A general analytical formula for the propagation of the new kind of power-exponent-phase vortex beam through a paraxial ABCD optical system is derived. With two different calculation methods, the evolution of the intensity distribution and phase contour of such a beam in free space is investigated. Some experiments are carried out to verify the theoretical predictions. Both of the theoretical and experimental results show that the beam’s profile can be modulated by the topological charge and the power order. In addition, the orbital angular momentum (OAM) density and the normalized OAM of such a beam are also studied.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Since the pioneering work of Nye and Berry  on phase singularities in optical fields revealed the existence of such interesting structures as phase dislocations and optical vortices, singular optics has been an active area of optical research. Optical vortex (OV) is characterized by sprial phase structure, hollow-core intensity distribution and intriguing orbital angular momentum (OAM) [2–4], which leads to a plenty of applications, such as optical trapping [5, 6], optical tweezers , quantum information processing, quantum cryptography [8, 9], and free-space optical communications .
Generally, a canonical optical vortex (COV) carries a sprial phase varying uniformly with azimuthal angle. The sprial phase is in the form of exp(ilθ), where l is an integer number and refers to the topological charge (azimuthal index) of the field, which is related to the OAM L of photons by the relationship L = lħ . Over the past decades, apart from the COV, several kinds of noncanonical optical vortices, such as nonsymetric vortex [11,12], fractional vortex [13–15], Mathieu vortex , modulated vortex , power-exponent-phase (PEP) vortex [18,19], remainder-phase optical vortex  and so on, have also been researched in order to explore the properties and applications of OAM.
PEP vortex (PEPV) is a kind of noncanonical vortex that characterized with power-exponent-phase [18, 19]. It has been illustrated that the PEPV beam carries OAM, which is determined by the topological charge . The PEPV beam has an asymetric intensity distribution and the intensity pattern would change during propagation , so that it may be useful in particle trapping and releasing . In this paper, we propose a new kind of noncanonical vortex, the phase of it in the source plane is characterized with new kind of PEPV, so we could call it new PEPV (NPEPV). The simplest and most popular way to generate the NPEPV beam is to employ a spatial light modulator (SLM) because SLM has the advantage of providing dynamic and programmable modulation. We experimentally generate the NPEPV beam by means of phase modulation through a SLM. The intensity properties and the phase properties of NPEPV beam in the case of free space are theoretically and experimentally studied.
It is assumed that the electric field of the NPEPV beam in the source plane has a form of20], respectively. Meanwhile, n denotes the power order of the beam. For the sake of convenience, A0 equals to 1.
The topological charge (TC) of a vortex may be quantified by the following integral :Eq. (1) is the TC of the NPEPV beam.
To distinguish the difference between COV, PEPV and NPEPV, we list the expression of phases:
Mathematically, there is a following equation,
According to Eqs. (3)–(4), in the case of n = 1, the PEPV and NPEPV reduce to the COV; in the case of m = 1, the NPEPV reduces to the PEPV. We compare the phases of these vortex beams with m = 3, n = 3, and these different phases are shown in Fig. 1.
It can be seen that the phase of PEPV usually does not have rotational symmetry, but COV and NPEPV do. These properties dominate the intensity pattern of these three kinds of beams. It is well known that the COV beam has the doughnut-like intensity pattern, while the PEPV beam possesses the c-like intensity pattern . So, the intensity pattern of COV beam has rotational symmetry, while PEPV beam does not. It could be inferred that the intensity pattern of NPEPV beam also has rotational symmetry.
Generally, in the situation of the paraxial approximation, the electric field in the transverse plane z = const > 0 through an ABCD paraxial optical system can be studied with the help of the generalized Collins formula 
And using the following formulas
The expressions of the intensity and phase of the NPEPV beam are given by
3. Experimental generation and propagation properties of the NPEPV beam through free space
Since transforming a laser beam into an arbitrary complex field could be done by means of phase modulation of SLM, which is a versatile and convenient way for modulating optical field, we use this method for generating a typical NPEPV beam and carry out experimental study of its intensity properties to verify the theoretical results. Moreover, we use the interferometric technique to verify the locations of the phase singularities. The experimental setup for generating the NPEPV beam and measuring its optical intensity and the locations of phase singularities under the situations of propagating through a free space is shown in Fig. 2. After a linear polarization He-Ne laser at 632.8 nm, a half-wave plate (HWP) and a polarization beam splitter (PBS) are placed. The HWP and the PBS serve to align the laser’s linear polarization state to the command polarization (accomplished by the PBS) and to control the power incident on the SLM (accomplished by the HWP in combination with the PBS). After passing through a beam expander (BE), the Gaussian beam is expanded, and the waist radius of the beam is set to be 2 mm. Then the beam was split into two paths, one is to illuminate on a SLM to generate the NPEPV beam, one is to be an interferometric beam. The direct way to generate the NPEPV beam is to impose the new kind of power-exponent phase into the Gaussian beam. In our scheme, a SLM (PLUTO, Phase Only) is controlled by a personal computer (PC1), which is used to input the holograph into the SLM, and the SLM is illuminated by the expanded laser beam. A personal computer (PC2) connects the CCD, which is used to measure the corresponding intensity. The beam reflected by the SLM is regarded as the NPEPV beam source. After reflected by a mirror reflector (MR), if an obstruction is placed, then the intensity of the NPEPV beam is detected by a charged-coupled device (CCD); if the obstruction is removed, then the interference pattern between NPEPV beam and Gaussian beam could be observed by the CCD. If a beam carried with phase singularities interferes with Gaussian beam, the fork interference pattern would arise around the phase singularities. So, the interference pattern could be used to figure out the informations (locations, sign and number of TC) about the phase singularities of the observed beam. To avoid the reflection of the SLM, the first-order diffraction of the beam from the SLM (adding the prism phase on the new kind of power-exponent phase) . The patterns of the holograph for generating the NPEPV beam of different topological charges and power orders are displayed in Fig. 3.
In the following, we will investigate the intensity properties of NPEPV beam propagating through a free space, in which the transfer matrix of distance z reads as 
Although the analytic expression of electric field is derived in the form of the expansion of the series, it remains error when we take the upper bound of the series in numerical calculation. If the optical system is a free-space system, we could use another calculation method  for comparison. We may rewrite the electric field in Cartesian coordinates. Then the Eq. (1) can be rewritten in the form of22]
Then the phase and the intensity of PEPV2 beam can be obtained, respectively, with
To compare the results of two methods, we convert the result with Eq. (10) in cylindrical coordinates to the result in Cartesian coordinates. The relation between the cylindrical coordinate system and the Cartesian coordinate system is given by the following formulas
The distribution of the intensity illuminated by the NPEPV beam at propagation distance z in free space are theoretically (with two different calculation methods) and experimentally shown in Fig. 4, where the dependencies on the power order n are illustrated. Meanwhile, the phase contour of the NPEPV beam at propagation distance z in free space are also shown in Fig. 5 both in theory and experiment. From these figures, we can see that the theoretical results calculated with two methods are well consistent with each other both in the intensity distribution and phase contour, and the experimental results verify the theoretical results. Although the theoretical results calculated with two methods are the same, we need to point out that the speed of calculation with Eq. (14) is much faster than with Eq. (12). Moreover, the intensity pattern of the NPEPV beam seems like fan blade, which is quite different with the distribution of the COV beam and PEPV beam. The COV beam has a “doughnut-like” profile and contains a circular dark core with zero amplitude along the optical axis, while the PEPV beam has a “C-like” profile and has an oval dark core . From Fig. 4, we will find that the NPEPV beam has more than one dark core and the number of dark cores is equal to the TC number m. From Fig. 5, it can be seen that the initially 3-charged vortex splits into 3 single-charge vortices. Furthermore, the distance between each dark core and optical axis is same, and the dark cores of these NPEPV beams shift away from the optical axis with the increase of the power order. The interference pattern shown in Fig. 5(c) verify the locations of singularities. Around the singularities, we can see the fork pattern and the pattern demonstrates the TC of each singularity is 1 with the same sign.
In order to see the influence of the power order n more explicitly, the intensity distribution and the phase contour of the NPEPV beam with large power order (n ≫ 1) are theoretically studied and displayed in Fig. 6. Compared to Fig. 4, it demonstrates that for a large power order, the intensity near by optical axis becomes brighter and three bright spots gradually disappear. When n is large enough [see Fig. 6(a) (n = 200)], the profile of the NPEPV beam takes approximately a Gaussian form. This phenomenon is same with that of PEPV beam and we could explain it by the property of the phase function in Eq. (1): when the power order n takes an extreme large value, the term of would approach to zero except the case that ϕ is very close to , thus the phase function in Eq. (1) would be almost a constant. According to Eq. (5), when the phase function is a constant, the beam’s profile would take a Gaussian form .
The theoretical and experimental results for the intensity distribution and the corresponding phase contour of the NPEPV beam with the power order n = 2 versus the TC m are shown in Fig. 7. In Figs. 7(a) and 7(b), the intensity distribution pattern of m-th NPEPV beam has m dark spots, which is resulted from the split of the singularity. Figure 7(c) is the theoretical results of the corresponding phase contour, while Fig. 7(d) is the experimental results of the interference patterns. From Fig. 7(c), the phase singularities are surrounded by the intensity blade, and the phase singularities are discrete on propagation, which is different from the phase transform of the COV beam. Figure 7(d) verifies the information of the phase contour, including the locations of singularities, the sign and number of the TC. The fork pattern around the singularities demonstrates the TC of each singularity is 1 with the same sign.
The propagation dynamics of NPEPV beam will be discussed now. In order to discuss the evolution of intensity distribution and phase contour, we simulate the NPEPV beam with different propagation distances z under the case of topological charge m is 3 and the power exponent number n is 3. Theoretical intensity distribution and phase contour of NPEPV beam with the TC m = 4 and power order n = 3 at different propagation distances z in free space are shown in Fig. 8. As the propagation distance increases, we can see the intensity of the area around the optical axis get brighter and the phase singularities get further away from the axis.
Some vortices would experience a rotation due to Gouy phase [25, 26]. Since the Collins formalism does not govern Gouy phase, so it may be not so appropriate to demonstrate the phenomenon of some vortices that could be influenced by Gouy phase significantly. However, the evolution of NPEPV is seem to be dominated by the phase factor . In the source plane, all the vortices of the NPEPV beam are on the optical axis, however, Eq. (12) shows that when z > 0 and ρ = 0, the electric field is not zero and the phase singularities are not on the axis, which means if the vortices leave the source plane, they begin to split. It is result from that the integral of azimuthal angle ϕ is not zero when z > 0 and ρ = 0. Moreover, from Figs. (5) and (7), we can conclude that as the TC m and power order n increase, the distance between the axis and phase singularities is becoming larger, however, the Gouy phase of the Gaussian beam in the experiment are the same. Thus, the evolution of NPEPV is determined by the phase factor.
4. OAM of NPEPV beam
It is known that some light fields, which are associated with the spiral wavefronts, can carry OAM. The proposed NPEPV beam is a type of light wave with spiral wavefront, which means it also carries OAM.
Figure 9 demonstrates the OAM density distribution (normalized) of NPEPV beam with different phases at the distancce z = 1.75m, where Figs. 9(a)–9(d) correspond to m = n = 1; m = n = 2; m = 3, n = 2; and m = n = 3; respectively. When a vortex is nested in the Gaussian beam [see Fig. 9(a)], the OAM density, which is related to the TC and the light intensity, distribute symmetrically around the central point. While for the NPEPV beam, as shown in Figs. 9(b) – 9(d), the OAM density distribution are totally changed, which varies by azimuthal angle ϕ. Meanwhile, the OAM density has rotational symmetry, whose center of rotation is on optical axis and the rotation angle is .
When the OAM and energy densities are integrated over the x–y plane, the ration of OAM to energy per unit length of beam can be calculated with 
To analyze the influence of parameters m and n on the OAM, we calculate the normalized OAM (which means the average OAM per photom posses) of the NPEPV beam, as shown in Fig. 10. It reveal that with change of the power order n, the normalized OAM also changes, which is different from that beam carrying PEPVs, whose normalized OAM remains a constant . It means the law that OAM is determined by the TC of COV and PEPV(L = lħ) cannot be applied to NPEPV.
In summary, we have proposed a new kind of noncanonical OV with the electric field expressed by Eq. 1. The analytical propogation formula for the NPEPV beam passing through a paraxial ABCD optical system is derived. On this basis, the propagation dynamics of NPEPV beam are theoretically studied. Meanwhile, we use another calculation method to verify the analytical formula. The intensity distribution and phase contour calculated by these two method are well consistent. The experiment is also carried out, and the experimental results agree well with the theoretical results. Both the theoretical and experimental results demonstrate that the TC m can change the “blade” number of the NPEPV beam, while the power order n influences the distance between the splitted phase singularities and the optical axis. The phase singularities are moving and discreate on propagation, and we explained the phenomenon.
Furthermore, the OAM density of the NPEPV beam is studied, and the influences of the parameters of the NPEPV beam on the OAM are also discussed. With the change of the power order n, the OAM also changes. It is expected that the proposed NPEPV beam and the corresponding conclusions can be useful for the extension applications of OVs, especially for particle trapping and rotating.
National Natural Science Foundation of China (NSFC) (11874321, 11474253); Fundamental Research Funds for the Central Universities of China (2018FZA3005).
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