. 9. Optical interference patterns with different topological charges arising in two rings.

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Finally, we observe the interference pattern when two rings with different topological charges are adjusted to the same size, as shown in Fig. 10(a)-(e). The intensity profile of interference pattern is petal-like bright spot, instead of ring-like, because the destructive interference occurs with the same intensity of two rings. The number of bright spots is determined by the topological charge within two rings, it is denoted as $|{{l_1} - {l_2}} |$ for different topological charges. The beam intensity decreases, when the number of the bright spots increases. Furthermore, a football-like structure can be observed when we adjust the size of two rings appropriately, as shown in Fig. 10(f). Those diversified light fields are potentials for the doppler velocimetry [44,45], the optical information encoding [46] and the cold atoms trapping [47].

 figure: Fig. 10.

Fig. 10. Intensity profile with different topological charges of two rings. (a)-(b) same size and same sign, (c)-(e) same size and different sign, (f) football-like intensity profile.

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5. Conclusions

We have demonstrated a reconfigurable DR-PVB generation with full manipulation of its amplitudes, radius, widths and topological charges, based on the phase-only SLM. We quantitively evaluate the quality and efficiency of the generated DR-PVB, in terms of correlation coefficient and generation efficiency. The correlation coefficients are beyond 0.8, proving that the intensity profiles of generated DR-PVB are in good agreement with that of ideal DR-PVB. In addition, we verify that the proposed DR-PVB generation scheme has a high efficiency. Meanwhile, we have experimentally verified the presence of optical vortex based on the optical interference between the generated DR-PVB and the Gaussian reference beam. Our characterization results indicate that two rings of the DR-PVB can take different topological charges, and the number of rings can be further extended. We believe that, the proposed generation scheme of reconfigurable DR-PVB has great potentials for the particle trapping and rotation together with the OAM multiplexing.

Funding

National Key Research and Development Program of China (2018YFB1801001); National Natural Science Foundation of China (61875061); Guangdong Introducing Innovative and Entrepreneurial Teams of “The Pearl River Talent Recruitment Program” (2019ZT08X340); Special Project for Research and Development in Key areas of Guangdong Province (2018B010114002).

Disclosures

The authors declare no conflicts of interests.

Data availability

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

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23. Z. Nie, Z. Li, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ring-shaped azimuthally polarized beam,” Opt. Lasers Eng. 59, 93–97 (2014). [CrossRef]  

24. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, and K. Miyamoto, “Transfer of Light Helicity to Nanostructures,” Phys. Rev.Lett. 110(14), 143603 (2013). [CrossRef]  

25. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. 3(1), 989 (2012). [CrossRef]  

26. S. Syubaev, A. Zhizhchenko, O. Vitrik, A. Porfirev, S. Fomchenkov, S. Khonina, S. Kudryashov, and A. Kuchmizhak, “Chirality of laser-printed plasmonic nanoneedles tunable by tailoring spiral-shape pulses,” Appl. Surf. Sci. 470, 526–534 (2019). [CrossRef]  

27. Y. Liang, S. Yan, M. He, M. Li, Y. Cai, Z. Wang, M. Lei, and B. Yao, “Generation of a double-ring perfect optical vortex by the Fourier transform of azimuthally polarized Bessel beams,” Opt. Lett. 44(6), 1504–1507 (2019). [CrossRef]  

28. S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018). [CrossRef]  

29. S. V. Karpeev, V. D. Paranin, and S. N. Khonina, “Generation of a controlled double-ring-shaped radially polarized spiral laser beam using a combination of a binary axicon with an interference polarizer,” J. Opt. 19(5), 055701 (2017). [CrossRef]  

30. S. N. Khonina, A. P. Porfirev, and A. V. Ustinov, “Diffractive axicon with tunable fill factor for focal ring splitting, ” Proc. SPIE 10233, 102331P (2017). [CrossRef]  

31. D. Deng, H. Zhao, Y. Li, and Y. Han, “Rotating structured light based on perfect vortex,” Appl. Phys. Express 14(3), 032004 (2021). [CrossRef]  

32. S. N. Khonina, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik - Int. J. Light and Electron Opt. 110(3), 137–144 (1999).

33. C. Rosales-Guzmán, X. Hu, A. Selyem, P. Moreno-Acosta, S. Franke-Arnold, R. Ramos-Garcia, and A. Forbes, “Polarisation-insensitive generation of complex vector modes from a digital micromirror device,” Sci. Rep. 10(1), 10434 (2020). [CrossRef]  

34. G. Wang, Y. Miao, Y. Li, X. Shan, and X. Gao, “Experimental and theoretical study of linearly polarized lorentz–gauss beams with heterogeneous distribution,” Chin. Opt. Lett. 19(2), 022602 (2021). [CrossRef]  

35. 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]  

36. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34(1), 34–36 (2009). [CrossRef]  

37. J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. 42(11), 2003–2008 (2003). [CrossRef]  

38. V. Arrizón, U. Ruiz, R. Carrada, and L. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef]  

39. T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016). [CrossRef]  

40. C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 modes on a single digital hologram,” J. Opt. 19(11), 113501 (2017). [CrossRef]  

41. A. M. Neto, A. C. Victorino, I. Fantoni, D. E. Zampieri, J. V. Ferreira, and D. A. Lima, “Image processing using Pearson's correlation coefficient: Applications on autonomous robotics,” IEEE Int. Conf. on Autonomous Robot Systems and Competitions, 1–6 (2013).

42. S. M. Barhli, M. Mostafavi, A. F. Cinar, D. Hollis, and T. J. Marrow, “J-Integral Calculation by Finite Element Processing of Measured Full-Field Surface Displacements,” Exp. Mech. 57(6), 997–1009 (2017). [CrossRef]  

43. R. L. Eriksen, V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Computer-controlled orientation of multiple optically-trapped microscopic particles,” Microelectron. Eng. 67-68, 872–878 (2003). [CrossRef]  

44. S. Qiu, Y. Ren, T. Liu, L. Chen, C. Wang, Z. Li, and Q. Shao, “Spinning object detection based on perfect optical vortex,” Opt. Laser. Eng. 124, 105842 (2020). [CrossRef]  

45. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light's orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]  

46. J. A. Anguita, J. Herreros, and I. B. Djordjevic, “Coherent Multimode OAM Superpositions for Multidimensional Modulation,” IEEE Photonics J. 6(2), 1–11 (2014). [CrossRef]  

47. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef]  

References

  • View by:

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  8. S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66(4), 043801 (2002).
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  10. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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  11. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
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  18. P. Bélanger and M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17(7), 1080–1088 (1978).
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    [Crossref]
  20. Y. Chen, Z. Fang, Y. Ren, L. Gong, and R. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015).
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  21. C. Zhang, C. Min, and X. Yuan, “Shaping perfect optical vortex with amplitude modulated using a digital micro-mirror device,” Opt. Commun. 381, 292–295 (2016).
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  22. J. García-García, C. Rickenstorff-Parrao, R. Ramos-García, V. Arrizón, and A. S. Ostrovsky, “Simple technique for generating the perfect optical vortex,” Opt. Lett. 39(18), 5305–5308 (2014).
    [Crossref]
  23. Z. Nie, Z. Li, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ring-shaped azimuthally polarized beam,” Opt. Lasers Eng. 59, 93–97 (2014).
    [Crossref]
  24. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, and K. Miyamoto, “Transfer of Light Helicity to Nanostructures,” Phys. Rev.Lett. 110(14), 143603 (2013).
    [Crossref]
  25. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. 3(1), 989 (2012).
    [Crossref]
  26. S. Syubaev, A. Zhizhchenko, O. Vitrik, A. Porfirev, S. Fomchenkov, S. Khonina, S. Kudryashov, and A. Kuchmizhak, “Chirality of laser-printed plasmonic nanoneedles tunable by tailoring spiral-shape pulses,” Appl. Surf. Sci. 470, 526–534 (2019).
    [Crossref]
  27. Y. Liang, S. Yan, M. He, M. Li, Y. Cai, Z. Wang, M. Lei, and B. Yao, “Generation of a double-ring perfect optical vortex by the Fourier transform of azimuthally polarized Bessel beams,” Opt. Lett. 44(6), 1504–1507 (2019).
    [Crossref]
  28. S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
    [Crossref]
  29. S. V. Karpeev, V. D. Paranin, and S. N. Khonina, “Generation of a controlled double-ring-shaped radially polarized spiral laser beam using a combination of a binary axicon with an interference polarizer,” J. Opt. 19(5), 055701 (2017).
    [Crossref]
  30. S. N. Khonina, A. P. Porfirev, and A. V. Ustinov, “Diffractive axicon with tunable fill factor for focal ring splitting, ” Proc. SPIE 10233, 102331P (2017).
    [Crossref]
  31. D. Deng, H. Zhao, Y. Li, and Y. Han, “Rotating structured light based on perfect vortex,” Appl. Phys. Express 14(3), 032004 (2021).
    [Crossref]
  32. S. N. Khonina, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik - Int. J. Light and Electron Opt. 110(3), 137–144 (1999).
  33. C. Rosales-Guzmán, X. Hu, A. Selyem, P. Moreno-Acosta, S. Franke-Arnold, R. Ramos-Garcia, and A. Forbes, “Polarisation-insensitive generation of complex vector modes from a digital micromirror device,” Sci. Rep. 10(1), 10434 (2020).
    [Crossref]
  34. G. Wang, Y. Miao, Y. Li, X. Shan, and X. Gao, “Experimental and theoretical study of linearly polarized lorentz–gauss beams with heterogeneous distribution,” Chin. Opt. Lett. 19(2), 022602 (2021).
    [Crossref]
  35. 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]
  36. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34(1), 34–36 (2009).
    [Crossref]
  37. J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. 42(11), 2003–2008 (2003).
    [Crossref]
  38. V. Arrizón, U. Ruiz, R. Carrada, and L. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007).
    [Crossref]
  39. T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016).
    [Crossref]
  40. C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 modes on a single digital hologram,” J. Opt. 19(11), 113501 (2017).
    [Crossref]
  41. A. M. Neto, A. C. Victorino, I. Fantoni, D. E. Zampieri, J. V. Ferreira, and D. A. Lima, “Image processing using Pearson's correlation coefficient: Applications on autonomous robotics,” IEEE Int. Conf. on Autonomous Robot Systems and Competitions, 1–6 (2013).
  42. S. M. Barhli, M. Mostafavi, A. F. Cinar, D. Hollis, and T. J. Marrow, “J-Integral Calculation by Finite Element Processing of Measured Full-Field Surface Displacements,” Exp. Mech. 57(6), 997–1009 (2017).
    [Crossref]
  43. R. L. Eriksen, V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Computer-controlled orientation of multiple optically-trapped microscopic particles,” Microelectron. Eng. 67-68, 872–878 (2003).
    [Crossref]
  44. S. Qiu, Y. Ren, T. Liu, L. Chen, C. Wang, Z. Li, and Q. Shao, “Spinning object detection based on perfect optical vortex,” Opt. Laser. Eng. 124, 105842 (2020).
    [Crossref]
  45. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light's orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref]
  46. J. A. Anguita, J. Herreros, and I. B. Djordjevic, “Coherent Multimode OAM Superpositions for Multidimensional Modulation,” IEEE Photonics J. 6(2), 1–11 (2014).
    [Crossref]
  47. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007).
    [Crossref]

2021 (2)

2020 (2)

C. Rosales-Guzmán, X. Hu, A. Selyem, P. Moreno-Acosta, S. Franke-Arnold, R. Ramos-Garcia, and A. Forbes, “Polarisation-insensitive generation of complex vector modes from a digital micromirror device,” Sci. Rep. 10(1), 10434 (2020).
[Crossref]

S. Qiu, Y. Ren, T. Liu, L. Chen, C. Wang, Z. Li, and Q. Shao, “Spinning object detection based on perfect optical vortex,” Opt. Laser. Eng. 124, 105842 (2020).
[Crossref]

2019 (2)

S. Syubaev, A. Zhizhchenko, O. Vitrik, A. Porfirev, S. Fomchenkov, S. Khonina, S. Kudryashov, and A. Kuchmizhak, “Chirality of laser-printed plasmonic nanoneedles tunable by tailoring spiral-shape pulses,” Appl. Surf. Sci. 470, 526–534 (2019).
[Crossref]

Y. Liang, S. Yan, M. He, M. Li, Y. Cai, Z. Wang, M. Lei, and B. Yao, “Generation of a double-ring perfect optical vortex by the Fourier transform of azimuthally polarized Bessel beams,” Opt. Lett. 44(6), 1504–1507 (2019).
[Crossref]

2018 (1)

S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
[Crossref]

2017 (4)

S. V. Karpeev, V. D. Paranin, and S. N. Khonina, “Generation of a controlled double-ring-shaped radially polarized spiral laser beam using a combination of a binary axicon with an interference polarizer,” J. Opt. 19(5), 055701 (2017).
[Crossref]

S. N. Khonina, A. P. Porfirev, and A. V. Ustinov, “Diffractive axicon with tunable fill factor for focal ring splitting, ” Proc. SPIE 10233, 102331P (2017).
[Crossref]

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 modes on a single digital hologram,” J. Opt. 19(11), 113501 (2017).
[Crossref]

S. M. Barhli, M. Mostafavi, A. F. Cinar, D. Hollis, and T. J. Marrow, “J-Integral Calculation by Finite Element Processing of Measured Full-Field Surface Displacements,” Exp. Mech. 57(6), 997–1009 (2017).
[Crossref]

2016 (3)

2015 (3)

P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of a Bessel beam,” Opt. Lett. 40(4), 597–600 (2015).
[Crossref]

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]

Y. Chen, Z. Fang, Y. Ren, L. Gong, and R. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015).
[Crossref]

2014 (3)

J. García-García, C. Rickenstorff-Parrao, R. Ramos-García, V. Arrizón, and A. S. Ostrovsky, “Simple technique for generating the perfect optical vortex,” Opt. Lett. 39(18), 5305–5308 (2014).
[Crossref]

Z. Nie, Z. Li, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of a sub-wavelength focal spot with a long transversally polarized optical needle using a double-ring-shaped azimuthally polarized beam,” Opt. Lasers Eng. 59, 93–97 (2014).
[Crossref]

J. A. Anguita, J. Herreros, and I. B. Djordjevic, “Coherent Multimode OAM Superpositions for Multidimensional Modulation,” IEEE Photonics J. 6(2), 1–11 (2014).
[Crossref]

2013 (5)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light's orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, and K. Miyamoto, “Transfer of Light Helicity to Nanostructures,” Phys. Rev.Lett. 110(14), 143603 (2013).
[Crossref]

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013).
[Crossref]

A. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

2012 (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. 3(1), 989 (2012).
[Crossref]

2009 (1)

2008 (1)

2007 (2)

2004 (1)

2003 (4)

W. M. Lee, X. Yuan, and D. Tang, “Optical tweezers with multiple optical forces using double-hologram interference,” Opt. Express 11(3), 199–207 (2003).
[Crossref]

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the Transfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[Crossref]

J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. 42(11), 2003–2008 (2003).
[Crossref]

R. L. Eriksen, V. R. Daria, P. J. Rodrigo, and J. Glückstad, “Computer-controlled orientation of multiple optically-trapped microscopic particles,” Microelectron. Eng. 67-68, 872–878 (2003).
[Crossref]

2002 (2)

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66(4), 043801 (2002).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

2000 (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

1999 (2)

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]

S. N. Khonina, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik - Int. J. Light and Electron Opt. 110(3), 137–144 (1999).

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel Optical Trap of Atoms with a Doughnut Beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuwand, 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]

1978 (1)

Abraham, E. R. I.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66(4), 043801 (2002).
[Crossref]

Ahmed, N.

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]

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Data availability

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

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Figures (10)

Fig. 1.
Fig. 1. Schematic DR-PVB: (a) transverse intensity profile, (b)cross-section intensity profile, (c) isophase surface profile under the condition of l1= l2 = 2, and (d) cross-sectional intensity profile with different topological charges.
Fig. 2.
Fig. 2. A sample hologram of DR-PVB
Fig. 3.
Fig. 3. Experimental setup to generate DR-PVB.
Fig. 4.
Fig. 4. Correlation coefficient between experimental and theoretical intensity profile of DR-PVB, (a) variable amplitude, (b) variable radius, and (c) variable beam width.
Fig. 5.
Fig. 5. Correlation coefficient variation of DR-PVB versus (a) amplitude, (b) radius, and (c) beam width.
Fig. 6.
Fig. 6. Efficiencies variation of DR-PVB intensity profile versus its (a) amplitude, (b) radius, and (c) beam width.
Fig. 7.
Fig. 7. Cross-sectional intensity profile of generated DR-PVB with different topological charges.
Fig. 8.
Fig. 8. Optical interference pattern under conditions of the same topological charges arising in two rings.
Fig. 9.
Fig. 9. Optical interference patterns with different topological charges arising in two rings.
Fig. 10.
Fig. 10. Intensity profile with different topological charges of two rings. (a)-(b) same size and same sign, (c)-(e) same size and different sign, (f) football-like intensity profile.

Equations (17)

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E ( ρ , θ ) = δ ( ρ w 0 ) exp ( i l θ )
E ( ρ , θ ) = exp [ ( ρ w 0 ) 2 Δ 2 ] exp ( i l θ )
E ( ρ , θ ) = A 1 exp [ ( ρ w 1 ) 2 Δ 1 2 ] exp ( i l 1 θ ) A 2 exp [ ( ρ w 2 ) 2 Δ 2 2 ] exp ( i l 2 θ )
E ( ρ , θ ) = ( a 1 + b 1 i ) ( a 2 + b 2 i ) = ( a 1 a 2 ) + ( b 1 b 2 ) i
a 1 = A 1 exp [ ( ρ w 1 ) 2 Δ 1 2 ] cos ( l 1 θ )
b 1 = A 1 exp [ ( ρ w 1 ) 2 Δ 1 2 ] sin ( l 1 θ )
a 2 = A 2 exp [ ( ρ w 2 ) 2 Δ 2 2 ] cos ( l 2 θ )
b 2 = A 2 exp [ ( ρ w 2 ) 2 Δ 2 2 ] sin ( l 2 θ )
A ( ρ , θ ) = | E ( ρ , θ ) | = ( a 1 a 2 ) 2 + ( b 1 b 2 ) 2
ϕ ( ρ , θ ) = angle ( E ( ρ , θ ) ) = { arctan b 1 b 2 a 1 a 2  when  a 1 a 2 > 0 π + arctan b 1 b 2 a 1 a 2  when  a 1 a 2 < 0 π 2  when  a 1 = a 2  and  b 1 > b 2 π 2  when  a 1 = a 2  and  b 1 < b 2
E ( x , y ) = A ( x , y ) exp ( i ϕ ( x , y ) )
h ( x , y ) = exp { i Φ [ A ( x , y ) , ϕ ( x , y ) ] }
Φ [ A ( x , y ) , ϕ ( x , y ) ] = f ( A ( x , y ) ) ϕ ( x , y )
f ( A ( x , y ) ) = 1 sinc 1 ( A ( x , y ) )
t g ( x , y ) = exp [ i 2 π ( u x + v y ) ]
Φ = mod { [ 1 sinc 1 ( A ( x , y ) ) ] [ ϕ ( x , y ) + 2 π ( u x + v y ) ] }
C = m n ( A m n A ¯ ) ( B m n B ¯ ) ( m n ( A m n A ¯ ) 2 ) ( m n ( B m n B ¯ ) 2 )

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