Abstract

We propose a novel and simple method for generating optical vortex with high topological charge (TC), merely using an asymmetrical pinhole plate illuminated by plane wave. N pinholes are arranged along a particular spiral line around the plate origin, with constant azimuth angle increment and varied radial distances. The radial differences introduce a constant variation of m/N wavelength to the optical paths from the N pinholes to the observation plane origin, and this increases the phases of the transmitting waves by progressively 2mπ/Nand totally2mπ. We numerically calculate the transmitted light field according to the Fresnel diffraction theory, and find the vortex with TC m around the observation plane origin. The experimental verifications are performed using some self-made asymmetrical pinhole plates fabricated by a femtosecond laser, with the high TC vortices both generated and detected in a Mach-Zehnder type interferometer. The experimental results coincide with the theoretical simulations well.

©2013 Optical Society of America

1. Introduction

The phase vortex of a light wave field has a helical phase variation of an integer multiple of 2π around the vortex core [1]. The core is called singular point due to its undefined phase and zero intensity. For a phase vortex of topological charge (TC) m with axially symmetric phase distribution exp(imθ), the orbital angular momentum of each photon of the light wave is m [2, 3]. Vortex light wave can trap or guide particles at its dark beam center [4,5], or drive particles to spin with the angular momentum transferred [6]. With these properties, optical phase vortices have found applications in many fields such as free-space communications [7], micromanipulations [8,9] and quantum optical information processing [10,11]. In recent years, the generation of optical vortices has attracted great interest and various methods have been proposed. The most common ones include the optical Fourier transform using spatial light modulators [12], diffraction by fork holograms [13,14], phase modulation with spiral phase plates [15] and wave-front division by pinhole screens [16]. Essentially, the phase vortices are generated by adjusting the optical path of the incident wave front [17], or by introducing optical path differences among the interfering multiple light beams that form the phase vortex [18,19]. It has been proven that pinhole screen interferometer is a convenient and feasible way for the generation and detection of optical vortices [20,21]. Berkhout et al. proposed a method for reading the TCs of Laguerre-Gaussian beams from the interference intensity patterns using symmetrical pinhole screens [2224]. Then, L. Shi et al. theoretically improved the method by distributing the pinholes in varied azimuthal angle increments with constant radial distance [25]. And F. Ricci et al. have analyzed the split of high-order vortex beams passing through a pinhole screen [26]. Up to now, the work in literature has made use of the azimuthal phase increase in a incident vortex beams in match of the azimuthal distribution of the pinholes, to obtain initial phase differences in the transmitted waves; or has engineered the required phase difference between each pinhole using optical light modulator [27]. But in the case of plane wave illumination, the generations of vortices with high TC are difficult because the initial phase differences cannot be conveniently achieved only through the azimuthal angle change of the pinholes, though in some cases vortex beams with ±1 TC may be obtained [21].

Basing on the quadratic relationship between the optical path (phase) differences and the in-plane positions of the pinholes on a screen, we propose a method for generating vortices with high TCs under plane wave illumination. Essentially different from previous papers [2227], the phase modulations are realized by adjusting the radial distances of the pinholes with constant azimuthal increment, where a spiral-curve distribution of the pinholes on the screen is needed due to the quadratic relationship. The shape of the spiral determines the total phase variation among the transmitting waves and then the TC of the vortices to be generated. The spiral pinhole arrangement is analogous with the previous vortex beam illumination schemes in obtaining the azimuthal phase increase. We numerically calculate the light fields behind these screens according to the Fresnel diffraction theory, and then validate the method by adopting self-made asymmetrical pinhole plates. The experimental results coincide with the theoretical simulations and high TC vortices are observed.

2. Theory and numerical simulations

As shown in Fig. 1, an asymmetrical pinhole screen lies in the x-y plane, and z is the distance to the observation plane X-Y. The N pinholes are distributed along a spiral curve with its center at the origin of plane x-y, and the azimuthal angle of the nth order pinhole Pn is azimuthally symmetric as αn=2nπ/N. For introducing a constant phase difference between the transmitting waves from neighboring pinholes, we need an increment between their optical paths to the observation plane origin lnln1=mλ/N, with m the expected TC and λ the wavelength of the incident plane wave. This may be realized by introducing a constant difference between the in-plane radial distance dn. In the far Fresnel diffraction field, dn is sufficiently small comparing with ln, then the optical path difference is approximately expressed as dn2/2zdn12/2zmλ/N in the first two terms of the Taylor expansion. This quadratic relationship determines a unique spiral distribution of the pinholes for providing a total phase difference of 2mπamong the transmitting waves at the observation plane.

 figure: Fig. 1

Fig. 1 Scheme of an asymmetrical pinhole screen. (a). Spiral distributed (solid red line) pinholes on a asymmetrical screen. The azimuth angle of the nth pinhole is uniformlyαn=2nπ/N, and the starting pinhole P0 with α0=0is also equivalent toαN=2π. The radial distance dn to the screen origin increases (or decreases) with n non-uniformly. (b) shows the phase variation of transmitting waves via the propagation, with the lines in red, yellow, green and blue respectively represent four different phase change of the incident plane wave. The observation plane is z away from the screen, with ln being the distance between its origin and the nth pinholePn.

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Neglecting the size of the isolated pinholes, the complex light field on the observation plane in the Fresnel diffraction theory can be expressed in the series form as

U(X,Y)=n=0N1exp{iπ[(Xxn)2+(Yyn)2]/λz}=n=0N1exp[iπ(X2+Y2)/λz]exp[i2π(Xxn+Yyn)/λz]exp[iπ(xn2+yn2)/λz],
where (xn,yn)is the Cartesian coordinate of thenthpinhole on the screen, the spherical term exp{iπ(X2+Y2)/λz} may be neglected in the small area under consideration, and it is dn2=xn2+yn2 that introduces the optical path differences. Thus Eq. (1) is approximated to be
U(X,Y)=n=0N1exp[i2π(Xxn+Yyn)/λz]exp(i2nmπ/N),
where the factor exp(i2nmπ/N)is exp(imαn) that represents the constant phase increment between neighboring pinholes. Equation (2) is equivalent to the Fourier transform of the asymmetrical pinhole screen with the phase increment of 2mπ/Npre-given somehow.

Figures 2(a)-2(f) show the calculated intensity and the phase patterns of light fields on the observation plane produced by even-N pinhole screens, where the normalized intensity patterns are given in gray scale maps and the phase patterns in the range of (π,π)are given in 8-interval colored maps. In the case of m = 0, the radial difference among the pinholes disappears, and this introduces no optical path differences between the transmitting waves. Equation (2) converts to the standard Fraunhofer diffraction of a symmetrical pinhole screen, as can be seen in Fig. 2(a) for N = 6, m = 0. For N = 6, m = 1 as given in Fig. 2(b), the phase around the origin has a total helical variation of2π, indicating the generation of a vortex with unity TC within the bright ring in the corresponding intensity map. This kind of vortex may also be produced with symmetrical pinhole plates of odd pinhole numbers [18,20,29]. But to our knowledge, optical vortices with higher TC cannot be produced with a pinhole system under plane wave illumination. Now the in-plane adjustment of pinholes makes this possible. The patterns in Fig. 2(c) and Fig. 2(d) for the screens with N = 6, m = 2 and N = 16, m = 2, respectively, indicates the generations of vortices with TC 2, although the intensity does not focus on a circle ring but on the vertices of a hexagon in the 6-pinhole case. Generations of higher order TCs are shown in Fig. 2(e) and Fig. 2(f) as examples, where the TCs are possessed by relatively larger bright intensity rings. It might be interesting to understand how the interference of waves from an odd pinhole number that generates the vortex of TC ±1 may affect the generations of vortices in our spiral pinhole arrangement. This may be well illustrated by given patterns in Figs. 2(g) for N = 5, m = 0 and (h) N = 5, m = 2, respectively. We see that no vortex appears at the center of the observation plane in the case of m = 0 with symmetrical pinholes, while the vortex array with TC ±1 appears on a circle around the center. In the case of N = 5, m = 2 with spiral pinholes, the vortex of higher TC appear at the origin of the observation plane as designed. This means that odd number of pinholes do not affect the generation of the expected vortex at the center of the observation plane.

 figure: Fig. 2

Fig. 2 Simulated distributions of the intensities and the phases of the light fields on the observation plane with the pinhole size neglected. For all calculations, the propagation distances are 10cm, the radial distances of the original pinholes are 1mm, and the observation areas are all 140μm × 140μm.

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Further simulations show that the limitation of the TC of the generated vortex isN/21 for an even-N-pinhole screen and is (N+1)/2 for an odd one. For the m'=Nm cases, the optical path differences between the pinholes are n(Nm)λ/N =nλnmλ/N. As the term nλ obviously contributes nothing to the phase variation, the light distributions are the same with the m'=m cases and the TCs are of opposite sign to the m'=m cases. This is seen in the patterns given in Fig. 3(a) for N = 6 and m = −2, in comparison with Fig. 2(c). Similarly, we can easily infer that the phase pattern repeats form'=N+m. Yet, the patterns in Fig. 3(a) may also be seen as the case for m = 4. For odd-N-pinhole screens, the m=N/2 cases are extraordinary. Although optical path differences are introduced between the pinholes, the phase difference between neighboring transmitting waves is constantly π and no traditional vortex appears around the origin, as can be seen in Fig. 3(b). Thus, a relatively enough pinhole number is needed for the generation of high TC vortices.

 figure: Fig. 3

Fig. 3 Simulated intensity and phase distributions on the observation plane. (a) shows the vortex with opposite sign to the one for N = 6, m = 2, with similar intensity pattern. (b) shows that no traditional phase vortex is found in N = 2m case.

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In practice, the size of the pinhole is a factor that influences the light distributions. Obviously, larger pinhole size is better for sufficient intensity transmission. But in the spiral system, pinholes of larger diameter produce smaller Airy disks, and this may lead to smaller overlap areas of them with the perfectness of the superposed intensity pattern being damaged. This is detrimental to formations of both the vortex charge and the intensity pattern. Considering the two contradictory effects of pinhole sizes, we should adopt an appropriate pinhole size to generate optical vortices. In simulation, Eq. (1) is then given as

U(X,Y)=U0(x,y)exp{i2π[(Xx)2+(Yy)2]/λz}dxdy,
where U0(x,y)is the amplitude transmittance function of the pinhole screen. Figure 4(a) shows the result for r=10μm, z = 10cm, and we see that both a regular vortex structure and a satisfactory intensity pattern is obtained. This is unavailable if the pinhole radius becomes 50μmwith the other parameters unchanged, for that the Airy disks are too small for their interference with each other to form good patterns. However, when the propagation distance increases, the Airy disks expand and better patterns are formed, as can seen in Fig. 4(b) for z = 1m. For a given pinhole screen for generation of high-order vortex, the total radial difference between pinholes is determined. Then different points on the pinhole itself may bring about phase difference for the transmitted wave due to the radial difference and too large pinholes will influence the formation of the vortices. This requires that the size of the pinholes should be moderated. Figure 3(d) gives the example patterns with r=200μm and z = 1m. We see a slight split of the high-order vortex in comparison with Fig. 3(c), and neither the satisfactory intensity nor a regular high-order vortex is obtained. Fortunately, the requirement for sufficient light power can be fulfilled by adopting screens with larger pinhole number.

 figure: Fig. 4

Fig. 4 Simulated light fields with pinhole size considered in. For all, the original radial distance is 1mm, pinhole number is 16, and m is 2. The calculated areas are 140μm × 140μmin (a), and 680μm × 680μmin the other three cases. The obvious deformation in (d) indicates a case with a too big pinhole diameter.

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3. Experiments

In experiment, the asymmetrical pinholes are fabricated on the plates which were cut from Coca cans by a femtosecond laser (Spitfire, Spectral Physics, wavelength 800nm, 1 KHz, 4mJ/pulse). The plates are mounted on a computer-manipulated 3-D motorized stage (resolution of 1μm) that runs following the encoded pinhole positions and the incident femtosecond laser is focused on the plates. By adjusting the exposure time, the defocusing distance, and the working laser power, we successfully get a good control of the pinhole sizes. Figure 5(a) and Fig. 5(b) show respectively the images of two examples of the pinhole plates we have fabricated with pinhole numbers N = 72 and N = 16. The plate in Fig. 5(a) has a pinhole diameter about 52μm, and the one in Fig. 5(b) about 36μm. The radial distance of the original pinhole for each example plate is 1mm and the expected distance between the pinhole plates and the observation plane is 1m.

 figure: Fig. 5

Fig. 5 Asymmetrical pinhole plates punched on aluminium plates. (a) 72-pinhole plate for m = 2, with pinhole radius approximately 26μm. (b) Plate for N = 16, m = 3 with pinhole radius r18μm. The radial distances of the original pinholes in both are all 1mm, and the observation planes are set to be 1m away.

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Using these pinhole plates, we generate the vortices and detect their phase distributions with a Mach-Zehnder type interferometer shown in Fig. 6. The vertically polarized input beam from a He-Ne laser with the wavelength 632.8nm is split into the reference and the object beams with approximately equal intensity by the beam splitter BS1. We first filter and expand the object wave using both spatial filter SF2, which consists of a microscope objective and a pinhole with radius 20μm, and lens L2 with focal length f2=5cm to appropriately enlarge the beam diameter. Then the asymmetrical pinhole plate is illuminated with this on-axis, normally incident plane wave. The pinhole plate is mounted on a 3D translation stage for convenient adjustment of its distance between the CCD (Cascade 1K, pixel size 8μm × 8μm, 1004 × 1002 pixels) that is placed about 1m away. We adopt another combination of the spatial filter SF1 and the convex lens L1 (f1=24cm) to expand and filter the reference beam, before the two wave parts arrive and combine at the beam splitter BS2. To control the light intensity received by the CCD, we plant a series of neutral attenuators A1 before BS1. Another series of neutral attenuators A2 is utilized to adjust the reference wave for an appropriate intensity compared with the object wave. Both the transmitting light field from the asymmetrical pinhole plate and its interference field with the reference wave are recorded by the CCD.

 figure: Fig. 6

Fig. 6 Schematic diagram of the experimental setup for generating and detecting phase vortices. M1-3 are mirrors, BS1 and BS2 are beam splitters. The spatial filters SF1 and SF2 are used to filter and expand the reference and the object beams respectively, and the lens L1 and L2 are used to convert the two beams to plane wave. A1 and A2 are neutral attenuators that adjust the light powers; Pin-P is the asymmetrical pinhole plate which is mounted on a 3D transmission stage.

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4. Experimental results

The first column in Fig. 7 shows the interferograms of the transmitting waves and the reference waves. The interferogram of two plane waves, as well known, is straight interference fringes with equal distances, and a single fringe splits into two/three at the point where the plane reference wave encounters a phase vortex with unity/two TCs. Thus, the interferogram can be seen as an indicator of the existence of phase singularities. In Fig. 7(a1), the straight fringe passing through the origin develops into two separated fringes within the red circle line, and this indicates the generation of a phase vortex with unity TC. Similarly, the splitting fringes of the interferograms within the circle lines in Figs. 7(b1) and 7(c1) reveal the phase vortices with TC 2 and 3 around the origin.

 figure: Fig. 7

Fig. 7 The Interferograms, the extracted intensity patterns and phase patterns on the observation plane that is 1m behind the asymmetrical pinhole plates. (a). N = 16, m = 1. (b). N = 72, m = 2. (c). N = 16, m = 3.

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The intensity and the phase distributions of the transmitting waves are extracted from corresponding interferograms following the phase reconstruction method [28]. For the interferogram formed by the transmitting light wave U(X,Y)and the reference waver(X,Y), the intensity is expressed by

I(X,Y)=|U(X,Y)+r(X,Y)|2=|U(X,Y)|2+|r(X,Y)|2+U(X,Y)r*(X,Y)+U*(X,Y)r(X,Y),
where ‘|...|2’ and ‘ * ’ represent modulus squared and complex conjugate respectively. As the reference wave is written asexp[i2π(f0XX+f0YY)]with spatial frequencies f0X and f0Y, the spatial spectrum of the interferogram after the Fourier transform of I(X,Y) is given by
If(fX,fY)=Bf(fX,fY)+Uf(fX+f0X,fY+f0Y)+Uf*(fXf0X,fYf0Y),
where Bf(fX,fY), Uf(fX+f0X,fY+f0Y), Uf*(fXf0X,fYf0Y) are the Fourier transforms of the first two real term, the third term and the last term in Eq. (4) respectively. Uf(fX,fY) is obtained following a translation of Uf(fX+f0X,fY+f0Y) in the spectrum coordinate. Performing the inverse Fourier transforms on Uf(fX,fY), we then obtain both the transmitting light intensity I(X,Y)=|U(X,Y)|2and its phase distribution φ(X,Y)=arctg[ImU(X,Y)/ReU(X,Y)] on the observation plane, as shown in the second and the third columns of Fig. 7 respectively.

It is obviously seen that dark intensity points are surrounded by quasi-circle bright rings in each intensity maps. With the increasing of TC m, the low-intensity areas expand from the origin. In fact, the bright ring is approximately the first maximum of the Bessel function of the first kind of order|m|, and it possesses most of the light intensity of the transmitting field. For clear indication, the normalized intensity distribution in Figs. 7(a2)-7(c2) along the vertical line passing through the vortex cores are presented in Fig. 8(a). We see that despite of the general accordance with the Bessel function, the intensity distributes in a somewhat non-centrosymmetric way, which is different from the standard kind of vortex beams. This deviation is due to the asymmetrical distribution of the pinholes that induces a spiral arrangement of the Airy disks on the observation plane. However, the levels of the abnormity are not such great and the intensity patterns are acceptable in applications, as the difference between the higher maximum and the lower one of the intensity curve is relatively small compared with the maximums. The curves in Fig. 8(b) represent the phase variations along a 0.2mm-radius circle around the origin of each phase map in Fig. 7. Fitted lines verify the generation of high TC vortices, for the phase increase or decrease linearly with the increasing azimuth angle and that the total variation is accurately2mπ. As is widely recognized and observed, high-order vortices are instable and they may split into single-charge vortices owing to any perturbation in experiments [19,26], i.e., slight off-axis and turbulence. And this is why serious high-order vortices are not directly observed in our paper. However, the total phase variation maintains, as demonstrated in the given results.

 figure: Fig. 8

Fig. 8 (a) Light intensities along the vertical line passing through the vortex cores. (b) Phase distributions around the origins. The curves for Figs. 7(a3-c3) are depicted by circle rings, squares and triangles represent respectively, and the linear-fits are shown by colored lines. Note that the phase regions are not(π,π), for clear indication of the total phase variation.

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5. Conclusions and discussions

In conclusion, we propose a novel method for generating high-TC vortex with asymmetrical pinhole screens under plane wave illumination. Basing on the quadratic relationship between the optical path difference and the in-plane positions of the pinholes on the screen, we distribute the pinholes along a spiral line with varied radial distances and constant azimuth angle increment to introduce phase variations to the transmitting waves. The light fields on the observation plane are numerically calculated in the Fresnel diffraction theory. Experimentally, asymmetrical pinhole plates with various expected TCs are self-made with a femtosecond laser, and then illuminated by plane waves. A reference wave is introduced for interferogram from which both the light intensity and the phase distributions are obtained. The experimental results are coincident with the theoretical predications, with satisfying intensity and expected TCs.

Comparing with previous works, the essential improvement in this paper is that the phase differences of the transmitting waves from the pinholes are not pre-given by incident vortex beam, but are introduced by the radial differences of the pinholes on the plates. This realizes the far-field phase variation by the adjustment of in-plane radial distances of pinholes.

It may be also interesting that the spiral pinhole arrangements is probed in generating fractional-m vortices, as the vortices of this kind has also been paid much attention on [30,31]. Our further simulations show that the results are similar with the ones reported in [17]. For the spiral pinhole arrangement with expected fractional-m TC, the generated vortex is approximately equal to the one with TC of the nearest integer to m, although the vortex split into several single-charge vortices with the same sign and the intensity pattern is deformed. When m is a half-integer, an additional single-charge vortex appears around the initial phase singularities.

Acknowledgments

National Natural Science Foundation of China (Grant No. 10974122) and Science and Technology Development Program of Shandong Province, China (Grant Nos. 2009GG 10001005) are gratefully acknowledged.

References and links

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2. 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]   [PubMed]  

3. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef]   [PubMed]  

4. K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index micro particles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999). [CrossRef]  

5. 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]  

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef]   [PubMed]  

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12. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]   [PubMed]  

13. A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

14. A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

15. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005). [CrossRef]   [PubMed]  

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17. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

18. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). [CrossRef]   [PubMed]  

19. R. K. Tyson, M. Scipioni, and J. Viegas, “Generation of an optical vortex with a segmented deformable mirror,” Appl. Opt. 47(33), 6300–6306 (2008). [CrossRef]   [PubMed]  

20. G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011). [CrossRef]  

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22. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008). [CrossRef]   [PubMed]  

23. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009). [CrossRef]  

24. G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010). [CrossRef]   [PubMed]  

25. L. Shi, L. H. Tian, and X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012). [CrossRef]  

26. F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012). [CrossRef]   [PubMed]  

27. J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51(12), 1872–1878 (2012). [CrossRef]   [PubMed]  

28. X. Y. Chen, Z. H. Li, H. X. Li, M. N. Zhang, and C. F. Cheng, “Experimental study on the existence and properties of speckle phase vortices in the diffraction region near random surfaces,” Opt. Express 20(16), 17833–17842 (2012). [CrossRef]   [PubMed]  

29. S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Appl. Opt. 46(32), 7862–7867 (2007). [CrossRef]   [PubMed]  

30. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011). [CrossRef]   [PubMed]  

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References

  • View by:

  1. P. Vaity and R. P. Singh, “Topological charge dependent propagation of optical vortices under quadratic phase transformation,” Opt. Lett. 37(8), 1301–1303 (2012).
    [Crossref] [PubMed]
  2. 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] [PubMed]
  3. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
    [Crossref] [PubMed]
  4. K. T. Gahagan and G. A. Swartzlander., “Simultaneous trapping of low-index and high-index micro particles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999).
    [Crossref]
  5. 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]
  6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
    [Crossref] [PubMed]
  7. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
    [Crossref]
  8. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29(15), 1796–1798 (2004).
    [Crossref] [PubMed]
  9. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
    [Crossref]
  10. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
    [Crossref] [PubMed]
  11. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
    [Crossref] [PubMed]
  12. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [Crossref] [PubMed]
  13. A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).
  14. A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).
  15. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005).
    [Crossref] [PubMed]
  16. M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
    [Crossref] [PubMed]
  17. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).
  18. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
    [Crossref] [PubMed]
  19. R. K. Tyson, M. Scipioni, and J. Viegas, “Generation of an optical vortex with a segmented deformable mirror,” Appl. Opt. 47(33), 6300–6306 (2008).
    [Crossref] [PubMed]
  20. G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
    [Crossref]
  21. G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066613 (2007).
    [Crossref] [PubMed]
  22. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
    [Crossref] [PubMed]
  23. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
    [Crossref]
  24. G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010).
    [Crossref] [PubMed]
  25. L. Shi, L. H. Tian, and X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012).
    [Crossref]
  26. F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012).
    [Crossref] [PubMed]
  27. J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Appl. Opt. 51(12), 1872–1878 (2012).
    [Crossref] [PubMed]
  28. X. Y. Chen, Z. H. Li, H. X. Li, M. N. Zhang, and C. F. Cheng, “Experimental study on the existence and properties of speckle phase vortices in the diffraction region near random surfaces,” Opt. Express 20(16), 17833–17842 (2012).
    [Crossref] [PubMed]
  29. S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Appl. Opt. 46(32), 7862–7867 (2007).
    [Crossref] [PubMed]
  30. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011).
    [Crossref] [PubMed]
  31. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
    [Crossref]

2012 (6)

2011 (2)

G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
[Crossref]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011).
[Crossref] [PubMed]

2010 (3)

G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010).
[Crossref] [PubMed]

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
[Crossref] [PubMed]

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

2009 (1)

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[Crossref]

2008 (3)

2007 (3)

2005 (1)

2004 (4)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
[Crossref] [PubMed]

W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29(15), 1796–1798 (2004).
[Crossref] [PubMed]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

2003 (1)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

2002 (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

1999 (2)

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]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

1992 (2)

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] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

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] [PubMed]

Almazov, A. A.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Barnett, S. M.

Baumgartl, J.

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010).
[Crossref] [PubMed]

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[Crossref]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref] [PubMed]

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] [PubMed]

Bekshaev, A.

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

Bekshaev, A. Y.

A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18(13), 13836–13841 (2010).
[Crossref] [PubMed]

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[Crossref]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Chen, X. F.

Chen, X. Y.

Cheng, C. F.

Cheong, W. C.

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Dholakia, K.

Elfstrom, H.

Flossmann, F.

Franke-Arnold, S.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Gahagan, K. T.

Götte, J. B.

Grier, D. G.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
[Crossref] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref] [PubMed]

Hirano, T.

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]

Jennewein, T.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

Joseph, J.

Khonina, S. N.

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

Kotlyar, V. V.

Kuga, T.

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]

Ladavac, K.

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

Lee, W. M.

Li, H. X.

Li, Z. H.

Liu, Y. Y.

G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
[Crossref]

Löffler, W.

McDuff, R.

Mourka, A.

O’Holleran, K.

Orlinska, O.

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

Padgett, M. J.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
[Crossref] [PubMed]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Paganin, D. M.

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066613 (2007).
[Crossref] [PubMed]

Pan, J. W.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

Popov, A. Y.

A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

Preece, D.

Ricci, F.

Ruben, G.

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066613 (2007).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Sasada, H.

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]

Scipioni, M.

Senthilkumaran, P.

Shanor, C.

Shi, L.

Shimizu, Y.

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]

Shiokawa, N.

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]

Singh, R. P.

Smith, C. P.

Soifer, V. A.

Spreeuw, R. J. C.

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] [PubMed]

Sviridova, S. V.

A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

Swartzlander, G. A.

Tian, L. H.

Tonomura, A.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
[Crossref] [PubMed]

Torii, Y.

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]

Turunen, J.

Tyson, R. K.

Tyurin, A. V.

A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

Uchida, M.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
[Crossref] [PubMed]

Vaity, P.

van Exter, M. P.

Vasnetsov, M.

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

Vaziri, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

Viegas, J.

Vyas, S.

Wang, P.

G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
[Crossref]

Wei, G. X.

G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
[Crossref]

Weihs, G.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

White, A. G.

Woerdman, J. P.

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] [PubMed]

Wright, E. M.

Xavier, J.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

Yuan, X.-C.

Zeilinger, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

Zhang, M. N.

Appl. Opt. (4)

Chin. Opt. Lett. (1)

J. Opt. A, Pure Appl. Opt. (2)

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A, Pure Appl. Opt. 11(9), 094021 (2009).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

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

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

Nature (1)

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
[Crossref] [PubMed]

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(71), (2004).

Opt. Commun. (4)

G. X. Wei, P. Wang, and Y. Y. Liu, “Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array,” Opt. Commun. 284(12), 2720–2725 (2011).
[Crossref]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002).
[Crossref]

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a ‘fork’ hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010).

A. Y. Bekshaev, S. V. Sviridova, A. Y. Popov, and A. V. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012).

Opt. Express (6)

Opt. Lett. (3)

Phys. Rev. A (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] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066613 (2007).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref] [PubMed]

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003).
[Crossref] [PubMed]

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]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Scheme of an asymmetrical pinhole screen. (a). Spiral distributed (solid red line) pinholes on a asymmetrical screen. The azimuth angle of the nth pinhole is uniformly α n =2nπ/N , and the starting pinhole P0 with α 0 =0 is also equivalent to α N =2π . The radial distance d n to the screen origin increases (or decreases) with n non-uniformly. (b) shows the phase variation of transmitting waves via the propagation, with the lines in red, yellow, green and blue respectively represent four different phase change of the incident plane wave. The observation plane is z away from the screen, with l n being the distance between its origin and the nth pinhole P n .
Fig. 2
Fig. 2 Simulated distributions of the intensities and the phases of the light fields on the observation plane with the pinhole size neglected. For all calculations, the propagation distances are 10cm, the radial distances of the original pinholes are 1mm, and the observation areas are all 140 μm × 140 μm .
Fig. 3
Fig. 3 Simulated intensity and phase distributions on the observation plane. (a) shows the vortex with opposite sign to the one for N = 6, m = 2, with similar intensity pattern. (b) shows that no traditional phase vortex is found in N = 2m case.
Fig. 4
Fig. 4 Simulated light fields with pinhole size considered in. For all, the original radial distance is 1mm, pinhole number is 16, and m is 2. The calculated areas are 140 μm × 140 μm in (a), and 680 μm × 680 μm in the other three cases. The obvious deformation in (d) indicates a case with a too big pinhole diameter.
Fig. 5
Fig. 5 Asymmetrical pinhole plates punched on aluminium plates. (a) 72-pinhole plate for m = 2, with pinhole radius approximately 26μm . (b) Plate for N = 16, m = 3 with pinhole radius r18μm . The radial distances of the original pinholes in both are all 1mm, and the observation planes are set to be 1m away.
Fig. 6
Fig. 6 Schematic diagram of the experimental setup for generating and detecting phase vortices. M1-3 are mirrors, BS1 and BS2 are beam splitters. The spatial filters SF1 and SF2 are used to filter and expand the reference and the object beams respectively, and the lens L1 and L2 are used to convert the two beams to plane wave. A1 and A2 are neutral attenuators that adjust the light powers; Pin-P is the asymmetrical pinhole plate which is mounted on a 3D transmission stage.
Fig. 7
Fig. 7 The Interferograms, the extracted intensity patterns and phase patterns on the observation plane that is 1m behind the asymmetrical pinhole plates. (a). N = 16, m = 1. (b). N = 72, m = 2. (c). N = 16, m = 3.
Fig. 8
Fig. 8 (a) Light intensities along the vertical line passing through the vortex cores. (b) Phase distributions around the origins. The curves for Figs. 7(a3-c3) are depicted by circle rings, squares and triangles represent respectively, and the linear-fits are shown by colored lines. Note that the phase regions are not (π,π) , for clear indication of the total phase variation.

Equations (5)

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U(X,Y)= n=0 N1 exp{iπ[ (X x n ) 2 + (Y y n ) 2 ]/λz} = n=0 N1 exp [iπ( X 2 + Y 2 )/λz]exp[i2π(X x n +Y y n )/λz]exp[iπ( x n 2 + y n 2 )/λz],
U(X,Y)= n=0 N1 exp [i2π(X x n +Y y n )/λz]exp(i2nmπ/N),
U(X,Y)= U 0 (x,y)exp{i2π[ (Xx) 2 + (Yy) 2 ]/λz}dxdy,
I(X,Y)= | U(X,Y)+r(X,Y) | 2 = | U(X,Y) | 2 + | r(X,Y) | 2 +U(X,Y) r * (X,Y)+ U * (X,Y)r(X,Y),
I f ( f X , f Y )= B f ( f X , f Y )+ U f ( f X + f 0X , f Y + f 0Y )+ U f * ( f X f 0X , f Y f 0Y ),

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