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Abstract
Diffraction patterns of optical vortex beams (VBs) by differently shaped apertures are used to determine their topological charge (TC). In this paper, we show by simulations and experiments that diffraction of a Laguerre-Gaussian (LG) beam by intersecting circular apertures can be used to reveal the TC. The presented aperture structure has the advantage of the measurement of fractional TC in addition to the integer, sensitivity to the sign of TC, and low sensitivity to adjusting apertures. Accordingly, in addition to the integer TC up to 8, the fractional TC is measured with a step of 0.1 by two intersecting circular apertures (TICA). By examining a wide range of similarity criteria between the diffraction pattern of the fractional TC and the pattern of the lower integer TC, three metrics for measuring the fractional TC are found. Furthermore, the determination of integer TC up to 6 for three intersecting circular apertures (THICA) is demonstrated.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recent developments in optical vortex beams (VBs) carrying orbital angular momentum (OAM) are paving the way for a wide range of research fields and applications such as optical tweezers [1,2], optical communications [3–5], quantum entanglement [6,7], astronomy [8,9] and microscopy [10]. Such beams contain a phase factor exp(iℓθ), where ℓ stands for topological charge (TC) and θ for the azimuthal angle [11,12]. Various approaches have been developed to generate OAM beams, such as spiral phase plates (SPP) [13–15], spatial light modulators (SLMs) [16–18], and the holographic fork gratings [19]. In general, VBs can carry both integer and non-integer values of ℓ, the latter being referred to as fractional vortex beams [20]. In recent years, VB with fractional TC compared to integer TC has attracted more attention as they are used to increase the capacity and speed of information transmission in optical communications [21,22]. It is noticeable that the detection and specification of vortex beams is as important as their generation, especially for optical communication applications.
To date, only a limited number of methods have been presented for measuring fractional vortex beams. However, several methods have been presented for successful measurement of integer TC based on interference and diffraction. For example, the study of the interference pattern of a vortex beam incident on a single slit [23] and double slits [24,25] are used to determine TC. Interference of VB with a plane wave [26] and the use of a cylindrical lens [27] have also been investigated for measuring OAM using the interference principle. The diffraction methods also reveal unique properties by illuminating apertures with different geometries with VBs. Most diffraction methods for measuring TC are based on counting bright or dark spots. For example, by investigation of the diffraction pattern of VBs from a triangular aperture TC is measured up to ℓ = ±7, but the measurement accuracy is very sensitive to the position of the aperture [28,29]. Furthermore, the value of TC is demonstrated for TC up to ℓ = ±10 with an isosceles triangular aperture [30]. In addition, diffraction patterns of VBs from a single slit [31], a hexagonal aperture [32], square [33], pentagonal [34], and a sectorial screen [35] have also been proposed to determine TC. Among the various shapes of diffraction apertures, circular shapes are also investigated for integer TC measurement. For example, a circular aperture was used to measure the value and sign of TC of a VB up to ℓ = ±3 [36]. In addition, an annular elliptical aperture has been developed to directly measure TC up to 3 [37]. However, not all TC measurement techniques are suitable and accurate for measuring fractional vortex beams. Therefore, methods such as Mach-Zender interferometer [38,39], dynamic double slit [40,41], spiral interferometer [42], and fiber grating tip [43] are developed for TC measurement of this type of VBs. The accuracy of fractional TC measurement for the spiral interferometer and fiber grating tip methods is 0.05 and 0.01, respectively. However, these methods are based on complicated configurations or require an additional SLM, highly accurate and expensive instruments. Recently, diffraction-based methods have been proposed to identify fractional TC. For example, diffraction of VB from an annular grating was employed to measure fractional TC from 0.1 to 1 [44]. Recently, by investigation of the diffraction pattern of VB from a crossed blades with a certain opening angel, the fractional TC of the VB was measured with an accuracy of 0.1 [45].
In this study, we present a method for determining the integer and fractional TC of a vortex beam based on the study of the diffraction pattern of a VB from a two intersecting circular apertures (TICA) structure. We have shown that the integer TC can be easily determined by counting the dark fringes in the diffraction pattern. In addition, we have investigated several similarity criteria for the diffraction pattern of VBs with fractional TC with lower integer TC diffraction pattern. While the TC of the fractional VB deviates from the lower integer TC, some of the studied similarity criteria show a monotonic increase or decrease. Therefore, these similarity criteria can be interpreted as a metric for fractional TC. In addition, analysis of the diffraction pattern of VB from the three intersecting circular apertures (THICA) yields another metric for the integer TC measurement. The presented method has the advantage of simplicity, low cost, and ease of use in the lab.
2. Theoretical model
The theoretical approach to study the diffracted field of VB at distance z from the diffraction aperture can be followed by calculating the modified Fresnel integral in cylindrical coordinates as follows [36]:
Fig. 1. A schematic representation of the introduced aperture structures (a) TICA and (b) THICA and the corresponding geometrical and coordinate parameters.
To generate VB with the desired OAM, the SLM is loaded with a forked grating with a certain TC number. The transmission function of the forked grating can be expressed as follows [47]:
3. Experiment and simulations
For the implementation of TICA and THICA structures in the experimental setup, it is assumed that the radii of the circular apertures are the same, and in the case of THICA, the apertures are placed at the corners of an equilateral triangle with side length a. In the experiment, the apertures are close to each other along the propagation direction of VB. Therefore, we can assume to a good approximation that VB sees both apertures simultaneously. In other words, we ignore the beam evolution after diffraction at the first aperture. Figure 2 shows a schematic diagram of the experimental setup for studying the diffraction pattern of VB at TICA.
Fig. 2. Schematic diagram of the experimental setup for investigating the diffraction pattern of VB passing through TICA apertures.
In the experimental setup shown in Fig. 2, A digital micromirror device (DMD) (Texas Instruments XR-325) is used as the light modulator, which has a resolution of 600 × 800 with a pixel size of 14 µm × 14 µm mm. A He-Ne laser with a wavelength of 632.8 nm passes through a beam expander that uses lenses L1 and L2 with focal lengths of 75 mm and 750 mm, respectively, to cover the area of the DMD. Mirrors M1 and M2 are mounted in adjustable holders to guide and steer the expanded beam to the center of the DMD chip. To generate VB with the integer or fractional TC, the designed forked grating according Eq. (5) is displayed on the DMD. Different orders of LG modes are generated in different orders of this forked grating, so by using an iris diaphragm, the order containing the VB with TC of +ℓ is selected. The lens L3 with a focal length of 300 mm is placed at a distance of 20 cm behind the circular aperture to adjust the radius of VB so that the ring of LG modes coincides maximally with the edges of the apertures. Each of the circular apertures is mounted on a XYZ stage with a movement accuracy of 10 µm for precise adjustment of the a parameter. Therefore, the circular apertures can be moved independently in three directions to create TICA structure. As mentioned earlier, the distance between the apertures in z direction is negligible. The lens L4 with a focal length of 50 mm is placed 15 cm after the TICA to transfer the intensity profile of the diffracted beam to the CCD chip with suitable size. Finally, diffracted beam is captured by a CCD camera (DMK23U274 from The Imaging Source Industrial) with resolution of 1600 × 1200 and pixel size of 4 µm × 4 µm. Furthermore, an intensity filter is used in front of CCD to avoid saturating of recording profile intensities.
To obtain a metric for TC measurement, the distance between center of apertures in TICA, a is varied and the diffraction patterns of VB are investigated. The radius of both apertures is constant and equal with R1 = R2 = 550 µm. It is observed that immediately increasing the a from zero (concentric apertures), bright and dark fringes are appeared in the center of the diffraction pattern. However, the fringes visibility is maximum for a=500 µm in the experiment and a=550 µm in the simulation. The diffraction patterns by variation of the TC of the VB obtained in the experiment and the simulation for TICA structure is shown in Fig. 3.
Fig. 3. Simulation and experimental diffraction patterns of VB from TICA with variation of TC. The distance between the apertures is a=500 µm to achieve maximum visibility of the central fringes.
To perform simulations, the diffraction patterns of VB with different TCs of TICA are numerically calculated by substituting Eq. (3) as the aperture function in Eq. (1) and shown in Fig. 3 in simulation rows. As can be seen from Fig. 3, the number of dark fringes in the central part of the diffraction pattern is always equal to the value of TC. Furthermore, the experimental results are in agreement with the simulations. However, a closer look at Fig. 3 shows that in some cases the fringes of the experimental intensity profiles have a different orientation rather than simulation fringes. It is worth noting that there are some unavoidable experimental errors and restrictions that can cause the simulation and experimental results to differ. For example, in the simulation, the incident LG mode to the TICA is perfect and symmetrical and the opening is evenly spaced from the center of the LG mode. The opt-mechanics precision restricts aperture adjustment and the DMD resolution limits the generation of the perfect LG mode.
The measurement of TC is possible up to ℓ=8 by counting the dark fringes, while for a higher number TC counting the fringes by eye is not easy. As mentioned earlier, with one circular aperture, a maximum of three TCs have been measured previously [36]. Here with two circular apertures, the countable number of TCs is increased to 8. According to this increase by increasing the number of diffraction apertures, the case of three apertures THICA is implemented to investigate further increases. For both the simulation and the experiment, three apertures with equal radius of R1 = R2 = R3 = 550 µm were chosen, and the center of the apertures is located in the corners of an equilateral triangle with a side size of a=500 µm. The results of the experiment and simulation are shown in Fig. 4.
Fig. 4. The simulated and experimental intensity profiles of diffraction of VBs with different TC from the THICA structures.
As can be seen from Fig. 4, the number of TC can be measured by counting the dark fringes between tines of forked shape diffraction pattern. However, it is not possible to count the dark fringes with the eyes when the number of TC in the experimentally recorded patterns is more than 6. Moreover, when comparing simulation and experiment, a rotation of the diffraction pattern can be seen in the recorded intensities. The physical reason could be the asymmetry of the placement of the apertures on the corners of the equilateral triangle with the accuracy of 10 µm of the XYZ stage. The adjustment of the parameter a for TICA is simple, since the variation of a certain a is done by varying each X- vernier of the XYZ stage by a/2 (see Fig. 1 and Fig. 2). However, in the case of THICA, the sides of the triangle do not lie along the XYZ verniers, and the directions and magnitude of each vernier variation are obtained from the trigonometric calculations.
The interesting feature of the TC measurement with the TICA structure is the low sensitivity to the geometric parameters of the apertures. Unlike other aperture shapes for measuring TC, which are too sensitive to the positioning in the VB transverse profile and the symmetry of the aperture [28,29], the TICA structure is flexible and easy to set up. To show the sensitivity of TICA with respect to the distance between the centers of two apertures, the diffraction pattern of a VB with ℓ=3 is shown in Fig.5a.
It can be seen from Fig. 5(a) that TC can be determined at different values of a, and from a= 400 µm to a=600 µm dark fringes can be easily counted by eye. In other words, even with a 20% inaccuracy in the aperture distance setting, the dark fringes are still countable. Moreover, the diffraction of TICA structures is sensitive to the sign of TC. As can be seen in Fig. 5(b), the orientation of the diffraction pattern is rotated when the sign of VB is changed from + to −.
Fig. 5. (a) The diffraction pattern of VB from the TICA structure when the distance between two apertures, a is varied. (b) The effect of the sign of TC on the diffraction pattern in the TICA structure.
For the feasibility of using the diffraction from the TICA for measurement of the TC in the fractional VBs, the diffraction pattern of this structure for variation of the TC from 1 to 2 with 0.1 step is recorded and illustrated in Fig. 6.
Fig. 6. Simulation and experimental results of the diffraction pattern of VB with fractional TC from the TICA structure.
As shown in Fig. 6, the diffraction pattern changes gradually when TC is increased from 1 to 2. Comparing the diffraction patterns for ℓ=1 and ℓ=2, it is clear that a bright lobe has formed in the right part of the pattern of ℓ=1. Therefore, the extent of this bright lobe, which is increased by increasing the TC value, can be studied as a measure of the TC measurement. In other words, for a fractional VB, the similarity of the diffraction pattern with the lower integer TC decreases as the fractional TC increases. This approach can be applied to any fractional TC between two consecutive integers TC, i.e., ℓ and ℓ+1. Various approaches are presented in different areas of image and data processing to determine the similarity of two images and data series. In order to find a metric to measure the similarity of the fractional VB diffraction pattern with the lower integer TC, we have investigated several criteria such as: correlation coefficient (CC) [47], spectral angle mapper (SAM) [48], 2D cross correlation [49], fidelity [50], structural similarity index [51], multi-scale structural similarity index [52], root mean squared error (RMSE) [53], and peak signal-to-noise ratio [54]. However, suitable similarity criteria should show monotonic behavior with variation of fractional TC. Examination of the above criteria reveals that SAM, RMSE, and CC show metric behavior for measuring TC. The equations governing these criteria are expressed as follows:
The normalized criteria defined according to Eq. (9) are 1 when TC of VB is set to ℓ=1, and 0 for TC of ℓ=2. As shown in Fig. 7, the chosen criteria for measuring the diffraction pattern of the fractional TC show metric behavior. Moreover, the simulation and experimental results show a good agreement. However, none of the investigated criteria shows a linear relationship with the fractional TC. Accordingly, the slope of the plots in Fig. 7 shows the sensitivity of the fractional TC measurement, with maximum sensitivity for CC occurring at ℓ=1.5 and for RMSE and SAM near ℓ=1. This approach to fractional TC measurement can be developed for finer variations of TC of less than 0.1 by using a higher resolution DMD for VB generation. It should be noted that the fractional vortex beams are difficult to precisely modulate TC even with high-resolution DMDs. In other words, due to the resolution of the DMD (with a resolution of 600 × 800), there will be an error between the input TC and the TC of the generated VB. To ensure that the generated VB contains an assigned fractional TC, in this work the experiments are performed for 0.1 steps.
4. Conclusion
In this work, a new diffraction-based method for measuring integer and fractional TC is presented. It is shown that by studying the diffraction pattern of a VB from the TICA and THICA structures, the integer TC can be determined up to 8 and 6, respectively. By applying appropriate similarity criteria to the diffraction pattern of VB with fractional TC and the diffraction pattern of the lower integer TC, metrics for measuring TC are also found. After a detailed investigation through different similarity criteria, CC, RMSE and SAM are revealed a metric behavior when the fractional TC is varied. Moreover, the sensitivity of the relative position of the apertures is varied experimentally and it is shown that the dark fringes corresponding to the number TC are countable even when the circular apertures are set imprecisely.
Disclosures
The authors declare no conflicts of interest.
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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