The general analytical formula for the propagation of the power-exponent-phase vortex (PEPV) beam through a paraxial ABCD optical system is derived. On that basis the evolution of the intensity distribution of such a beam in free space and the focusing system is investigated. In addition, some experiments are carried out, which 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 of the PEPV beam.
© 2016 Optical Society of America
Nowadays, considerable interest has been exhibited in the propagation and interaction of optical beams through random media, and characterization of the propagation properties offers a good reference point for evaluating the effect of a beam in practical environment. In some circumstances, it is desirable to have a laser beam with a different irradiance distribution from the one originally coming out of the laser cavity, such as laser welding, laser processing, and laser medical applications.
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, the research on optical vortices has been investigated extensively both in theory and experiment for its wide practical applications, such as astronomical, micromanipulation of particles and atom guiding [1–4]. Optical vortices are a peculiar type of beam which has one or more singularities on the transverse beam profile [5–7] and an azimuthal phase dependence , where is an integer number and refers to the topological charge (azimuthal index) of the field , which is related to the orbital angular momentum (OAM) of photos by the relationship . Over the past decades, apart from the canonical vortices, which carry a spiral phase that varies uniformly with azimuthal angle, several kinds of noncanonical vortices, such as nonsymmetric vortices [9, 10], Mathieu vortices  and fractional vortices [12–14], have also been investigated in order to explore the properties and applications of OAM. It has been reported that the OAM could be used for quantum computing  and moving particles .
Power-exponent-phase vortex (PEPV) beam is a kind of noncanonical vortex that characterized with power-exponent-phase [17, 18]. It has been illustrated that the PEPV beam carries OAM, which is determined by the topological charge . The simplest way to generate the PEPV beam is to employ a spatial light modulator (SLM) . This method is more interesting and popular because it has the advantage of providing dynamic and programmable modulations. In this paper, we experimentally generate the PEPV beam by means of phase modulation through a SLM, that is, we directly impose a power-exponent phase into a Gaussian beam in such way. And the intensity properties of PEPV beam in the cases of free space and focusing system are theoretically and experimentally studied.
It is assumed that the electric field of the PEPV beam in the source plane has a form of
In the situation of the paraxial approximation, the electric field in the transverse plane through an ABCD paraxial optical system can be studied with the help of the generalized Collins formula [19, 20]Eq. (2).
And using the following formulas 
The expression of the intensity of the PEPV beam is given by
3. Experimental generation and propagation properties of the PEPV beam through free space and a focusing system
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 PEPV beam and carry out experimental study of its intensity properties to verify the theoretical results. The experimental setup for generating the PEPV beam and measuring its optical intensity under the situations of propagating through free space and a focused system is shown in Fig. 1. A straightforward way for producing the PEPV beam is to impose a power-exponent phase into a Gaussian beam, in such a way as to modulate the complex field and optical intensity distribution. In our scheme, the reflective phase SLM is controlled by a personal computer (PC1), which is used to input the holograph into the SLM, and is illuminated by a linearly polarized He-Ne laser beam with . In addition, the beam is conditioned by a beam expander, and the waist width of the beam is set to be 1mm. The beam reflected by the SLM is regarded as the PEPV beam source. Then the intensity of the beam is detected by a charge-coupled device (CCD) after propagating through the distance . In order to study its focusing properties, a thin lens with focal length is seated before the CCD, which is located at the focal plane. A personal computer (PC2) connects the CCD, which is used to measure the corresponding intensity. The separation between the SLM and the thin lens is . To avoid the reflection of the SLM, we select the first-order diffraction of the beam from the SLM. The patterns of the holograph for generating the PEPV beam of different power orders and topological charges are displayed in Fig. 2.
In the following, we will investigate the intensity properties of PEPV beam propagating through free space and a focused system. We first consider the case of free space, in which the transfer matrix of distance reads as 
The distributions of the intensity illuminated by a PEPV beam at propagation distance in free space are theoretically and experimentally shown in Fig. 3, where the dependencies on the power order are illustrated. Figures 3(e)-3(h) represent the experimental results, which are well consistent with the theoretical calculation results shown in Figs. 3(a)-3(d). From these figures, we find that the intensity of the PEPV beam is closely determined by . Moreover, being different from the intensity distribution of the traditional canonical vortex beam, which has a “doughnut-like” profile and contains a circular dark core with zero amplitude along the optical axis, the intensity pattern of the PEPV beam seem like the letter “C”, as Figs. 3(a) and 3(e) depicted. In Figs. 3(b) and 3(f), the “C-like” pattern evolves into a thwart-wise “U-like” pattern, and the PEPV beam has an oval dark core. When the power order is gradually increased to 10, the intensity pattern would gradually evolve into a skew “L-like” pattern, or saying the “boomerang-like” pattern. Furthermore, the dark cores of these PEPV beam shifts away from the optical axis with the increase of the power order.
In order to examine the influence of the power order more explicitly, the intensity distributions of the PEPV beam with large power orders () are theoretically studied and displayed in Fig. 4. Comparing Fig. 4(a) with Fig. 3(d), we can find that for the larger power order, the upper part of the “L-like” pattern would be brighter and wider, while the lower part of the pattern becomes narrower and shorter. When is large enough [see Figs. 4(b, c)], the upper part of the pattern would dominate the whole pattern and the lower part would vanish. In addition, it should be noticed that the center of the beam returns back to the optical axis. Particularly, the profile of the PEPV beam with takes approximately a Gaussian form. This phenomenon could be well explained by the property of the phase function appearing in Eq. (1): when the power order takes an extreme large value, the term of would approach to zero except the case that is very close to , thus the phase function in Eq. (1) would be almost a constant. According to Eq. (2), when the phase function is a constant, the beam's profile would take a Gaussian form . Hence, the PEPV beam would evolve into a Gaussian beam when is great enough.
The theoretical and experimental results for the intensity distributions and the corresponding phase contour of the PEPV beam with the power order versus the topological charge are shown in Fig. 5. In Figs. 5(a) and 5(e), the patterns of optical intensity distribution are a “C-like” optical arc with an intensity spot at its upper end. As the topological charge increases, the pattern of the optical intensity enlarges, and the energy of the beam starts to concentrate on the intensity spot. Meanwhile, the intensity of the lower part is weaken. Therefore, the topological charge could modulate the size of the optical pattern. In Figs. 5(i-l), it is obvious that the phase singularities are surrounded by the intensity arc, and the phase singularities are discrete on propagation, showing a different picture from the pattern of phase of the beam source and the canonical vortex.
As another particular example, we will now consider the focusing properties of the PEPV beam. The corresponding transfer matrix between the source plane and the observed plane in the focusing system is 
The focused intensity distributions of the PEPV beam are shown in Fig. 6. It can be seen from Fig. 6 that the main part of the focused intensity likes a kidney in shape, and a bright optical intensity spot is formed inside the “kidney-like” pattern. In addition, such “kidney-like” patterns are accompanied by some dark cores on the left side, and some dim optical spots and lines might appear besides the “kidney-like” pattern.
4. Error analysis
In order to evaluate the error between the results of numerical simulation and experiment, we can consider the error of the following equation, which is a part of the Eq. (5):
Since the computer could not sum up infinite terms, it is necessary to set the upper summation limits for the indices , , and .
4.1 Selecting the suitable upper summation limit for
According to the properties of Gamma function and Stirling’s formula, when is great enough, we have
First let us discuss the part containing in Eq. (12), which can be rewritten as
Theoretically, the inequation need to be satisfied to make sure the summation is convergent. According to the parameters in the experiments, we can find out that , thus we could set the maximum summation index as 100.
The error of this part is
4.2 Selecting a suitable upper summation limit
Approximately, we have
Since would be close to zero when is great enough, and we only choose the terms that dominate the simulation results, we would care about the ratio between and , the maximum value of . Thus, would be helpful for selecting a suitable (see Fig. 7).
According to the results, we can find that when , the ratio between and is less than . Thus, we set as 15 in the simulation, and it would be accurate enough.
4.3 Choosing maximum values for indices and in Eq. (6)
Defining , and Eq. (6) becomes
When both and are great enough, according to the Stirling’s formula, we have
Similarly, in order to make the summations are convergent, both terms and should be less than 1. As a result, it is carried out that and . Thus, we set as 80 and as 300.
We can estimate the error produced by the last summation term with an approximation:
Another approximation for estimating the first term of the error is:
The approximation for estimating the second term is:
We sum up the three terms and find that the total error is approximately equal to , which is negligible and all the we take into consideration in our simulation are much greater than .
4.4 Estimating the error caused by limitation of and the relative error of Eq. (5)
The error caused by the range of is (neglect the error caused by limitation of , which is quite small):
As a result, since the magnitude of the results of Eq. (5) is approximately , we could see that the relative error of the simulation is approximately 2%.
We will discuss the propagation dynamics of PEP vortices now. In order to discuss the evolution of intensity distribution and phase contour, we simulate the PEPV beam with different propagation distances under the case of topological charge is 4 and the power exponent number is 3.
Theoretical distributions of intensity and phase of a PEPV beam with topological charge and power order at different propagation distances in free space are shown in Fig. 8. In Fig. 8, we can see that the intensity distribution of is a “C-like” pattern. As the propagation distance increases, the lower part of the pattern would become dimmer, and the pattern would gradually evolve into a “bean-like” intensity spot. In addition, the phase singularities are getting farther and farther away from the axis.
In some other researches on vortex beams, vortices would experience a rotation due to Guoy phase [23, 24]. Since the Collins formalism does not govern Guoy phase, it may be not so suitable for illustrating the phenomenon of some kinds of vortex beams that could be influenced by Guoy phase significantly. However, in our research, the evolution of PEP vortices is more likely to be determined by the phase factor . Since all the PEP vortices are on the optical axis when they are in the source plane, and Eq. (5) shows that when and , the electric field is not zero and the phase singularities are not on the axis, which means that the vortices would not on the optical axis once they leave the source plane. That is due to the fact that the integral of azimuthal angle is not zero when and . In other words, the initial motion of the phase singularities is determined by the phase factor. In addition, in Fig. 5, we can see that as the topological charge increases, the average distance between the axis and phase singularities is likely to be greater, however the Guoy phase of the Gaussian beam in the experiment are the same. Thus, the phase factor should be the major contributor of PEP vortices’ evolution.
In conclusion, we have derived an analytical propagation formula for the PEPV beam passing through a paraxial ABCD optical system. On this basis, the properties of the intensity of such a beam on propagation in free space and the focused system are investigated respectively. In addition, we also carried out the generation of the PEPV beam experimentally and measured its optical intensity distribution in these two cases. Experimental results are well consistent with the theoretical predictions, and it is shown that the power order of the beam can change the shape of the PEPV beam and the topological charge governs the size of the beam’s pattern. In the far field (or in the focal plane), the intensity distribution would present a “kidney-like” shape accompanied with several dark cores. Furthermore, the phase singularities are moving and discrete on propagation, and we discuss the reason of this phenomenon.
Due to this kind of unique propagation properties, the PEPV beam could be used in optical tweezers, optical manipulation and medical applications and have some advantages. One of the advantages of PEPV beam is that it could trap a group of tiny particles and then release them just through adjusting the parameters, such as the power and the topological charge . For example, since PEPV beam’s profile is a ring-like pattern when is 1, tiny particles could be trapped by PEPV beam with power . When these particles are going to be released, a gap could be produced on the optical ring through increasing PEPV beam’s power , so that these particles could be pushed out of the optical trap by the non-uniform gradient force of the PEPV beam. In addition, since the location of the gap could be determined, the direction of these particles’ escaping motion could be estimated or also determined.
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11274273 and 11474253) and the Fundamental Research Funds for the Central Universities (2016FZA3004).
References and links
1. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989). [CrossRef]
3. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000). [CrossRef]
4. C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009). [CrossRef]
5. A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008). [CrossRef]
6. Z. S. Sacks, D. Rozas, and G. A. Swartzlander Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15(8), 2226–2234 (1998). [CrossRef]
7. Y. S. Rumala and A. E. Leanhardt, “Multiple-beam interference in a spiral phase plate,” J. Opt. Soc. Am. B 30(3), 615–621 (2013). [CrossRef]
8. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
12. I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004). [CrossRef]
14. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef] [PubMed]
15. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005). [CrossRef] [PubMed]
18. W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014). [CrossRef]
19. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
20. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, Beijing, 2000).
21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980).
22. J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Dekker, 2003), 999–1013.
24. Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015). [CrossRef]