Novel measurement and approximation methodologies for studying orbital angular momentum (OAM) modes in radio beams, i.e., electromagnetic beam modes having helical phase fronts, are presented. We show that OAM modes can be unambiguously determined by measuring two electric field components at one point, or one electric field component at two points.
 It is well known that electromagnetic (EM) radiation can carry both linear momentum and angular momentum [see, e.g., Jackson 1998; Schwinger et al., 1998]. This fact has been exploited for a long time in radio systems that use polarization diversity. As predicted already by Poynting , and later verified experimentally by Beth , the polarization of light, i.e., the spin part of the angular momentum (SAM), can cause the rotation of a mechanical system. It is only recently that the orbital part of the angular momentum (OAM) has found practical use [see Gibson et al., 2004; Allen et al., 1992]. The applications to date have mostly been within the optical and to some extent microwave regimes [see Allen et al., 2003]. The basic physical properties of the EM fields are the same also in low-frequency radio, and numerical experiments performed by Thidé et al.  have shown that low-frequency radio beams generated by a circular antenna array can carry electromagnetic OAM. A comprehensive, systematic numerical study of OAM-generating circular antenna arrays was performed by Mohammadi et al. .
2. Theory and Background
 If we let E and B be the electric and magnetic vector fields, r the position vector, and ε0 the vacuum permittivity, respectively, the total angular momentum (AM) with respect to the origin is given by the volume integral,
 Polarization is the classical manifestation of the quantum mechanical (photon) spin and we will therefore refer to the intrinsic, polarization-dependent, rotation as spin angular momentum. If we introduce the SAM mode number s, then s = ±1 corresponds to right- and left-hand circularly polarized modes, respectively; s = 0 correspond to a linearly polarized mode.
 The extrinsic rotation is referred to as orbital angular momentum with mode numbers denoted by l. The parameters l and s can be both positive and negative integers. The angular momentum is the composition of OAM and SAM such that the angular momentum mode number j = l + s. In this paper we study OAM modes only, and will therefore consider linearly polarized fields, i.e., s = 0 and j = l.
 Choosing the z axis along the beam-axis direction, the angular momentum mode number can be explicitly calculated by normalizing the AM with respect to the beam energy Jackson :
In order to calculate the angular momentum mode explicitly, we need all three components of the electric and magnetic fields,
Note that j is only an integer for pure OAM and SAM modes.
where N ≫ 1 is the number of dipoles, D is the diameter of the array, and k = 2π/λ is the angular wave number. We see then that we indeed have an azimuthal phase dependency in equation (5). However, whereas the Laguerre-Gaussian beams create pure OAM states [Allen et al. 2003], the radio beams considered here have no pure states, but fluctuate around the intended OAM value as a function of position in the beam. Because of this, we will henceforth also call the OAM modes generated in this way “phase modes” [see Chireix, 1936; Knudsen, 1953; Josefsson and Persson, 2006]. In addition, as will be shown later, the side lobes have, or appear to have, a completely different OAM state than the main lobe.
 These modes correspond to the dominant OAM mode in the radio beams, but to avoid confusion we will still use the mode number l to indicate both the intended phase mode and the numerically estimated mode. Note that when an OAM beam is endowed with multiple OAM modes, the numerically estimated mode, according to equation (2), will be noninteger.
 Since the phase structure can be characterized by its phase gradient, a beam OAM state can be implicitly measured by estimating this quantity, a method that is significantly simpler than the explicit measurement of the full 3-D field as required by equation (2).
3. Simulation Method
 We have produced our simulated data (EM fields) by using the Maxwell equation solver NEC2++ and in-house software for simulation data preprocessing, but our results should be reproducible with most electromagnetic simulation tools. The calculations were performed in cubic 3-D volumes, sliced such that the z axis becomes the normal to the intersecting xy sampling planes, defined by a Cartesian coordinate system; see Figure 1. In each plane, the OAM (phase mode), is calculated from the 3-D electric and magnetic fields according to equation (2). This method of estimating the angular momentum, i.e., to calculate the OAM in intersecting xy planes and sum them over a limited volume, was also used by Berry .
 OAM-carrying fields can be generated by a plane, phased, circular antenna array where the N array elements, distributed equidistantly around the periphery of the circle, are phased such that the phase difference between each element is δϕ = 2πl/N. In our simulations, we have chosen to orient the individual dipoles such that they all point in the same direction. The array then defines the xy plane with the z axis normal to this plane as shown in Figure 1.
 The radiation patterns of OAM-carrying beams are characterized by an intensity null along the beam axis, which can be understood mathematically from the Bessel function occurring in the array factor; see equation (5). As expected, the radiation pattern widens for higher OAM modes; see Figure 2. For larger array diameter the beams become narrower but sidelobes develop; see Figure 3.
 In the simulations, conventional far-zone approximations of the fields cannot be applied before the numerical estimations of j has been performed since for transverse EM fields, r · E⊥ = 0 and r · B⊥ = 0
In Figure 4 we see that OAM is indeed carried all the way out to the far zone.
4. OAM Far-Field Approximation
 Since the total angular momentum is conserved, it will be transported to infinity, but as indicated by equation (6) the far-field approximation must be reconsidered in this case. From equation (4), we see that not all of the field components need to be estimated. Under optimum circumstances, we can simplify the measurement even further by making use of a far-field approximation suitable for OAM modes. This is not the same as for the intensity (Poynting) flux, where, when applying the far-field approximation, only the 1/r terms are kept, which means that the radial part of the EM field is discarded. Since OAM is produced by the interaction of the far-zone components, ��(1/r), and intermediate-zone components, ��(1/r2), the latter components must not be discarded.
 Let us approximate the radial field component with the z component in Cartesian coordinates, such that k ≈ k. This is valid as long as we measure with a sufficiently small angle with respect to the center of the generating OAM circular array and the beam axis. Then the electric field can be approximated by
and the magnetic field by using the Fourier transformed Faraday's law (k × E = ωB)
i.e., we need to measure only two components of the electric field to determine the OAM mode of a radio beam along the z axis.
 In Figure 4 we see the calculated behavior of the OAM modes, evaluated using equation (2) with Jz computed using the complete field (equation (3)), and the approximation (equation (9)). If we consider the estimate of the OAM mode using the complete fields (dashed lines in Figure 4), we find that the point at which an integer mode is achieved shifts toward higher z values with increasing array diameter. However, with increasing vertical distance from the array the approximation of the OAM mode (solid lines in Figure 4) converges faster toward the exact OAM mode as the array dimensions increase, and thus the estimate of the point where a pure OAM mode is achieved possesses a much smaller shift toward higher z values.
5. Measurement Methods
 In this section we present two different methods for estimating the OAM of a radio beam:
 1. The first method is the single-point method, which estimates the magnetic field from the electric field using the OAM far-field approximation, such that equation (9) can be used to calculate the particular OAM mode of a beam at each point in space.
 2. The second method is the phase gradient method, which explicitly utilizes the helical phase structure and estimates the OAM mode by measuring the phase gradient, by approximating it by a two-point phase measurement in the xy plane; see Figure 7.
5.1. Single-Point Method
 The single-point method or the far-field approximation method (case 1 above) utilizes a two-component field sampling of the electric field, one vertical and one transverse. The transverse component must point such that the polarization matches the polarization of the field. If we once again consider a circular array with electric dipoles pointing in the y direction, then we measure Ey and Ez and approximate the Bx component of the magnetic field as −Ey/c. In Figure 4 the OAM mode has been calculated using the full E and B fields (dashed lines), and with the approximation method just described (solid lines).
 Clearly, the approximation method works better for collimated beams. Notice that in the plot labeled D = 6λ in Figure 4, the red lines (approximation method) stabilize closer to the vertical distance from the array at which the full 3-D sampling stabilizes (blue lines) than in the case of the smaller D cases.
 We also see that the approximation method works well globally, or on average, e.g., when integrating over many sample points (as used to compute the red lines). Unfortunately, the method does not work sufficiently well locally, i.e., when unaveraged, e.g., when computed for a single point. In Figure 5, we see the local behavior of the OAM mode, calculated with full 3-D field information. The corresponding behavior for the approximation method is not shown. It behaves well in a region of a few λ in size from the beam axis, but fails further away from the beam axis.
5.2. Phase Gradient Method
 As already mentioned, a distinct characteristic of the OAM-carrying beams is their helical phase fronts. The number of branch cuts of the helix where the phase jumps from π to −π radians depends on the OAM mode of the beam. A beam with an OAM mode l = 1 has one branch cut, l = 2 has two, and so forth. This can be seen in Figure 6. Since the phase front of the beam exhibits such a distinct behavior, it is possible to determine the OAM mode by analyzing the phase change, i.e., the phase gradient [cf. Berkhout and Beijersbergen, 2008]. An approximation to the phase gradient would be to measure the phase difference between two points on a circle or a circle segment with the circle center on the beam axis. This method works regardless of how the dipoles are oriented, but if they are not parallel, the geometric angles between them also must be considered. A similar method, were two points were sampled on a circle, was used as an OAM filter for optical interferometers by Elias .
 The method is illustrated in Figure 7. That is, the phase samples are ϕ1electric, ϕ2electric, and the angle of the circle segment is, β, then the estimated OAM mode can be calculated from
In Table 1 this method is applied to the OAM generating array of 16 electrically short antennas, equidistantly spaced in a circle with array diameter D = 6λ, generating OAM modes l = 1, 2, 3, 4, 5 and 6.
Table 1. Estimated OAM Mode, by a Two-Point Sample of the Electric Phase Gradient δϕelectric = ϕ1electric − ϕ2electric in a Plane 200λ From the Antenna Arraya
Relative Error (%)
Both points are at a radius of 20λ from the beam axis, separated by an angle of β ≈ 2.87. The array consists of 16 electrically short antennas, equidistantly spaced in a circle with array diameter D = 6λ, generating OAM modes l = 1, 2, 3, 4, 5, and 6. The relative error is given by ∣lideal − lestimated∣/lideal.
l = 1
l = 2
l = 3
l = 4
l = 5
l = 6
 The OAM mode is estimated by measuring only the phase of the y component of the electric field at two points on a circle in a plane at a vertical distance, 200λ away from the antenna array. The points are on a circle sector with a radius of approximately 20λ and an angle of approximately 2.9° with respect to the beam axis; see Table 1. Note that the specific values have only been chosen as an example. As long as the phase singularity at the beam null is avoided, most clearly visible in Figure 6 for D = λ and l = 6, this method of estimating the OAM mode in a radio beam is particularly effective and straightforward. Care must be exercised when choosing the angle of the circle sector, since this angle will restrict the measurability of the highest OAM mode.
 Consider a circle sector of 180°, spanned by only two sample points. As follows from the Nyquist sampling theorem, the measured phase difference of beams carrying positive odd OAM modes, l = 1, 3, 5, 7,…, would then ideally be 180°, and the measured phase difference of positive even modes, l = 2, 4, 5, 6,…, will all be 360°. Hence it will be impossible to distinguish precisely which OAM mode the radio beam carries, only whether it is odd or even. If we would also consider negative OAM modes, nothing could be said except that the beam possibly carries OAM. In order to estimate the maximum OAM mode l, the sampling antennas must be placed in a circle sector with an angle β such that
 The number of antennas on the circle or circle sector will impact the reliability of the measurement. More antennas (or spatial samples) would mean better statistics and smaller errors. This will, however, not be discussed further in this paper.
 Notably, the sidelobes of the array appear to carry a different OAM mode than the main lobe. This effect is observable both in Figures 5 and 6. Considering the phase plots and the local OAM plots visualizing data from OAM-generating arrays with diameter larger than λ and comparing them to the radiation patterns, we see a correlation between the sidelobes, the locally estimated OAM mode, and the phase structure. The plots in Figure 5 are a local estimate of the OAM mode at a distance of 25λ from the antenna array. The close distance to the antenna array results in a wide simulation field of view, thus encompassing a number of side lobes and illustrating that the sidelobes can appear to carry a different dominating phase mode than the main lobe, resulting in these kaleidoscope-like structures. This may be real, or could be the result of the off-perpendicular (in relation to the sidelobes) calculations used in these simulations. At larger distances, if the same sample area is used, the sidelobes will be excluded from the cubical simulation box and only the main lobe will remain inside it; see Figure 8. Note also that there is a strong correlation between the integrated OAM number and the local behavior. The simulation volume is a rectangular box, and each integrated value is proportional to the sum of all local OAM values in the sample area. Hence the integrated OAM number curve levels out when the sample area is capturing most of the main lobe; see Figures 4 and 8.
6. Application to Half-Wave Dipoles
 Although the orientation of individual finite dipoles in an antenna array may be the same as in an identical array composed of short dipoles, the application to, for example, half-wave (λ/2) dipoles is not without complications. The finite shape and size of λ/2 dipoles introduces an asymmetry in the antenna array which is reflected in the beam pattern, in the local OAM modes, and to some extent in the phase structure. The complication manifests itself as asymmetries in the radiation pattern, which gives rise to strong variations in the local estimated OAM mode; see Figure 9.
 In essence, when using λ/2 dipoles, care must be exercised to ensure that the array is able to create the desired mode. For instance, from our simulations we have observed that a larger number of dipoles is needed than the theoretical lower limit of ∣l∣ ≥ N/2. Our conclusion is that one should at least use two to four dipoles more than this limit. Although this may not be a general result, it has been tested for dipole arrays with up to sixteen λ/2 dipoles.
 From Figure 10, we see that the approximation calculation takes much longer (in terms of vertical distance from the array) to stabilize and become useful than in the short dipole case. Note that the λ/2 dipoles can be replaced by almost any other type of radiator. Each radiator type will, however, create different results with respect to the distance required for the integrated OAM mode to stabilize. As shown by Josefsson and Persson , the directive properties of the radiating elements influence the radiation pattern.
6.1. Measuring Two Electric Components at a Single Point
 The plots in Figure 10 show that the single-point method fails to estimate the correct OAM mode unless the diameter of the antenna array is significantly larger than the size of the antenna elements. In many cases, particularly for higher OAM modes, the OAM approximation method yields the wrong OAM mode, and hence is not a reliable method when applied to higher-order OAM beams generated by λ/2 dipoles. However, the method can be used under certain circumstances. If the beam in question is highly collimated, such that the vertical component is a good approximation of the radial component, and if the beam carries an OAM mode with a small ∣l∣, the method can be used with some caution.
6.2. Measuring Phase Gradients
 The phase structure is surprisingly resilient to the asymmetry created by λ/2 dipoles, in contrast to the local OAM estimate; see Figure 11. Also notable is that the diameter of the antenna array must be larger than λ. When the diameter is very large compared to the size of the radiator, the asymmetry presented by the dipoles becomes less significant and hence a more symmetrical beam appears.
 The phase gradient method has been applied to a 10-element λ/2 dipole, circular array with diameter D = 5λ. Just as for the electrically short dipoles, the phase gradient is estimated by considering the phase at two points spanning a circle sector with approximate radius of 20λ and an angle of β ≈ 2.9°, in a plane perpendicular to the beam axis, at a vertical distance of 200λ. As can be seen in Table 2, the method works well with a relative error (∣lideal − lestimated∣/∣lideal) of 10%. Had a smaller β been chosen, the relative error would have been lower. However, if the sampling antennas are placed too close to each other there will be antenna coupling issues that will degrade the phase measurements. For a given measurement system, using this method, the maximum measurable OAM mode is thus given by the maximum phase error and the Nyquist rate.
Table 2. OAM Mode, Estimated by Performing a Two-Point Sample of the Electric Phase of the Ey Component at a Plane 200λ From the Antenna Arraya
Relative Error (%)
Both points are at a radius of 20λ from the beam axis, separated by an angle of β ≈ 2.87. The array consists of 10 λ/2 dipoles, equidistantly spaced along a circle with array diameter D = 5λ, generating OAM modes l = 1, 2, 3, 4. The relative error is to ∣lideal − lestimated∣/lideal.
l = 1
l = 2
l = 3
l = 4
 We have presented and investigated numerically two different methods for measuring the orbital angular momentum in radio beams. Both of these require two samples, but in different geometries, and a known beam axis. In both cases the given method is significantly simpler than if one would measure the full E and B fields.
 The single point method, based on a single-point measurement of two electric field components, works well for symmetrical fields produced by electrically short dipoles. It does not perform as well when the fields are generated by λ/2 dipoles. In general, the single point method works best on fields with low OAM mode numbers.
 The phase gradient method, which measures one component of the electric field at two separated points and approximates the gradient of the phase of that component, works well both for fields generated by small dipoles as well as for fields generated by λ/2 dipoles. The errors of this method are insignificant and compare well with the calculation of the OAM state using the complete EM field.
 Although Figure 11 suggests that the OAM values in the sidelobes differ from that of the main lobe, it is possible that the OAM of the sidelobes is in fact the same as in the main lobe. This possibility exists because the side lobes and the main lobe are cut at different angles by the sample volume, which again in our case has square sides parallel to the plane of the antenna array. According to Berry , a cylindrical OAM beam sampled in this way will yield the correct OAM number. However, our beam is not cylindrical, but rather conical. Hence the correct geometry for sampling each lobe would be to sample in a volume not parallel to the array but perpendicular to the particular propagation axis of the sampled lobe [Elias, 2008].
 There are numerous possible radar and RF experiments utilizing these measurement methods. For instance, if an ionospheric radar or high-frequency transmitter has sufficient number of receivers it would be possible to study the phase structure of the backscattered radiation and compare with simulation results to study OAM properties of the ionosphere. Experiments have been performed using the HAARP HF transmitter in Alaska in which OAM beams were generated and used to produce stimulated electromagnetic emissions (SEE), but the phase structure of the emissions and the backscatter was not studied [Leyser et al., 2009]. More recently, the Jicamarca 50 MHz radar was used to generate pulsed phase modes and to receive backscatter from the equatorial electrojet for comparison with standard measurements without phase modes (J. Chau, private communication, 2009).
 We thank Anders Ahlén, T. B. Leyser, and Erik Nordblad for useful comments and discussion.