## Abstract

We propose a new type of noncanonical optical vortex, named “power-exponent-phase vortex (PEPV)”. The spiral focusing of the autofocusing Airy beams carrying PEPVs are experimentally demonstrated, and the physical mechanism is theoretically analyzed by using the energy flow and far field mapping. In addition, the influences of the parameters of PEPVs on the focal fields and orbital angular momenta are also discussed. It is expected that the proposed PEPVs and the corresponding conclusions can be useful for the extension applications of optical vortices, especially for particle trapping and rotating.

© 2014 Optical Society of America

## 1. Introduction

Autofocusing beams, of which the abruptly autofocusing property may exhibit greatly potential applications in biomedical treatment and optical micromanipulation, have been an attractive theme for both theoretical and applied research in recent years [1–9]. A typical autofocusing beams is often referred to autofocusing Airy beam (AAB), which is described by the radially symmetric or circular Airy function. It has been shown that an AAB undergoes abrupt autofocusing in free space, where the energy suddenly increases right before the focal point [1–3]. Based on this property, many basic researches, such as abrupt polarization transitions [7, 8] and non-linear intense light bullets [10], have been reported, in addition to the applications on practical trapping [5, 11]. In comparison with apparatus used in optical tweezers, the implementation of optical manipulations with AABs only need an objective lens with weaker magnification.

Another structured light beam, commonly known as optical vortex (OV), is characterized by helical phase structures, hollow-core intensity distributions and intriguing orbital angular momenta fractional (OAM) [12–15], which therefore promise plentiful applications in particle trappings [16–18]. Moreover, relying on their topological structures, interesting linear and nonlinear evolution dynamics of OVs have been demonstrated [19–21]. Generally, a canonical OV carries a spiral phase varying uniformly with azimuthal angle. To exploring novel features and applications of OAM, many variational OVs, such as nonsymmetric OVs [22, 23], Helico-conical optical beams [24–26], fractional OVs [27, 28], Mathieu OVs [29], modulated OVs [30] and so on, have been proposed. On the other hand, introducing canonical OVs into AABs has been reported to present new optical properties [6–9]. For example, two opposite OVs can collide and annihilate when nested in an AAB [6]. Furthermore, we recently have demonstrated that canonical OVs can also change the polarization states of the vector AABs [7, 8]. However, the focusing properties of AAB carrying a noncanonical OV, as we have known, have not been reported yet.

In this paper, we propose a new type of noncanonical OV, characterized with power-exponent-phase, so we name it power-exponent-phase vortex (PEPV). The spiral autofocusing of the AABs carrying PEPVs is experimentally demonstrated, and the physical mechanism is theoretically analyzed with the energy flow and the far field mapping.

## 2. Power-exponent-phase vortex (PEPV)

In general, a canonical OV has a complex exponential term, exp(i*ψ*), where the phase function is expressed by

*θ*is the azimuthal angle, ranging from 0 to 2π;

*l*is the topological charge, which determines the number of 2π-phase shifts that occurs across one revolution of

*θ*, and the sign of

*l*determines the handedness of helix.

Equation (1) can describe the general spiral phase distribution of OV, which changes uniformly with *θ*, as shown in Fig. 1(a). Now we construct a new type of noncanonical OV by writing the phase function as

*l*still denotes the topological charge of the vortex, and

*n*determines the power order of the spiral phase, which can be either an integer or a fraction. Such an OV can be named as PEPV. Figure 1(b) shows a PEPV with

*l =*7 and

*n =*4. In contrast to the canonical vortex, the phase variation of PEPV with

*θ*is gradually intensified due to the power-exponent phase term. To introduce PEPV to the incident beam, the most convenient way is to imprint the phase directly by using a phase spatial light modulator (PSLM) to form a helical phase plate, the same way as that has been done for noncanonical OVs [24, 25].

## 3. AABs carrying PEPVs

Now we analyze the dynamics of AABs carrying PEPVs. The electric fields of AAB superimposed by a PEPV in cylindrical coordinates can be expressed as

*r,θ*) denote the polar coordinates,

*a*is the decaying parameter,

*r*

_{0}and

*ω*are the radius and width of the main lobe of AABs, respectively;

*ψ*denotes the phase function given by Eq. (2).

To experimentally generate the AABs with the desired phase function, we utilize computer-generated holography similar to that used in the prior demonstrations [7,8,19], as also shown in Fig. 2. Here, a computer-generated hologram (CGH) is displayed on a PSLM, with its intensity profile computed as the interference field between the AABs and a plane wave, i.e. |*u*(*r,θ*) + exp(i*fx*)|^{2}, where *f* determines the spatial frequency of the CGH. After passing through the PSLM, the expanded beam is filtered via a 4f system consisting of lenses L1 and L2 and forms the desired AABs. Placing the CCD at different distances, we observe the propagation characteristics of AABs carrying PEPVs.

Figure 3 depicts the experimental results of the intensity distribution near the focal points of AABs carrying different phases. Where, Fig. 3(a) shows the intensity profile of the generated AAB, and Fig. 3(b) represents the focal field of the AABs without any additional phase (*l = n =* 0), which consists of a bright spot similar to the zero-order Bessel-like pattern [5] formed via the abrupt autofocusing of the AABs. For the case of study, where *l = n =* 1, as shown in Fig. 3(c), the AABs are nested with a single charged vortex, and evolved into a hollow spot [6–8]. When the AABs carry a PEPV [as indicated in Fig. 3(e), when *l* = 8, *n* = 2], the autofocusing process will be totally changed, and it forms the spiral spot. This resembles to an Archimedes spiral at the focal region, which is much similar to the focal field of Helico-conical optical beams [24, 25]. To show evolution of the AABs with PEPV, the beam propagation method (BPM) is utilized to numerically simulate the propagation process. Figure 3(d) represents the simulation result of the side view of the AAB propagation at *y*-*z* plane. It can be seen that the light energy gradually converged and the intensity presents a bit oscillation when closing to the focal point. This can reveal the spiral focusing to some extent. Actually, the formed spiral focal spot will rotate during propagating at the focal region, resembling the cork-screw path. To demonstrate clearly this phenomenon, we simulate the propagation process of the AAB carrying a PEPV near the focal point (the parameters are selected as *l* = 12, and *n* = 2), as shown in Fig. 4, where Figs. 4(a)–4(d) correspond to the intensity profiles at different propagation distances. The rotation direction is indicated by the white arrow head in Fig. 4(a). It can be clearly seen that the intensity profile of AAB follows the cork-screw path, and evolves into a spiral spot.

For better understanding the formation of spiral focal fields, we analyzed the energy flow [31, 32] of AABs. Figure 5 depicts the energy flow of AABs at the input planes (top) and before the focal points (bottom). As shown in Fig. 5(a), for the AAB without any additional phase (*l = n =* 0), the light energy flow to the central region whether at the initial position or just before the focal point, behaving as focusing. For the AAB with a single charged vortex (*l = n =* 1), as shown in Fig. 5(b), the light energy circulates along a ring while shrinks gradually during propagation. This illustrates that the light spirals around the propagation axis while focusing. Due to the balanced energy cycle, the light intensity distribution keeps axisymmetric, i.e. a donut-shaped spot. For the AABs with a PEPV, the cycle of the energy flow is broken, and the energy will concentratedly flow to the certain region [see the regions with weak energy flow in the top of Figs. 5(c) and 5(d)]. On the other hand, it can be seen that the energy flow tends to shrink into the center because of the focusing effect. These lead to that the energy behaviors a spiral-like flow. Due to the spiral energy flow of PEPV, the light gradually evolves to a spiral-shape spot.

It is worth mentioning that during the propagations of AABs, although the intensity profiles vary a lot, the phase distributions keep unchanged essentially. Figure 6 illustrates four examples of the phase distributions near the focal points of the AABs with different initial phase. It reveals that besides the additional spherical phase, the phase distributions near the focal points are roughly identical to the initial ones.

## 4. Theoretical explanation

To theoretically explain the spiral focal fields, we look into the far-field propagation of AABs, which can be described by Fraunhofer diffraction integral. For the AAB with a PEPV, the far-field envelope is described by

*x*,

*y*) and (

*x'*,

*y'*) are the Cartesian coordinates,

*k*is the wave number,

*z*is the coordinate along the propagation direction, and

*u*(

*x'*,

*y'*) corresponds to the light field given by Eq. (3), of which the phase term is

*ψ*(

*r*,

*θ*) described by Eq. (2). Equation (4) can be considered as the Fourier transform of

*u*(

*x'*,

*y'*) if the propagation distance is far enough [25].

Under the approximation of slow-varying amplitude and phase, the field mapping can be provided by the local spatial frequencies, which can be achieved with the method of stationary phase and is defined as [25, 30]

*f*,

_{x}*f*) is the orthogonal coordinates in frequency domain. Applying this theory to Eq. (2), we obtain

_{y}According to Eq. (6), the frequency mapping (*f _{x}*,

*f*) to far field can be plotted, as shown in Fig. 7(a), where the parameters are selected as

_{y}*n =*3,

*l =*7, and

*r =*1. The curve represents an Archimede spiral, which meets the shape of the corresponding focal field of AABs with PEPV [see Fig. 7(b)].

## 5. Influences of the parameters on the focal fields

Now we examine the influences of the parameters *l* and *n* on the focal fields of the AABs carrying PEPVs. Figure 8 demonstrates the experimental results of the focal fields (the intensity distributions at the plane 30 cm away from the lens L2) of AABs carrying PEPVs (*l =* 8), where Figs. 8(a)–8(h) correspond to *n =* 2, 3, 4, …, 9, respectively, with the corresponding spatial frequency mappings inserted as white lines. It reveals that as the power order *n* increases, the focal spot shrinks with the tail shortened and the curved line stretches itself gradually. This phenomenon can be also seen from Eq. (2): with the increase of *n*, the phase gradients concentrate to the interval of *θ* near 2π, leading to the locally high gradient of phase. As a result, the light energy flow more quickly to the certain region. With increasing the parameters *n*, the light concentrates faster, and the focal spot became more shrinked. The experimental results are coincident with the corresponding frequency maps calculated from Eq. (6).

Figure 9 demonstrates the experimental results of the focal fields of AABs with PEPVs (*n* = 2), where Figs. 9(a)–9(h) correspond to *l =* 3, 4, 5, …, 10, respectively, with the corresponding spatial frequency mappings inserted as white lines. It can be seen that with the increase of the topological charge *l*, the spiral focal spot is wholly enlarged, but the shape remains the same. From Eq. (2), it can be also seen that the topological charge *l* never influences the distribution of the phase function, but changes the whole phase gradient. Thus, the energy flow distribution keeps wholly unchanged. While the rotating flow is speeded up with increasing *l*, and the focusing flow is accordingly reduced. These result in the wholly enlarged spiral focal spot. The conclusions of PEPVs are same as that of the canonical vortices [33]. Moreover, Eq. (6) also reveals that the frequency mapping is proportional to *l*. It means that *l* merely changes the size of the frequency mapping, as also shown in Fig. 9.

## 6. OAM of AABs carrying PEPVs

It is known that some light fields can carry OAM, which are associated with the spiral wavefronts. Actually, the proposed PEPV is a type of light wave with spiral wavefront. Thus, it also carries OAM [34].

The OAM density of light field about the *z* axis in spatial space can be expressed as

**E**and

**H**are the electric and magnetic fields, respectively;

*r =*(

*x*

^{2}+

*y*

^{2})

^{1/2};

*S*and

_{x}*S*are the components along the

_{y}*x*and

*y*axes of the Poynting vector (

**S**

*=*

**E**×

**H)**, respectively.

Figure 10 demonstrates the OAM density distributions of AABs with different phases at the focal points, where Figs. 10(a)–10(d) correspond to *l = n =* 0, *l = n =* 1, *l =* 12, *n =* 2, and *l =* 12, *n =* 3, respectively. It can be easily seen that, the AAB without OV [see Fig. 10(a)] does not carry OAM. When a vortex is nested in the AABs [see Fig. 10(b)], the OAM density, which is related to the topological charge and the light intensity, distributes symmetrically around the central point. While for the AABs with PEPVs, as shown in Figs. 10(c) and 10(d), the OAM density distributions are totally changed, and follow the spiral lines. It would be specially mentioned that the points on the spiral lines, such as points A and B in Fig. 10(d), have almost the same OAM, although with different distances from the central point.

To analyze the influence of parameters *l* and *n* on the total OAM, we calculate the normalized OAM of the AABs with different PEPVs, as shown in Fig. 11. It reveals that with change of the power order *n*, the OAM essentially remains a constant, with a small perturbation caused by the computing error. More importantly, the OAM is tightly related to the topological charge *l*, and perfectly matches the theory about the canonical vortex, i.e. *J = lħ* [13]. Namely, the law that OAM is determined by the topological charge of OV can be also applied to PEPV.

## 7. Conclusions

In summary, we have proposed a new kind of noncanonical OV with the phase distribution expressed by Eq. (2). The propagation dynamics of AABs carrying PEPVs is experimentally demonstrated. It reveals that the AAB evolves into a spiral spot during its autofocusing process when carrying a PEPV. To explain the physical mechanism, the energy flow of AABs and the far field mapping are theoretically analyzed, and the conclusions conform to the experimental results. Furthermore, the influences of the parameters of PEPVs on the focal fields and the OAM are also discussed. It is found that, with the increase of the power order *n*, the intensity distribution is more concentrated; meanwhile the spiral focal spot is wholly enlarged and the shape is unchanged with the increase of the topological charge *l*. The OAMs of the AABs carrying PEPVs are merely related to the topological charge. It is expected that the proposed PEPVs and the corresponding conclusions can be useful for the extension applications of OVs, especially for particle trapping and rotating.

## Acknowledgments

This work was supported by the 973 Program (2012CB921900), the National Natural Science Foundation of China (61205001 and 61377035), Natural Science Basic Research Plan in Shaanxi Province of China (2012JQ1017), the Northwestern Polytechnical University (NPU) Foundation for Fundamental Research (JC20120251).

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