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 and totally. 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
The phase vortex of a light wave field has a helical phase variation of an integer multiple of 2π around the vortex core . 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 , the orbital angular momentum of each photon of the light wave is [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 . With these properties, optical phase vortices have found applications in many fields such as free-space communications , 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 , diffraction by fork holograms [13,14], phase modulation with spiral phase plates  and wave-front division by pinhole screens . Essentially, the phase vortices are generated by adjusting the optical path of the incident wave front , 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 [22–24]. Then, L. Shi et al. theoretically improved the method by distributing the pinholes in varied azimuthal angle increments with constant radial distance . And F. Ricci et al. have analyzed the split of high-order vortex beams passing through a pinhole screen . 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 . 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 TC may be obtained .
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 [22–27], 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 is azimuthally symmetric as . 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 , with m the expected TC and the wavelength of the incident plane wave. This may be realized by introducing a constant difference between the in-plane radial distance . In the far Fresnel diffraction field, is sufficiently small comparing with , then the optical path difference is approximately expressed as 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 among the transmitting waves at the observation plane.
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 asEq. (1) is approximated to beEquation (2) is equivalent to the Fourier transform of the asymmetrical pinhole screen with the phase increment of pre-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 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 of, 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 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 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.
Further simulations show that the limitation of the TC of the generated vortex is for an even-N-pinhole screen and is for an odd one. For the cases, the optical path differences between the pinholes are . As the term obviously contributes nothing to the phase variation, the light distributions are the same with the cases and the TCs are of opposite sign to the 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 for. Yet, the patterns in Fig. 3(a) may also be seen as the case for m = 4. For odd-N-pinhole screens, the cases are extraordinary. Although optical path differences are introduced between the pinholes, the phase difference between neighboring transmitting waves is constantly 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.
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 asFigure 4(a) shows the result for , 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 with 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 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.
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 ) 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 , and the one in Fig. 5(b) about . 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.
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 20, and lens L2 with focal length 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 8 × 8, 1004 × 1002 pixels) that is placed about 1m away. We adopt another combination of the spatial filter SF1 and the convex lens L1 () 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.
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.
The intensity and the phase distributions of the transmitting waves are extracted from corresponding interferograms following the phase reconstruction method . For the interferogram formed by the transmitting light wave and the reference wave, the intensity is expressed byEq. (4) respectively. is obtained following a translation of in the spectrum coordinate. Performing the inverse Fourier transforms on , we then obtain both the transmitting light intensity and its phase distribution 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, 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 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 accurately. 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.
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 . 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.
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.
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