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Abstract
In this paper, we designed a binary phase mask (PM) with specific phase modulation characteristic and analyzed the spatial spectrum of the beam passing through the PM. In the case where the difference of phase modulation between two lattices of the binary PM is not equal to π, we found the spatial spectrum has the central spot (direct current component, DC component) except for the central eight strong symmetrical spots and many outer weak symmetrical spots. Based on the multiple-beam interference, the propagation-invariant vortex with a square array can be realized by interference of the eight plane waves with the same wave vectors along the optical axis from the central eight symmetrical spots via the modulated phase values of the central eight symmetric spots. The vortex arrays have two kinds of vortex with an opposite topological charge of l=±1. The helix can be formed with a square array by the interference of the vortex array and the plane wave along the optical axis from the DC component. The helix with outstanding helical intensity distributions have two screw directions, which coincide with the phase distribution of the optical vortex with the square array. The energy efficiency of this method can reach more than 80%. The simulation results demonstrate the feasibility of this method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex beam have the typical orbital angular momentum per photon for its special helical phase distribution [1,2]. It can be applied in the fields of image processing [3–8], electron acceleration [9,10], optics communication, et. al. The vortex beam can be produced by mode selector in the cavity [11], mode conversion by cylindrical lens [12], spiral phase plate [13,14], liquid crystal spatial light modulator [15–18], et. al. The researchers have also proposed many methods for produce vortex arrays, such as, multiple-beam interference [19–24], helical phase spatial filtering based on typical 4f optical system [25], fractional Talbot effect [26], diffractive optical elements worked as the beam splitter [27–30], et al. The multiple-beam interference is much better than other methods for its high energy efficiency and propagation-invariant characteristic [31]. In addition, by interference the propagation-invariant vortex array with one plane beam along optical axis, it can be formed the helix array. The helix have the special spatial helical intensity distribution. The vortex array and helix array can be applied in the fields of material processing [32–35], micro-particle manipulation and sorting [36,37], et.al.
2. Theoretical analysis and simulation results
Based on the multiple-beams’ interference theory, here we proposed an efficient method to generate the vortex and helix with square array by use of one two-dimensional (2D) binary phase mask (PM). Figure 1 is the optical system diagram. L1 is the beam expander and L2 is the beam collimator. We can obtain the collimated wide beam via the laser passed through the beam expander and collimator in turn. L3 and L4 are the Fourier Lens with the same focal length of f. One binary PM was set on the front focal plane of L3 and we can obtain the spatial spectrum of the beam passing through the binary PM on the back focal plane of L3. One filter was set on the back focal plane of L3 and the front focal plane of L4. The filter can not only filter the spatial spectrum, but also modulate the phase distribution of the spatial spectrum. The charge coupled device (CCD) was set behind L4 and it was used to record the intensity pattern of the output optical field.
Figure 2 show the phase modulation characteristics of the binary PM and the spatial spectrum of the beam passing through the PM. The binary PM shown in Fig. 2(a) has the periodic distribution in four directions, which include the horizontal, vertical and the ± 45 degrees from horizontal to vertical directions [38]. This binary PM can be used for generate the vortex and helix with square array. The binary PM shown in Fig. 2(a) has two kinds of lattices with different gray levels, which represent the two different phase modulation values ±φ on the incident beam. Ideally, when one uniform plane wave with amplitude A passed through the binary PM, the optical field can be simply expressed as
Fig. 2. The phase modulation characteristics of the binary phase mask with φ=0.25π and the spatial spectrum of the beam passing through the phase mask.
Fig. 2(b) shows the spatial spectrum of the binary PM with φ=0.25π shown in Fig. 2(a). From Eq. (1), it is obviously that the amplitude of the DC component correspond to the central spot is Acosφ. In addition to the central spot, there are central eight strong symmetrical spots with the same intensity next to the central spot and many outer weak symmetrical spots in the spatial spectrum. It is evidently there is no central spot for φ=π/2.
By analysis, the components correspond to the central eight strong symmetrical spots shown in Fig. 2(b) have the same amplitude as
At first, we only permit the central eight symmetric spots passing through. Figure 3 shows the phase distribution of the eight spots which can be used to generate the vortex with square array. The phase values of these eight symmetric spots can be realized by modulate the spatial spectrum of the binary PM. L4 can transformed the central eight symmetric spots into eight symmetric plane waves with the same wave vectors along optical axis. For the case of Fig. 3, the complex amplitude of the interference pattern of the eight symmetric plane waves can be deduced as
Fig. 3. The phase distribution of the eight symmetric spots which can be used for generate the vortex with square array.
Figure 4 shows the simulation results of the interference pattern expressed by Eq. (3). Figure 4(a) is the normalized intensity distribution of the interference pattern and Fig. 4(b) is the phase distribution of the interference pattern. It is evident that it formed the optical vortex with square array. From Fig. 4(a) and (b), we can see the vortex arrays have two kinds vortex with the opposite topological charge of l=±1. The minimum unit of vortex array is 2×2 lattices.
Fig. 4. The simulation results of the interference pattern of the eight symmetric plane wave from the eight symmetric spots shown in Fig. 3.
Figure 5 shows the contour of the normalized intensity distribution of the single vortex shown in Fig. 4. The difference of values between the adjacent contours is 0.1. Figure 6 shows the phase difference between the single vortex and an ideal optical vortex with l = 1, which has the linear phase gradient along azimuthal direction. From the inner to outer of Fig. 6, the 1st solid line, 2nd solid line, 3rd solid line, …, nth solid line correspond to the 1°, 2°, 3°, …, n° (π/180, 2π/180, 3π/180, …, nπ/180) respectively. From Fig. 6, we can see that the single optical vortex is very close to an ideal optical vortex.
Fig. 6. The phase difference between the single vortex shown in Fig. 4 and an ideal optical vortex with l = 1.
Then we permit the central spot (DC component) passing through. L4 can transform the central spot into the plane wave along optical axis. The total complex amplitude of the interference pattern of the vortex array and DC component can be written as
Figure 7 shows the simulation results of the intensity distribution of the helix with the square array for different z. Here we set φ=0.224π, λ=632.8 nm, kr=0.01k, the size of the images is about 0.5mm×0.5 mm. Figure 7(a)-(f) correspond to z = 110 mm, 112 mm, 114 mm, 116 mm, 118 mm and 120 mm respectively. It is obvious that the helix have the outstanding helical intensity distributions along optical axis. Meanwhile, we found the helix have two type screw directions, rotate clockwise and counterclockwise. This result coincide with the above analysis that the vortex have two kinds vortex with the opposite topological charge of l=±1.
Fig. 7. The intensity distribution of the helix with square array at different z. (a) z = 110 mm, (b) z = 112 mm, (c) z = 114 mm, (d) z = 116 mm, (e) z = 118 mm, (f) z = 120 mm.
Regardless of the energy loss of the optical system, ideally, the energy efficiency of the optical system can be calculated as
3. Conclusion
We proposed a method to generate the vortex and helix with square array. It has the higher energy efficiency than 80%. We obtained the spatial spectrum of the PM with the central spot except the central eight symmetrical spots and many outer weak symmetrical spots by modulate the phase distribution of the binary PM. It can be formed the propagation-invariant vortex with square array by interference of the eight plane wave with the same wave vectors along optical axis from the central eight symmetrical spots with the specific phase distribution. The vortex arrays have two kinds of vortex with the opposite topological charge of l=±1. Then, it can be realized the helix with square array by interference the vortex array with the plane wave along optical axis from the DC component. The helix with the outstanding helical intensity distributions have two type screw directions, rotate clockwise and counterclockwise along the optical axis. The simulation results demonstrate the feasibility of this method. We can also expanded this method to the electron beam [39–43], extreme ultraviolet or acoustic wave [44–46]. This method can be widely used in many fields, such as material processing [32–35], micro-particle manipulation [36], optical sorting [37], telecommunications [47] and et. al.
Funding
National Natural Science Foundation of China (11504096); Natural Science Foundation of Shandong Province (ZR2017MA047); Doctoral Foundation of University of Jinan (XBS1407, XBS1611); School Scientific Research Foundation of University of Jinan (XKY1407, XKY1706).
Disclosures
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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