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The diffraction from a linear grating having a quadratic curvature on its lines is proposed for vortex beam characterization. Three types of transmission functions are considered for the gratings, including pure amplitude, pure phase, and hybrid amplitude and phase profiles. The first-order diffraction of the vortex beam through such gratings is only a set of elongated intensity spots. The number of spots determines the value of the topological charge (TC), and the sign of the TC can be distinguished with the elongation (and rotation) direction of the spots. This method is effective because over a given diffraction order, all of the energy of the beam transfers into the bright bar-like spots. Another advantage of the method is its ease of use, because it is not sensitive to the relative location of the beam axis and the grating center. Using a spatial light modulator, we provide different hybrid amplitude and phase linear gratings with a quadratic curvature on their lines and demonstrate the effectiveness of the proposed method experimentally. Also, the same justification was done in the diffraction of vortex beams from printed pure amplitude gratings. We have also investigated the effect of lateral shearing between the amplitude and phase parts of the hybrid grating on the resulted diffraction pattern. It is shown that for given values of lateral shear, one of the first-order diffraction patterns is eliminated, and the intensity of the other one is maximized. Finally, we present the key results of the diffraction of optical vortices from annular amplitude and phase gratings and from phase objects having linearly increasing phase functions along the radial direction. It is shown that the diffraction of optical vortices only from the parabolic-line linear gratings is insensitive to the off-axis value of the beam and grating centers.

We analytically, numerically, and experimentally determine a topological charge (TC) of the sum of two axisymmetric off-axis Laguerre-Gaussian (LG) beams with the indices (0, m) and (0, n). In particular, we find that at m = n, the combined beam has TC = n, which suggests that the sum of two identical off-axis LG beams has the TC of an individual constituent LG beam. At m < n, the TC of the sum is found to take one of the following four values: TC1 = (m + n)/2, TC2 = TC1 + 1, TC3 = TC1 + 1/2, and TC4 = TC1 – 1/2. We also establish rules for selecting one of the four feasible values of TC. For the sum of two on-axis LG beams, TC of the superposition equals the larger constituent TC, i.e. TC = n. Meanwhile following any infinitesimally small off-axis shift, TC of the sum either remains equal to the pre-shift TC or decreases by an even number. This can be explained by an even number of optical vortices (OV) with TC = –1 instantly ‘arriving’ from infinity that compensate for the same number of OV with TC = +1 born in the superposition. We also show that when two LG beams with different parity are swapped in the superposition, the topological charge of the superposition changes by 1. Interestingly, when superposing two off-axis LG beams tilted to the optical axis so that their superposition produces a structurally stable beam, an infinite number of screw dislocations with TC = +1 are arranged along a certain line, with the total TC of the superposition equal to infinity.

In a recent study, we have reported a simple, efficient, and robust method that is based on diffraction in an amplitude parabolic-line linear grating for determination of the topological charge (TC), l , of an optical vortex beam [J. Opt. Soc. Am. B 37, 2668 (2020)JOBPDE0740-322410.1364/JOSAB.398143]. Here, we present a demonstration of the application of that method for characterization of a pair of superposed vortex beams having different winding numbers. It is shown that, when two vortex beams, described by Laguerre–Gaussian beams with winding numbers l 1 and l 2 and radial index p = 0 , impinge on-axis and collinearly on a diffraction grating having a quadratic curvature on its lines, with a simple analysis of the resulted diffraction patterns at the zero and first order, the TCs and their signs can be determined. The zero-order diffraction pattern shows an interference pattern of the beams. For close values of l 1 and l 2 , it has a petal-like pattern in which the number of spots is equal to | l 1 − l 2 | . It is also found that the first-order diffraction pattern depending to the signs of the beams’ TCs shows two different forms. If l 1 and l 2 have the same signs, the first-order diffraction pattern is only a set of elongated intensity spots. When the signs of l 1 and l 2 are opposite, the resulted pattern is a ( l 1 + 1 ) by ( l 2 + 1 ) slanted checkered-like matrix of bright spots. In addition, in this work, we use a simple, novel, and initiative method to generate and combine on-axis and collinearly a pair of vortex beams. Finally, a supporting theoretical study is presented that fully confirms the experimental results and simulation of propagation.

In this work, we report the characterization of a Laguerre-Gaussian (LG) beam with given values of topological charge (TC) and radial index in a simple, efficient, and robust experimental diffraction scheme. The beam diffracts from an amplitude parabolic-line linear grating and the resulting diffraction patterns at zero- and first-order reveals the values of the TC, l, and radial index p of the incident LG beam using a simple analysis. The zero-order diffraction pattern consists of p + 1 concentric intensity rings and the first-order diffraction pattern contains an (l + p + 1) by (p + 1) two-dimensional array of intensity spots. The experimental scheme is robust since it is not sensitive to the relative locations of the impinging beam axis and the grating center, and is efficient since most of the energy of the output beam is in the diffraction order of interest for LG beam characterization. The measurement is also simple since the intensity spots of the array are placed exactly over straight and parallel lines. Both experimental and simulation results are presented and are consistent with each other.