Cosine-type apodized spiral zone plate to handle the topological charge of a vortex beam

By taking advantage of spiral petal-like zone plates, we have recently demonstrated that a spiral element’s geometric shape impacts its charge. Keeping it in mind, we aim to control the topological charge of a vortex using phase modification of a spiral zone plate without transforming its geometrical shape. This is achieved by apodizing a spiral zone plate having p spiral zones with a radial grating with spatial frequency m. Having implemented some gratings with different m, we realized that an optical vortex is produced whose topological charge and its handedness depend on p and m. Consequently, a constant phase replaces on-axis screw dislocation under the situation that p takes odd multiples of m. Studies are carried out for large and small pairs of (p, m) and various gaps between the couple.


Introduction
Beam carrying orbital angular momentum (OAM) known as optical vortices have a helical wavefront with a phase dependence on azimuth as exp(ip ) , where is the azimuthal angle, p refers to the topological charge, and the sign of p shows the handedness of the helical wavefront (Curtis and Grier 2003). Since introducing this unique beam, many techniques were presented and developed to produce it owing to its fascinating behavior. Some of These methods including cylindrical-lens mode converters (Courtial and Padgett 1999), computer-generated holograms (Sabatyan et al. 2016;Sabatyan and Fatehi 2018;Sabatyan 2019), spiral phase plates (SPPs) (Kotlyar 2005;Cheong et al. 2004;Grigorea and Craciun 2021), tube laser (Tian et al. 2021), metamaterials (Jin et al. 2017;He et al. 2021;Pan and Li 2021), and refractive wrinkled axicon (Sanchez-Padilla et al. 2016). Among them, spiral zone plate (SZP) plays a prominent part in generating optical vortices even when structured as photon sieves Sabatyan 2020, 2021).
As interest in using optical vortices increases, handling the topological charge may be of great significance. This has been discussed and considered, for example, using spin-orbit interaction of light passing through curved nanoslits (Brasselet 2013;Brasselet et al. 2013) and petal-like SZP (Golbandi and Sabatyan 2021). Regarding it, we have recently demonstrated a new element "azimuthal modulated Bessel SZP (AMBSZP)" which is created by apodizing a SZP by using a Bessel function. We see that it is clearly capable of achieving high performance in complex beam shaping and structuring Sabatyan 2018, 2019).
Based on the last method, we are about to investigate the possibility of controlling topological charge using apodizing SZP through azimuthal trigonometric functions. To this end, the azimuthal cosine function assumed as cos (m ) , in which and m denote azimuth angle and an integer number, respectively, is inserted into SZP structure, so the resultant element is called cosine spiral zone plate (CSZP). Having analyzed the diffractive behavior of the element, we show that CSZP is capable of generating an on-axis optical vortex carrying charge equals the difference between the number of SZP spiral arms p and an odd multiple of m, i.e. J = p − (2n + 1)m if the condition 2nm < p ≤ 2(n + 1)m is met where n is an integer number. Plus, the sign of J indicates the vortex handedness. Moreover, it is demonstrated that the on-axis phase singularity dies off as the ratio of p to m becomes an odd number i.e. p m = odd . Eventually, the theoretical predictions are verified by the corresponding experiments.

Theoretical discussions and computational result
Basically, the transmittance of a combination of an azimuthal Cosine function via a spiral zone plate (CSZP) may be readily described as Here, is the wavelength of the incident light, f is the focal length, and (r, ) is the polar coordinate. Moreover, m and p are positive integer numbers that refer to angular frequency and the number of SZP spiral arms, respectively. In addition, Bin(x) is defined as As stated earlier, apodizing an SZP by a cosine function, the SZP is turned into an azimuthal composite element called CSZP composed of 2m segments with phase difference concerning the neighbors (Taheri Balanoji and Sabatyan , 2019 . To shed light on the idea, some samples of CSZP are illustrated in Fig. 1, which are built as a combination of some SZPs carrying topological charge p = 6 through p = 10 and cosine function with m = 8 . It is worth mentioning to note that the created CSZPs have p − m number of split spiral arms around the center that denote the charge of the produced vortex employing CSZP, as we will see later in this section. As usual, the scalar diffraction theory of Fresnel-Kirchhoff in the polar form was considered in order to survey the focusing properties of CSZP. So, the amplitude of a diffracted plane wave at a distance z from CSZP may be computed as in which ( , ) and (r, ) are the polar coordinates of the observation and CSZP planes, respectively. Now, by substituting Eq. (1) into Eq. (3) and considering that the square phase expression in r 2 cancel each other out at the focal plane (z = f ) , One may acquire the following relation for the amplitude of a diffracted plane wave at a focal plane (z = f ) Concerning the Jacobi-Anger expansion (Weber and Arfken 2003) (3) Equation (4) is then simply transformed as following The second integral of Eq. (6), may be computed as Eventually, by taking advantage of J − (x) = (−1) J (x) , we derive the following relation It is obvious that the second term becomes zero at the origin ( = 0 ) when p = m , so the phase vorticity on the optical axes is fully died out, as we will demonstrate later. Besides, we may justify Eq. (8) as a superposition of two vortex beams carrying charges T 1 = p − m and T 2 = p + m (Sabatyan 2019;Huang et al. 2016). So, based on behavior of the Bessel function, we come to the result that the first term in Eq. (8) is more effective around the optical axis because of having smaller order, i.e., T 1 < T 2 . As a result, the first exponential function significantly impacts the produced vortex's focusing behavior and topological charge. As a further clarification, it is not difficult to show that Eq. (8) can be altered after a few mathematical simplifications as follows Given the amplitude transmittance, numerical computations are carried out through the FFT-based Fresnel-Kirchhoff integral (Sabatyan and Elahi 2013): ) is known as free space impulse response, and FT denote for Fourier transform.
Prior to discussing simulation and experimental results, we would like to point out that all samples considered for numerical computations and experimental verification have the same radii R = 4 mm and focal lengths f = 500 mm, respectively, illuminated by a plane wave with the wavelength of = 632.8 nm. They were printed using the high-resolution imagesetter "Phoenix 2250" with a 3600 dpi resolution. After being spatially filtered and collimated, the He-Ne laser light is incident normally on the printed samples. Afterward, a CCD camera captures diffraction patterns at the focal plane as shown schematically in Fig. 2. Furthermore, the tilt technique is used to measure the charge of the generated vortices Sabatyan 2020, 2021;Golbandi and Sabatyan 2021).
Our first step is to investigate how m and p affect the amplitude and phase of a focused beam. Regarding Eqs. (8) and (9), one can also obtain petal-like beams by coupling two oppositely helical beams of the same topological charge, namely, T 1 | p=0 = −m and T 2 | p=0 = +m . The focused intensity map, then, is obtained considering Eq. (10) for typical samples of CSZP having different m = 1 and 3 with identical p = 0 , respectively. Accordingly, the results are shown in the first and second rows of Fig. 3 for the pair of (p, m) =(0, 1) and (0, 3), denoting simulation and the corresponding experimental works. As we see, petal-like beams with the number of petals equal to |T 1 | p=0 + |T 2 | p=0 = | − m| + |m| = 2m are generated (Sabatyan and Rafighdoost 2017).
Aside from its ability to create petal-like beams, CSZP also exhibits a considerable ability to produce optical ring lattice beams. The generation of these beams occurs by superimposing two twisting helical beams of varying topological charges over each other (Sabatyan and Rafighdoost 2017). In our case, CSZP may generate such beams when T 1 < 0 or p < m , and ||T 1 | − |T 2 || > 1 are met. To prove this, some simulations were conducted using typical samples with p = 2 and various m = 4 and 6 shown in Fig. 3 for pairs (p, m) =(2, 4) and (2, 6). Taking Fig. 3 as an example, it is evident that the number of lattices is determined by |T 1 | + |T 2 | = |p − m| + |p + m| = −(p − m) + (p + m) = 2m.
The next step is devoted to the impact of m and p on the phase structure of the created on-axis generated optical vortex. To this end, the same calculations were carried out for some samples constructed with identical m = 2 and diversity of p under the condition That being the case, we find out that the handedness of the vortex and also its charge are tailorable by the period of cosine function (m) and is given by p − m regarding Fig. 4 for pair of (p, m) = (1, 2) Through (8, 2). Accordingly, when m = 2 with p = 1 , as shown in Fig. 4 for pair of (p, m) = (1, 2) the total topological charge of the on-axis generated optical vortex is J| (p=1,m=2) = p − m = −1 and its handedness is counterclockwise. Also, for m = 2 and p = 3 the total topological charge of the on-axis generated optical vortex is clockwise and equals J| (p=3,m=2) = p − m = +1 , as depicted in Fig. 4 for pair of (p, m) = (3, 2). In addition, as we predicted, the on-axis vortex disappears when p is the same as m, as illustrated in Fig. 4 for pair of (p, m) = (2, 2). Considering other intervals such as 2 ) and also 3 × ( 2 ) and carrying out the same simulations, it becomes clear that the charge of generated vortices takes integer value between 0 and 2, as shown in Fig. 4. Note that the intervals are specified by where n is a positive integer. So that a general relation to measure the value and sign of the charge is given as L = p − (2n + 1)m.
In the same way, some other samples of CSZP constructed by considerably larger values of p and m within the interval given by Eq. (11) are considered. Following the examination of the results shown in Fig. 5, we conclude that the rule stays the same, i.e., the produced vortices carry the charge with integral values between 0 and p − (2n + 1)m , which would eliminate the on-axis singularity if the ratio p m = (2n + 1) is fulfilled. Due to the replacement of spiral zones around the origin with straight ones, as clarified in Fig. 1. Despite this, we must remember that the distance between singularities grows when increases for identical J and a given value of p and m.
Up to now, p and m considered to build CSZP had a slight difference in value, now we will see if the significant difference follows the rule, too, i.e., the generated vortex has the charge determined through p − (2n + 1)m . For this reason, some samples of CSZP built by ps and ms are assumed and simulated to have a large difference in value. Concerning the results shown in Fig. 6, it is evident that the rule is satisfied, for example, the vortex generated by the sample with p = 9 and m = 5 carries charge 4 as depicted in Fig. 6.
As expressed before, the ratio of SZP arms p to the radial grating period m is crucial in dealing with singularity, such that if it is set to an odd number, the on-axis screw (11) 2nm < p ≤ 2(n + 1)m, Fig. 3 The focused diffraction patterns of typical samples of CSZP constructed for pair of (p, m) =(0, 1), (0, 3), (2, 4), and (2, 6). The first and second rows are the simulations and corresponding experimental results dislocation disappears, and a constant phase distribution replaces it. To make it more clear, the simulation results for typical samples of CSZP with a variety of p and m are chosen. So, the ratio p/m is an odd number as shown in Fig. 4 for the pair of (p, m) = (2, 2) and (4, 2), Fig. 5 for pair of (p, m) = (30, 30) and (90,30), and also Fig. 6 for pair of (p, m) = (15, 5). Figures demonstrate that an on-axis bright spot is observed rather than a dark region. In this case, no singularity is detected, because the phase distribution around the optical axis is constant.  (29, 30) , (30, 30) , (31, 30) , (32, 30) , (88, 30) , (89, 30) , (90, 30) , (91, 30) , and (92, 30) . The simulation part includes the focused intensity patterns of CSZP (first column), tilting CSZP (second column), and the corresponding phase map (third column). The experimental part includes the recorded focused diffraction patterns (first column) and their corresponding detected topological charges through tilting CSZP (second column) As we know, the Poynting may be computed by the following relation for a scalar field (Kumar and Viswanathan 2013) where Φ is phase and I is intensity of the field. Given Eqs. (10) and (12), the Poynting vector map, intensity and phase distribution of focused beams of some typical CSZP samples with different pair (p, m) = (5, 7), (9, 7), (7, 7) , and (21, 7) have been computed under plane wave illumination and illustrated in Fig. 7. As expected, the samples generates optical vortices carrying charge − 2, and + 2 as well as two beams of constant phase are formed around the origin, respectively as shown in Fig. 7a through d. According to figures, we see that poynting vectors oriented azimuthally for vortex beams, so their handedness is counterclockwise and clockwise depending on the sign of charges − 2 and + 2 as depicted in Fig. 7a and b, respectively. The two other beams, however, have radial Poynting vectors as indicated in Fig. 7c and d.

Conclusion
In summary, we introduced CSZP which is realized by imposing triangular azimuthal phase change on the spiral zone plate. It was demonstrated that the technique allows us to handle the topological charge of a vortex. We saw that CSZP produces two vortex beams carrying charges T 1 = p − m and T 2 = p + m . As a result, the following cases may be obtained while superposition of them: • Petal-like beams with a number of petals equal to 2m are generated, under the condition p = 0 or |T 1 | = −|T 2 |. • Optical ring lattice beams with a number of lattices equal to 2m are generated, when T 1 < 0 or p < m and ||T 1 | − |T 2 || > 1. • Generating vortices carrying a topological charge which depends on the difference between p and m, not just on p. • Producing on-axis optical vortices carrying charge equals J = p − (2n + 1)m if 2nm < p ≤ 2(n + 1)m for n = 0, 1, 2, … . • An on-axis screw dislocation of the phase is replaced by a constant phase when p takes an odd multiple of m. . The focused intensity patterns of CSZP (first column) and tilting CSZP (second column) shows at the right side for simulation (first row) and the experimental (second row). The corresponding phase maps are given at the top of each vector fields