APPLIED PHYSICS LETTERS 99, 204102 (2011)
Experimental verification of photon angular momentum and vorticity with radio techniques Fabrizio Tamburini,1 Elettra Mari,2 Bo Thide´,3 Cesare Barbieri,1 and Filippo Romanato1,4,a) 1
Department of Physics and Astronomy, University of Padova, via Marzolo 8, IT-35131 Padova, Italy CISAS, Centro Interdipartimentale di Studi e Attivita` Spaziali G. Colombo, University of Padova, Via Venezia 15, IT- 35131 Padova, Italy 3 Swedish Institute of Space Physics, Angstro¨m Laboratory, Box 537, SE-75121 Uppsala, Sweden 4 LaNN, Laboratory for Nanofabrication of Nanodevices, Venetonanotech, via Stati Uniti 4, IT-35100 Padova, Italy 2
(Received 21 May 2011; accepted 17 October 2011; published online 16 November 2011) The experimental evidence that radio techniques can be used for synthesizing and analyzing non-integer electromagnetic (EM) orbital angular momentum (OAM) of radiation is presented. The technique used amounts to sample, in space and time, the EM field vectors and digitally processing the data to calculate the vortex structure, the spatial phase distribution, and the OAM spectrum of the radiation. The experimental verification that OAM-carrying beams can be readily generated and exploited by using radio techniques paves the way to an entirely new paradigm of radar and C 2011 American Institute of Physics. [doi:10.1063/1.3659466] radio communication protocols. V That electromagnetic (EM) fields can carry not only energy and linear momentum but also angular momentum over very large distances has been known for over a century. However, it is less than twenty years ago that optical methods for the manipulation of light in well-defined EM orbital angular momentum (OAM) eigenstates were developed,1 thereby enabling and initiating a systematic utilization of photon OAM in science2–5 and technology,6–8 down to the single photon level. In classical electrodynamics terms, the electromagnetic field (E, B) in a volume V of free space, where the dielectricÐ permittivity is 0, carries the total linear momentum p ¼ e0 V dx3 ðE BÞ, associated with translational dynamics Ðand force action, and the total angular momentum J ¼ e0 V dx3 ðx x0 Þ ðE BÞ, associated with rotational dynamics and torque action about the moment point x0 . To leading order in distance from the source r ¼ jx x0 j, the conserved physical observables p and J are independent of r. This is why both of them can carry information over arbitrarily large distances.9 It is worth noticing that the linear momentum p from a radiation source may tend to zero at infinity, while the angular momentum J from the same source may tend to a non-zero constant there. This is typically the case of an OAM structured electromagnetic beam whose field intensity falls off sufficiently rapidly with distance from the beam axis. The total angular momentum can be written as J ¼ SþL, where the spin angular momentum S is associated with the two states of wave polarization, whereas the orbital angular momentum L, which is intimately related to beam vorticity and phase singularities, spans a state space of a denumerable infinite dimension. A pure L eigenstate is characterized by a quantized topological charge m 2 Z in the phase term expðimuÞ, where u is the azimuthal angle around the beam axis. In most cases in nature, however, the total OAM is a noninteger value a 2R. Such a non-integer OAM state can
a)
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be decomposed into a Fourier series superposition of orthogonal OAM states10 expðiauÞ ¼
1 expðipaÞ sinðpaÞ X expðimuÞ ; p am m¼1
m 2 Z: (1)
This decomposition reflects the facts that topological charges of vortices always have integer values and that each photon of the beam can take any integer value corresponding to an OAM of mh (Ref. 11). Numerical experiments have shown how low-frequency radio beams can be readily prepared in pure OAM eigenstates and their superpositions.12 The results from these numerical experiments are confirmed in the laboratory experiments reported in the present letter. We performed this laboratory experiment in the large ˚ anechoic antenna chamber of the Angstro ¨ m Laboratory of the Uppsala University, Sweden. The chamber is electromagnetically as well as acoustically/vibrationally shielded. The measurements were performed at 2.4 GHz (12.49 cm wavelength), and the source was realized with a commercial off-the-shelf (COTS) seven-element Yagi-Uda antenna for 2.4–2.5 GHz WLAN communications, fed with a continuous 0.01 W monochromatic signal from a signal generator. The radio beam was reflected off a discrete eight-step staircase phase mask that represents a discrete approximation of a non-focusing spiral reflector designed for a total 2p phase shift spiral; see Fig. 1. The reflector had N discrete jumps and a surface pitch gr (in the right-handed sense). At the wavelength k the reflected beam acquires a total OAM value evaluated as8 a¼
2gr N þ 1 : k N
(2)
For gr ¼ 6:22 cm, k ¼ 12.49 cm (m ¼ 2.40 GHz) and N ¼ 8 as in the experiments reported here, this formula predicts that the reflected radio beam had a total non-integer OAM value of a 1.12.
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To determine the phase in the far zone, two identical COTS WLAN dipole antennas, both oriented along x-direction transverse to the radio beam axis, were used in a interferometric configuration set up. One of the receiving antennas was held at a fixed position far from the singularity, Ex0, and conventionally taken at the maximum of Ex in order to maximize the signal response. The other antenna was moved around to sample the field signal at different fixed grid points. The difference of two antenna signals, DEx ¼ Ex Ex0 , has been mapped out (Fig. 2(c)) and compared with the corresponding fields difference simulation (Fig. 2(d)). The raw experimental data show the presence of a strong modulation of the DEx field characterized by a change of the sign along the direction of the step (45 ) and roughly centred on the singularity point. The experimental and simulated phase distributions (Figs 2(e) and 2(f) respectively) have been extracted from the DEx field (Figs. 2(c) and 2(d)) and the Ex field intensity map (Figs. 2(a) and 2(b)) by using a simple analytical formula (the phase u is obtained from the Ex and from the field difference maps FIG. 1. (Color online) The spiral reflector that imposes OAM to the antenna beam was made from styrofoam blocks, trimmed into a staircase-like structure with eight steps. Each staircase of the structure introduced a local, successive, discontinuous phase step of p/8 radians, except in the 45 direction where the phase step was þ2p radians.
The horizontal transverse component of the electric field of the reflected, twisted radio beam, Ex, was detected by using a single COTS electric dipole antenna for WLAN. The EX was sampled in correspondence of fixed square grid points of 3 cm lattice period, reported on a plane placed in the far zone, about 55k away from the non-focusing reflector. The spatial map of Ex (Fig. 2(a)) shows the existence of OAM spatial field distribution in the radio beam represented by a sub-wavelength hairpin shaped configuration centred around a central minimum (x ¼ 12 cm, y ¼ 9 cm). In the ideal case of a beam endowed with a pure state OAM m ¼ 61, the typical fingerprint of a field intensity is a perfect doughnutshaped intensity map. In our case the Ex field intensity distribution is slightly suppressed in the þ45 direction as an effect of the non-integer character of a and of the corresponding imperfect phase matching over the full circuit around the central minimum. We have taken into account the expected non-integer OAM value a ¼ 1:12 in the simulation of Ex map (Fig. 2(b)) obtaining a quite satisfactorily agreement with our experimental findings.
DE2 E2 E2
using the following relation, cos u ¼ x2Ex E0x0x x .) Notwithstanding some spurious signal in the experimental map, the major features of the spatial phase distribution are clearly recognizable. Because in our case the OAM state number is not too different from the pure state with a ¼ 1, many of the phase map features are similar to the phase map phase term expðiuÞ. Namely, it can be observed the clockwise increase of the phase around the singularity point that eventually results in the 2p phase step across the 45 direction, i.e., along the direction corresponding to the mask cut (see Fig. 1). As expected, the symmetry axis of the vortex structure and the phase singularity are both aligned. In the numerical far zone propagation, we have not taken in account the effects of the atmosphere turbulence as it is well known to be negligible in the radio wave propagation. Our simulations show that the monochromatic and coherent beam propagates in vacuum without any modification of the line phase singularity and without decaying into a cascade of pairs of vortex-antivortex like expected in a turbulent inhomogeneous medium. These features of phase map dependence clearly indicate the expðiauÞ electromagnetic OAM vortex structure harboured in the reflected radio beam. The OAM (spiral) spectrum of the experimental phase map has been computed13 and reported in Figs. 2(g) and 2(h)
FIG. 2. (Color online) Map (a) Experimental map of the transverse electrical field intensity obtained by probing the radio beam in a plane perpendicular to the beam axis. (b) Simulation of the field intensity. The colour scale is normalized to the maximum value measured of 7.5 lV/cm. The vortex singularity corresponds to the deep field minimum. Difference of two antenna signals: (c) experimental and (d) simulation. The maximum value measured was 9.5 lV/cm. (e) Experimental and (f) simulated phase maps obtained from the interference of two antenna signals. The phase step across the direction 45 indicates the presence of the vortex structure. For all the maps the position is expressed in cm. Histograms of OAM spectrum weight: experimental (g) and simulated (h).
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for the experimental and simulated data respectively. The OAM components m ¼ 0, m 6 1, and m 6 2 are the most relevant in particular for the experimental map, whereas the other higher OAM components (jmj ¼ 3, 4,…,10) give only minor contributions. Both the experimental and simulation OAM spectra show an asymmetric dominance towards the negative OAM components that represent an effect of the harbored vorticity, determining the direction of the phase gap rotation.6 The presence of a range of integer OAM components in the experimental spiral spectrum confirms that the radio beam was not in a pure OAM state but, in accordance with Eq. (1), was a superposition of several orthogonal states, independent of the other and all at the same frequency. Each of these orthogonal OAM eigenstates of the total non-integer OAM beam can propagate, be detected, demodulated, and characterized individually, thus enabling new methods and techniques for wireless communications based on topological diversity. The use of spiral spectra will reveal both the OAM content of an EM wave and the additional information of the spatial orientation of the source. This is a distinctive advantage over conventional experiments in which only products of EM field components (intensities, correlations, averages) are measurable. The emergence of completely new classes of fundamental as well as applied OAM-based studies in disciplines ranging from astrophysics4,7,10,14–17 to high spectral efficiency wireless communications, both classically18,19 and quantum mechanically,20,21 can therefore be envisaged. B. T. acknowledges the financial support from the Swedish Research Council (VR) and the hospitality of the University of Padova and the Nordic Institute for Theoretical Physics (NORDITA), Stockholm. F.T., E.M. and C.B. gratefully acknowledge the support of the CARIPARO Foundation within the 2006 Program of Excellence. F.T. and E.M. wish to acknowledge the help from the Vortici e Frequenze
Appl. Phys. Lett. 99, 204102 (2011)
group–Orseolo Restauri, and the kind hospitality of Uppsala University. F.R. and F.T. acknowledge the Padua University project 2011 Study of orbital angular momentum. The ˚ Angstro ¨ m Laboratory antenna chamber was funded by a grant from the Knut and Alice Wallenberg Foundation. 1
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