## Abstract

We present a method to measure the skew angle of the wave-fronts in an optical vortex, which is directly related with the energy flux. It is based on the analysis of the evolution on propagation of the near-field diffraction pattern generated by a single-slit, consisting of two main lobes that shift in opposite directions depending on the vortex helicity. The transverse displacement of each lobe as a function of the propagation distance allows to quantify the energy circulation. Analytical, numerical and experimental results are compared, showing good agreement. We illustrate the method for the case of Bessel beams, although we also discuss other types of helical beams, such as Laguerre-Gauss and Mathieu beams.

© 2013 Optical Society of America

## 1. Introduction

The study of optical vortices (OVs) has been the focus of great attention in the last two decades due to both, their topological and dynamical properties, which have led to important applications in several areas of physics. Regarding the topology of their wavefronts, OVs exhibit a screw dislocation or phase singularity along their propagation axis [1], characteristic that may be used, for instance, to store quantum information [2]. In terms of their dynamical properties, OVs are often associated with the presence of orbital angular momentum (OAM) carried by the optical field [3, 4]. Although this is not a property of the vortices per se [5], the most familiar propagation modes with circular symmetry and embedded vortices on-axis, such as Laguerre-Gaussian (LG) and Bessel beams (BBs), do possess OAM, which can be transferred to matter [6–8]. Other beams whose geometry departs from the circular, as the case of elliptical Mathieu beams (MBs) with helical phase [9], have also been used to rotate matter [10]. The topological charge or singularity strength of an OV is the integer number *l* of phase cycles of 2*π* in a closed contour around the vortex core. It might be positive or negative depending on the handedness of rotation of the wavefronts. In this work, we will deal with three types of optical fields: BBs, MBs and LG beams, which will be referred to, altogether, as helical beams. As a consequence of the rotating phase, helical beams are also characterized by a circulating energy flow in the angular direction or optical current [5], responsible for the torques exerted on small particles by these beams.

Due to the myriad of applications of helical beams, not only for rotating matter but also for microscopy [11], quantum information storage and transmission [2, 12], optical lattices [13], nonlinear optics [14–16] and astronomy [17], to name but a few, there have been big efforts dedicated to their experimental generation [18–21] and characterization. The determination of the topological charge and handedness have been the most common goal. There are techniques based on the interference of a helical beam with a reference plane wave [4]; others are based on Young’s experiment with different portions of the same beam [22]. On the other hand, methods to infer the properties of OVs based on diffraction have also been extensively developed, specially in the last few years. For example, diffraction of a vortex through a half-plane or a single slit allows to determine its handedness [23–25]. More recently, diffraction by a triangular aperture was shown to give rise to truncated lattice patterns containing information about both, the magnitude and sign of the singularity strength [26]. Many other apertures, such as annulus, either circular [27] or elliptical [28], iris diaphragms [29], squares [30] and refractive elements like axicons [31] have also been used to investigate diffraction of OVs.

Only few experimental works, however, have focused on the energy circulation in helical beams, even when this has been subject of considerable interest from the theoretical viewpoint [32, 33]. For instance, a proposal by Bekshaev is based on the analysis of the distortions produced in an oblique section of a beam due to the energy circulation compared to the geometric-optics expectation, where the shift of the center of gravity of the beam intensity is the characterization parameter [33]. Although ingenious, this analysis does not seem very useful in practice in the optical domain, due to the high accuracy required for the measurements. In contrast, experimental studies in this direction have relied so far on the use of a Shack-Hartmann sensor to obtain a full map of the transverse energy flow [34, 35].

In this work, we demonstrate that it is possible to obtain quantitative information about the energy circulation in helical beams by means of diffraction through a single-slit. The pattern consists of two main lobes that shift outwards in opposite directions depending on the vortex helicity. By analysing the evolution of this pattern along the propagation direction in the near-field, we can determine the local skew angle of the Poynting vector. In order to discriminate the effects of simple diffraction from those arising due to the energy circulation, we compare the case of the helical beam with a non-rotating beam of similar characteristics. A comparison among analytical, numerical and experimental results is established. As a case of study we focus our attention on Bessel beams, but we also discuss some results for other helical beams.

## 2. Energy circulation in helical beams

Optical vortices are often linked with the concept of orbital angular momentum (OAM). Nevertheless, as Berry pointed out recently [5], this association does not have general validity, since the angular momentum is a quantity defined with respect to a given position in a reference frame, whereas wave vortices can be located at arbitrary positions within the wave field. In contrast, a characteristic that is indeed related with any vortex is the twist of the energy current (Poynting vector, in the case of vector light) about the singularity, which gives the time-averaged force exerted by the wave field on matter [5]. In that sense, although the topological charge of a vortex is the most relevant quantity for applications related with multiplexing of quantum information, for instance, the energy circulation takes the main role for other applications related with wave-matter interaction. An example of this fact is found in reference [36], where it was demonstrated for the case of acoustic vortices that the torque exerted on an object by the vortex depends on the acoustic intensity vector (energy circulation) in the region of the space filled by the object.

In the case of paraxial helical light beams with circular cylindrical symmetry, the time-averaged Poynting vector can be written as [3, 7]

Here *C*_{0} = *cω*^{2}*ε*_{0}/2 for free space propagation, (*ρ⃗*, *φ⃗*, *z⃗*) are the unit vectors in circular cylindrical coordinates, *l* is the topological charge and *u* = *u*(*ρ*, *z*) represents the complex amplitude of the optical field. The radial component
$\u3008{S}_{\rho}\u3009=\rho z/\left({z}^{2}+{z}_{R}^{2}\right)$ for LG beams [3], with *z _{R}* denoting the Rayleigh range, and it is zero for BBs [7]. The parameter

*σ*accounts for the polarization state, taking the values of 1 or −1 for left- or right-handed circular polarization, respectively, while it vanishes for linear polarization. We will only consider here the case of linear polarization. The case of circular polarization deserves a detailed study by itself, since the energy circulation will be modified by the second term in the angular component in Eq. (1), specially in the case of nonparaxial beams. In fact, the experimental method proposed here could be useful for detecting these differences for different polarization states.

Figure 1 shows the calculated transverse energy circulation at a given *z* plane for three examples of helical light beams: (a) a Bessel beam, (b) an LG mode and (c) a helical Mathieu beam. The analytical expression of the Poynting vector for Mathieu beams looks considerably more complicated than Eq. (1), and depending on the parameters describing the beam (topological charge *l* and ellipticity), it may exhibit a single vortex of charge *l* or *l* single-charged individual vortices along the interfocal line [9].

Our aim is to determine the values of the angles (tan *α _{ρ}*) = 〈

*S*〉/〈

_{ρ}*S*〉 and (tan

_{z}*α*) = 〈

_{φ}*S*〉/〈

_{φ}*S*〉. If we look at a vertical line passing through the center of the beam, we have that

_{z}*S*=

_{ρ}*S*and

_{y}*S*=

_{φ}*S*. As we shall see in the next section, we will achieve this goal by studying the diffraction of a helical beam through a single slit and its evolution in propagation in the near-field.

_{x}## 3. Near-field diffraction of an optical vortex by a single-slit

A plane wave impinging with a small inclination on a screen with an aperture gives rise to the diffraction pattern of the aperture with a transverse shift that is proportional to the inclination angle. It is not surprising then, that the diffraction pattern produced by a vortex passing through a single slit consists of two main lobes that shift in opposite directions outwards from the slit axis [24, 25], since the helical wavefront in that case has inverse inclination in diametrically opposite points with respect to the vortex core. For helical beams, the local inclination of the wavefronts is related with the direction of the energy flux. Therefore, our idea is to calculate the local inclination of the wavefront by measuring the lateral shift of the diffraction lobes as a function of the propagation distance in the near field. For this purpose we use the experimental setup shown in Fig. 2. In order to illustrate the method, we chose the example of helical Bessel beams with *l* = 6 and *l* = 9, for which we establish a comparison among analytical, numerical and experimental results, but we also make numerical studies for LG modes and MBs. A CW laser at 532*nm* is reflected by a phase-only spatial light modulator, which displays a phase hologram of the desired BB, following the generation method described in [20]. A standard 4*f*-system of spatial filtering is used to clean the generated beam that impinges on the slit, and the diffraction pattern is observed at different planes after the slit, along the *z*-axis, with a CCD camera mounted on a micrometric translation stage.

From each of the obtained images we track the position of the intensity peaks of the two main lobes at different propagation distances. Because the maximum intensity points may shift due to reasons alien to energy circulation (such as diffraction ripples and a slight misalignment of the translation stage) we need a reference beam for comparison, with no energy circulation but with a similar intensity distribution to that of the helical beam in the region of the slit. In our experiments these are a cosine and a sine Bessel modes, for *l* = 6 and *l* = 9 respectively. In Fig. 3 we show the experimental images (green) and numerical simulations (gray-scale) of the incident beams (left-most column) and the near field diffraction patterns at different planes *z*. The numerical simulations were performed by solving the Fresnel diffraction integral at different *z*-planes with an iterative algorithm. It is easy to notice how the upper and lower lobes move in opposite directions from the slit axis over propagation for the helical beam, whereas the diffraction pattern is symmetric about the slit for the case of non-rotating beams. This is an indication of the energy circulation around the OV.

The experimental results for the positions of the intensity peaks of the upper and lower lobes of the rotating mode, calculated with respect to those of the non-rotating mode, as a function of *z* are shown in Fig. 4. The slope of the linear fit (red lines) for the shift in *x* is the value of (tan *α _{φ}*) = 〈

*S*〉/〈

_{x}*S*〉 for the portion of beam selected by the slit. The theoretical curves are shown in green for comparison. A difference in the Y-intercept of the experimental plots may occurs due to an uncertainty in the estimated position

_{z}*z*= 0, corresponding to the plane of the slit, since the optics associated with detection do not allow us to reach that point. The theoretical value of the slope is calculated in general as:

*y*

_{0}is the

*y*coordinate of the intensity maximum. In particular, for the case of paraxial Bessel beams, Eq. (2) reduces to (tan

*α*) =

_{φ}*l*/(

*ky*

_{0}), which can be also derived directly from Eq. (1). As expected due to the propagation invariance property of BBs, the shift in

*y*, related with (tan

*α*) = 〈

_{ρ}*S*〉/〈

_{y}*S*〉, is negligible.

_{z}The experimental, numerical and theoretical values of (tan *α _{φ}*) = 〈

*S*〉/〈

_{x}*S*〉 for the Bessel beams of orders 6 and 9 are summarized in Table 1. We found a fairly good agreement of the experimental and numerical values with respect to the theoretical values; differences are below 6% for the

_{z}*l*= 6 mode and below 11% for the

*l*= 9 mode. The difference in the experimental results for the upper and lower lobes might be due a minor misalignment of the slit or to a lack of perfect homogeneity of the intensity around the beam profile.

This technique can also be applied to other kinds of beams, but some further considerations must be taken into account. For circular beams, like LG and BBs, it is clear that in principle, due to the rotational symmetry, the diffraction by a single slit oriented along any arbitrary diameter provides enough information to extrapolate the energy circulation around the whole circumference. In contrast, this method has a more limited application for non-circular beams, like the elliptical MBs for instance. In that case, the orientation and position of the slit would be additional parameters to build up a full map of the energy circulation, which may become a very hard task. On the other hand, for beams that are not propagation invariant, as in the case of LG modes, the plane *z* at which the slit is placed is an important parameter.

We made a numerical analysis of a MB diffracted by a vertical slit passing through its center and of an LG mode with the slit plane at the beam-waist. We compared the numerical results (see Fig. 5 and Table 2) with the corresponding analytical results and, as in previous cases, we found good quantitative agreement. The numerical and theoretical results show a difference below 2% for a MB of order *l* = 6 and parameter of ellipticity *q* = 12 [9], and below 10% for a single-ringed LG mode (radial index *p* = 0) with *l* = 6. For the LG beam, we performed the analysis over *z* distances within less than half the Rayleigh range. In this case, as in the case of non-diffracting beams, (tan *α _{ρ}*) = 〈

*S*〉/〈

_{y}*S*〉 is negligible and hence we only observed the energy circulation, with no effects due to beam divergence. To determine whether this method can be successfully applied to propagation regions where the wave fronts of LG beams become affected by the spherical curvature would require further investigation.

_{z}## 4. Discussion and conclusions

We have presented a technique to determine, quantitatively, the energy circulation in an optical vortex by analyzing the evolution on propagation of the near-field diffraction pattern produced by a single slit. Based on a numerical and experimental study, we evaluated the method by comparing our results with theoretical calculations, and found a maximum error of about 10% or less. Among the sources of error in our results are the spatial resolution of the CCD camera in the experiments and the resolution of the numerical grid in the simulations, since for some cases the pixel size might become comparable with the relative displacement of the intensity peaks we are tracking from plane to plane. The width of the slit is an important parameter. Although a narrow slit enables the selection of a portion of beam where the energy flux has uniform direction, so that the trajectory of the lobes will be almost linear, it will produce broader lobes and stronger diffraction effects than a wide slit, making it more difficult to determine the positions of the points of interest. In our experiments the slit has a width of 0.1*mm* ≈ 188*λ* and for the case of the BB of order 6 it is
$\frac{1}{5}$ the diameter of the innermost ring (for the BB of order 9 it is
$\frac{1}{7}$). According to our numerical study, for a given set of parameters defining the beam transverse size, the larger the value of the topological charge *l*, the better the results, since the measured shifts are also larger, allowing higher accuracy. In the experiments this is not always the case, since there are other factors that can affect the results, such as slight inhomogeneities in the intensity distribution of the generated beams, which are usually larger for higher order beams. However, we want to stress that we successfully applied our method to beams that are smaller than 1*mm* in diameter, which is a beam size that cannot be analyzed with standard Shack-Hartmann sensors.

This method can be used to compare the energy circulation of different vortices, depending on their topological charge, characteristic size, total optical power or even polarization state. It could also be a straightforward way to analyze the energy flux as a function of the wavelength in the case of polichromatic vortices [37], or to determine the local energy flux in arrays of vortices. Finally, it is important to point out that, in contrast with refraction-based techniques like the Shack Hartman detector, our method can be applied not only to optical waves, but also to other kinds of waves, such as acoustic vortices [36, 38–40], electron-wave vortices [41], vortices in Bose-Einstein condensates [8] or X-ray vortices [42].

## Acknowledgments

Authors acknowledge support from DGAPA-UNAM, grant IN100110, and from CONACYT Mexico, grants 132527, 186368 (K. Volke-Sepúlveda) and 323560 (R. A. Terborg). We are also very grateful to Ms. Laura Perez-Garcia for her valuable help in the experiments.

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