## Abstract

A modification of the helical phase profile obtained by eliminating the on-axis screw-dislocation is presented. Beams with this phase possess a variety of interesting properties different from optical vortex beams. Numerical simulations verify analytic predictions and reveal that beams with this phase have intensity patterns which vary as a function of the phase parameters, as well as the propagation distance. Calculations of the Poynting vector and orbital angular momentum are also performed. Experiments verify the intensity profiles obtained in simulation.

©2009 Optical Society of America

## 1. Introduction

Phase fronts containing screw dislocations possess singularities wherein all phase-values are realized along the dislocation lines. This phase gradient singularity is referred to as an optical vortex (OV) [1–16]. The phase profile common among optical vortices is expressed as

Here *φ* is the azimuthal angle while *ℓ* determines the topological charge, the total number of 2π phase shifts around a closed loop along the azimuth. The phase of the wave changes continuously by *ℓ*2π in one complete revolution around the optical axis with an indeterminate phase resulting along the dislocation center. Equiphase surfaces in the phase profile form a periodic helicoidal structure of pitch *ℓλ* [1–7]. The phase singularity located along the dislocation center is responsible for the characteristic annular intensity profiles of OV beams [4–12].

Due to their helical phase front structure, these beams have been shown to possess a rotating Poynting vector [16–19] as well as the ability to impart orbital angular momentum (OAM) of *mħ* per photon on birefringent and partially-absorbing particles resulting in their translation around a circuit centered on the beam’s axis [20–22].

Dynamic properties arising from the helical phase structure have been exploited using other beam profiles and more complex configurations. Optical ferris wheels – optical ring lattices with tunable rotation rates have been generated by interfering two co-propagating Laguerre-Gauss (LG) beam modes that are equally polarized. These have been studied in the context of trapping ultracold and quantum degenerate atomic samples [23].

Modifications of the helical phase profile have also been proposed. Higher-order Bessel beams possessing the phase expression $\ell \phi -\genfrac{}{}{0.1ex}{}{2\pi}{{\rho}_{\mathbf{max}}}\rho $ have been investigated particularly for its capacity to impart OAM as well as its potential for atom trapping given the merit of nondiffraction [24]. Similarly, helico-conical beams characterized by the phase expression $\ell \phi (K-\genfrac{}{}{0.1ex}{}{\rho}{{\rho}_{\mathbf{max}}});\phantom{\rule[-0ex]{.5em}{0ex}}K\in \left\{0,1\right\}$ have been generated. Owing to the nonseparability of this phase expression compared to that of higher order Bessel beams, helico-conical beams are observed to form spiral intensity profiles instead of the characteristic annular intensity patterns of OV beams. Moreover, a rotation of intensity profiles of helico-conical beams synchronized with switching in the direction of the tail of the spiral have been observed as the focal plane is approached. These beams offer potential applications which include positioning of asymmetric particles and the accumulation of small particles towards the center of the spiral [25, 26].

Investigation of OV’s and various spatial phase structures could further enhance previous knowledge concerning properties of optical beams, particularly in terms of the propagation dynamics, and energy flow of the light field, as well as open new potential applications in various disciplines including imaging and optical micromanipulation.

In this work, a modification of the helical phase profile is presented. Beams possessing this new phase are characterized in terms of the phase parameters and the propagation distance *z*. Within the bounds of the paraxial approximation of the scalar diffraction theory, an analysis is presented detailing the features and dynamics of this new class of beams. This preliminary treatment does not explore further the vortex dynamics which include birth and evolution of OV’s and beam propagation very near the focal plane.

Results are derived mostly from a mathematical treatment of the wave propagation phenomenon of beams with the modified helical phase, as well as from a numerical model simulating free-space linear propagation. Experiments serve to verify some results obtained from theory.

## 2. Bored helical beams

The new phase is obtained by replacing the helical structure with a uniform profile within a circular region which is centered along the screw dislocation of the phase. This phase structure, referred to as bored helical (BH) phase is characterized by the step function

where the ratio *ρ _{REL}* denotes the bore radius

*ρ*in proportion to the outer radius

_{i}*ρ*as shown in Fig. 1. In this study, the topological charge

_{o}*ℓ*is limited to integer-order values. More specifically, |

*ℓ*|∈

**N**. The characteristic function

*χ*of a set

_{A}*A*is given by [27]

while the sets *I* and *O* are as follows:

$$O\u2254\{\rho \in {\mathbf{\mathbb{R}}}^{+}\cup \left\{0\right\}:\rho <{\rho}_{o}\};{\rho}_{o}\in {\mathbf{\mathbb{R}}}^{+};{\rho}_{i}<{\rho}_{o}.$$

The set *O*\*I* is thus

The BH phase defined by Eq. (2) describes a helical phase profile with edge dislocations along the circuit of radius *ρ _{i}* replacing the screw dislocation previously found on-axis. In particular, a BH phase of charge

*ℓ*possesses |

*ℓ*| equally-spaced π – phase point discontinuities along the radius

*ρ*. This phase distribution will be shown to generate vortices despite the absence of the screw dislocation.

_{i}Free-space linear propagation of beams possessing the BH phase was modeled numerically using the Split-Step algorithm. Numerical implementation of the Fresnel approximation of the Huygens integral was performed using Fourier analysis. The propagated wave *u*(*x*, *y*, *z*)is obtained by convolving the initial disturbance *u*(*x*, *y*, *z*=0) with the transfer function

The simulation carried out in Matlab utilizes the Fast-Fourier Transform (FFT) algorithm in implementing the convolution Due to the periodic nature of the FFT algorithm, aliasing effects at the boundaries can be observed. This was addressed by zero-padding, to ensure the field *u*(*x*, *y*, *z*) vanishes at the edges. The model simulates (linear) propagation of a He-Ne laser (*λ*=632.8nm) in free space. The two-dimensional transverse wavefront and intensity profiles were observed as variations in the parameters $(\ell ,{\rho}_{\mathrm{REL}}=\genfrac{}{}{0.1ex}{}{{\rho}_{i}}{{\rho}_{o}},z)$ were introduced.

Beams possessing the BH phase described by Eq. (2) are observed to form transverse intensity patterns distinct from beams with helical phase. As shown in Fig. 2, it is found that for *ρ _{REL}* values near unity, in particular for {

*ρ*∈[0,1]: ~0.6≤

_{REL}*ρ*≤~0.9}, transverse intensity profiles viewed at a distance

_{REL}*z*are observed to form intense arms. These arms extend from the center, spiraling outwards perpendicular to the propagation axis. Alternatively, for

*ρ*values within the range [~0.1, ~0.3], BH beams instead form into symmetric patterns resembling polygons which encapsulate

_{REL}*ℓ*distinct OV’s distributed uniformly within the profile. The intermediate

*ρ*interval serves as a transition wherein beams are observed to form quasi-ring structures containing inner intense-arm formations. In all cases, the number of intense-arm or outer ring formations equal the integer-order topological charge

_{REL}*ℓ*of the BH phase.

In general, it is observed that as *ρ _{REL}* is decreased, interference of the wavefronts results in the intensity distribution of the beam gradually shifting from the center towards the outer ring structures. The limiting case of which being

*ρ*=0, where the BH phase reverts to the corkscrew topology of a charge

_{REL}*ℓ*helical phase.

Interestingly in Fig. 2, the rotational sense of the spirals reverses as one goes from *ρ _{REL}* value of 0.84 to 0.48, even though the encoded topological charges have the same sign and the intensity patterns were obtained at the same propagation distance. This is evident most specially in

*ℓ*=1. It seems that the rotational sense is not determined by the sign of the topological charge alone but also of the value of

*ρ*. An analysis similar to Soskin [6,16] and Berry [4] may give insights on this observation since the on axis screw dislocation is now replaced by an edge dislocation that varies its value around the circle with radius

_{REL}*ρ*from 0 to 2

_{i}*πℓ*.

In addition to its dependence on the phase parameters, it is observed that BH beam patterns are a function of its propagation distance *z* (Fig. 3). In contrast to the propagation invariant LG modes however, BH beams are, in general, structurally unstable upon propagation. Moreover, the variation in the intensity pattern as the beam propagates gives the impression of rotation. Such rotation is unlike the rigid rotation of intensity profiles discussed in Refs. [7] and [16]. We have tracked the rotation by looking at the zeroes of the intensity patterns. Simulations reveal that the rotation rates of BH beams vary as a function of its phase parameters *ℓ* and *ρ _{REL}* as well. Beams with identical charges but differing

*ρ*values exhibit dissimilar rotation rates. It is observed that beams with smaller

_{REL}*ρ*values rotate more quickly than beams with larger cavity radii (Fig. 4). In particular, it was found that beams with

_{REL}*ρ*values corresponding to the transition interval possess rotation rate greater than intense arms BH beams. Similarly, for beams of identical

_{REL}*ρ*values, it is observed that rotation rate monotonically decreases with the magnitude of charge |

_{REL}*ℓ*| (Fig. 5). In all cases, angular displacements asymptotically approach a maximum value as the wave propagates. This behavior suggests an association with the Gouy phase shift. Further details pertaining to the particular role of the Gouy effect as well as vortex dynamics in relation to beam profile evolution is outside the scope of this paper.

Experimental confirmation of results obtained from both analysis and the model was performed using the setup described in Fig. 6. BH beams were generated by use of computer-generated holograms (CGH) possessing the phase described by Eq. (2). The holograms were constructed following the method proposed by Bazhenov *et al* [5]. Interference of the beam with BH phase (object wave) with a plane wave (reference wave) whose wave vector subtends the propagation axis of the object wave was performed numerically. The resulting ideal interference pattern may be described by

where *Λ* denotes the spatial period of the plane wave in the transverse plane [26]. The calculated interference pattern illustrating a sinusoidal diffraction grating is then converted to a suitable image file, scaled accordingly, and printed. A variety of CGH’s corresponding to different *ℓ* and *ρ _{REL}* were constructed. The incident beam was expanded and collimated so that it uniformly illuminates the CGH’s. Experiments verify the intensity profiles obtained in the numerical simulation.

Initial experiments concerning propagation of the beam were performed by mechanically displacing the diffracting lens (f=*f _{1}*) parallel to the input beam. This method simulates propagation by shifting the focal plane of the diffracting lens instead of mechanically translating the viewing plane (CCD). Figure 10 shows a sequence illustrating propagation of a

*ℓ*=3,

*ρ*=0.6 BH beam. This verifies qualitatively the unstable behavior of BH beams upon propagation as well as the decrease in rotation rate as the beam departs the focal plane.

_{REL}## 3. Calculations of Poynting vector and the orbital angular momentum

Let the initial wavefront be given by

where Φ*BH _{ℓ}* is the BH phase (Eq.(2)) with cavity radius

*ρ*,

_{i}*σ*denotes the initial waist of the truncated Gaussian amplitude profile, and

*C*the normalization constant. The initial beam waist

*σ*is chosen to be larger than the outer radius

*ρ*to ensure the beam varies only slightly within the region

_{o}*O*. Outside this region, the transmission is zero. From Eq.(8), the propagated field can be obtained, within the paraxial limit, using the Huygens-Fresnel integral [28, 29]

$$+{i}^{\mid \ell \mid}\mathbf{exp}\left(i\ell \phi \right){\displaystyle \colorbox[rgb]{}{$\underset{0}{\overset{+\infty}{\int}}$}}{\chi}_{o\setminus I}\left({\rho}^{\prime}\right)\mathbf{exp}\left[(-\genfrac{}{}{0.1ex}{}{1}{{\sigma}^{2}}+i\genfrac{}{}{0.1ex}{}{k}{2z}){\rho}^{\prime 2}\right]{J}_{\mid \ell \mid}\left(\genfrac{}{}{0.1ex}{}{\mathrm{k\rho}}{z}{\rho}^{\prime}\right){\rho}^{\prime}d{\rho}^{\prime}\}$$

$$={\phantom{\rule[-0ex]{.2em}{0ex}}}^{I}u(\rho ,z)+{\phantom{\rule[-0ex]{.2em}{0ex}}}^{O}u(\rho ,\phi ,z).$$

Insofar as the sets *I* and *O*\*I* are disjoint, the two integrals * ^{I}u*(

*ρ*,

*z*) and

*(*

^{O}u*ρ*,

*φ*,

*z*) in Eq. (9) do not overlap. The separation of the integral occurs when the separation in the radial coordinate of the phase term of the integrand factor

*u*(

^{BH}*ρ*,

*φ*,

*z*=0) is considered. Expressed in this manner, the field

*u*(

^{BH}*ρ*,

*φ*,

*z*) at some

*z*>0 may be interpreted as the interference of two co-propagating wavefronts, with only the

*(*

^{O}u*ρ*,

*φ*,

*z*) component possessing a helical phase structure. Note that the

*(*

^{O}u*ρ*,

*φ*,

*z*) term includes

*χ*

_{O/I}. The presence of the plane-phase component in the wavefront is responsible for the decomposition of a single

*ℓ*-charged (|

*ℓ*|>1) vortex into nearly-situated |

*ℓ*| unit-charged vortices of the same sign (identical helicity) [12 - 14, 16]. This accounts for the characteristic intensity profiles observed in simulations and experiment. An examination of Eq. (9) would also show that the relative amplitude of the plane- and helical-phase components

*(*

^{I}u*ρ*,

*z*) and

*(*

^{O}u*ρ*,

*φ*,

*z*) is dictated by the cavity radius

*ρ*. Larger

_{i}*ρ*values would imply greater contribution of the plane-phase wave

_{i}*(*

^{I}u*ρ*,

*z*) giving rise to intensity arms, while as

*ρ*approaches zero, the intensity profiles of BH beams revert to those of OV beam patterns.

_{i}For a linearly-polarized field, the time-averaged Poynting vector may be calculated within the paraxial approximation to possess angular and axial components given by

The nonzero angular component 〈** S**〉·

*φ*̂ is attributed to the helical phase structure of the beam. This implies a spiraling Poynting vector trajectory similar to OV beams.

The angular component of the Poynting vector ** S** also gives rise to a nonzero component of the angular momentum density of

*u*(

^{BH}*ρ*,

*φ*,

*z*) associated with the transverse plane. This is given by [17, 18]

The total angular momentum of a BH beam relative to its energy per unit length **Ξ** is thus

$$=\genfrac{}{}{0.1ex}{}{\ell}{\omega}{\displaystyle \colorbox[rgb]{}{$\underset{0}{\overset{2\pi}{\int}}$}}{\displaystyle \colorbox[rgb]{}{$\underset{0}{\overset{+\infty}{\int}}$}}{\mid {\phantom{\rule[-0ex]{.2em}{0ex}}}^{O}u(\rho ,\phi ,z=0)\mid}^{2}\mathrm{\rho d}\mathrm{\rho d}\phi ,$$

where power conservation [30] is invoked to obtain the final expression in Eq. (13). Noting that

the orbital angular momentum (OAM) imparted by BH beams is thus smaller than that imparted by OV beams for any *ρ _{i}*>0. Eq. (13) also exhibits that a plane-phase wavefront such as

*does not contribute to the OAM expression of the beam [14], in spite of the fact that these field components do influence the Poynting vector expression [13, 16].*

^{I}u## 4. Conclusion

The bored helical phase is a modification of the well-known helical phase obtained by eliminating the region *I*, thereby replacing the on-axis screw-dislocation with several edge-dislocations. Beams with this phase possess a variety of interesting properties different from OV beams.

Intensity patterns of the BH beams are easier to generate compared to the patterns that are obtained when LG modes of different waists and/or of different topological charges are interfered. The former requires only a change in the hologram while the latter needs extra optical equipment and a more delicate alignment.

The distinctive intensity patterns and propagation dynamics of BH beams may find potential applications in the field of optical trapping and micromanipulation, and in fabricating micrometer sized 3D structures in parallel as in Ref. [31]. While particular details concerning the feasibility of the use of such beams as a trapping source are still the subject of research, BH beams appear as a promising tool which can further improve current trapping setups.

## Acknowledgments

The authors are grateful to Dr. Raphael Guerrero for the use of the optical setup at the Photonics Research Laboratory, Ateneo de Manila University. Mr. Columbo Enaje prepared the holder of the computer generated holograms. The authors also thank the School of Science and Engineering, Ateneo de Manila University, and the Philippine Council for Advanced Science and Technology Research and Development (DOST-PCASTRD) for the grant. Dr. Darwin Palima shared some valuable insights.

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