Survey of Techniques for Achieving Topological

Diversity A study and some future research directions

Rudra Pratap Singh

Student member, IEEE

Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati

Guwahati 781039, India

Email id: rp.singh@iitg.ac.in

P.G. Poonacha

Senior member, IEEE

International Institute of Information Technology

Bangalore

Bangalore 560100, India

Email id: poonacha.pg@iiitb.ac.in

Abstract—Twisting of Radio waves is a hot topic that is being worked upon which utilizes the many OAM (Orbital Angular Momentum) states to transmit data at the same frequency. Twisting of antennas is done to transmit the various orthogonal OAM states that can be used as basis functions in N-dimensional signaling. The utilization of various orthogonal states is termed as “Topological” diversity. The technique has been experimentally implemented and demonstrated and the idea of OAM is already well understood and used in the optical domain. In this paper, we present the techniques used to achieve topological diversity and its prospects in future Wireless Communications.

I. INTRODUCTION

Various diversity techniques such as Space Diversity, Time diversity, Frequency diversity, Polarization diversity and Angle diversity have been used have been used to transmit data efficiently. Due to an increase in the number of users and a fixed frequency range to be used for wireless communication, the world is facing the problem of frequency bandwidth allocation. Topological diversity has been proposed recently as a great potential of improving the spectral efficiency (capacity) of radio transmissions [9]. The signals are transmitted using different orthogonal Orbital Angular Momentum (OAM) states. The twisting of Radio waves which accounts for the phase front of the transmitted wave by providing independent phase states and thus additional degrees of freedom, i.e. utilizing the OAM states for increasing the data rate, is studied and discussed in the context if it could be the revolution in Multi-Gigabit data transfer. As it works for carrier signals with the same frequency, it helps in conserving bandwidth along with a high data transfer rate.

The paper is organized as follows. Section II introduces Orbital Angular Momentum and generation and detection of EM beam has been discussed. Section III discusses about Orbital Angular Momentum in Radio domain. The benefits of OAM transmission and its scope in future wireless communication have been discussed in Section IV.

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II. ORBITAL ANGULAR MOMENTUM

A. An introduction

One particular property used in modern physics, i.e. by using the properties of molecules and photons for a laser beam, is the OAM (Orbital Angular Momentum). The transfer of information encoded as OAM states of a light beam has been demonstrated in [1]. Light beams can carry both Spin Angular Momentum (SAM) and Orbital Angular Momentum (OAM) with the property of circular polarization and helicity of their phase fronts respectively [2]. The OAM states can be used as channels that are orthogonal to each other, without increasing the frequency bandwidth. So, a beam comprising of l helical phase fronts or l OAM states is described by a phase term ( ) carrying an OAM of per photon, where l is any integer value.

The quantum nature of OAM was demonstrated by Mair in [3]. Optical spin can be described in the two-dimensional basis of right and left circular polarization with an angular momentum of ±h per photon, respectively. OAM has an infinite number of eigen states, corresponding to the values of l =0, ±1, ±2… and therefore, the number of bits the OAM of a single photon can represent is unlimited. This is what makes OAM a promising parameter with the help of which information can be encoded into these various orthogonal states. Light carrying OAM can be described in the terms of Laguerre-Gaussian (LG) mode [4], which is explained in the next section.

B. Laguerre-Gaussian Modes

According to [5], in order to get our EM beam with OAM, we have to search for a solution of the Helmholtz equation with an azimuthal ( ) dependence. The commonly used expression for a Gaussian beam is

( )

( ) * (

)+

[

(

)

] *

( )+ (1)

As shown in [5], using eqn.1 as a starting point and using the Helmholtz equation, the expression for Laguerre- Gaussian beam can be obtained as,

( )

( ) [

(

)

]

* ( | | ) (

)+ *

( )+

*√ (

( ))+

| |

| |

(

( )) (2)

where ,

z = distance from the transmitting antenna, = phase, k = wave number, l = number of twists in the helical wavefront, p = number of radial nodes in the mode,

( )= beam waist, ( ) i.e. the beam width at z = 0 (will be minimal), = Rayleigh range

is a generalised Laguerre polynomial defined as

regular solution to equation given below

( )

(3)

The coefficient is obtained by requiring that very mode

transmits the same amount of power and is given by

√

( | |) (4)

Here the phase front appearance will depend on the value of l. If l=0, it will resemble the Gaussian beam. For higher values of l, the beam has phase fronts that look like intertwined spirals. When moving around the axis one rotation we will cross l number of planes because ( ) will be

satisfied l times when = . The sign of l is also important since it will result in a right hand spiral for negative l and left hand spiral for positive l. The phase front for l = 0 does not rotate at all which is the normal transmission beam. A spiral needs l wavelengths to complete a rotation. It means that the rotation of the phase fronts will be higher for lower values of l. In Fig.1 phase fronts for different values of l are plotted.

Fig.2 [10] shows how Laguerre-Gaussian laser beams are generated from planar laser beams with the help of transparent spiral phase plates which bring in a linear phase delay with

azimuthal angle. OAM state k implies a 2 k phase delay over one revolution. Three different phase plates are illustrated in gray, for OAM state 0, 1 and 2. The colored surfaces are contour surfaces indicating where the phase of the laser beams is zero.

For p=0, eqn.3 equals 1 for all ls, so the intensity of a LG mode is a ring of radius proportional to | | It is shown below that for fixed p, the following principle for orthogonality is satisfied

( ) ∫ ( ) ( )

{∫ | |

(5)

Clearly, different OAM states for fixed p are mutually orthogonal and as such they can be used as basis functions for OAM modulation. The maximum number of OAM states that can be used depends on atmospheric turbulence. For N = 2L + 1, l takes values in the set {-L,-L + 1,…,-1,0,1,…,L}. With the increase in the number of dimensions i.e., the number of OAM eigen states, the total data rate of the system can be increased.

Figure 1: Phase fronts for beams with different orbital angular momentum, reaching from orbital angular momentum zero in the upper left-hand corner to an orbital angular momentum of three in the lower right hand corner. (Taken from [5])

Figure 2: The different OAM states i.e. for l=0, l=1 and l=2 respectively (Taken from [10])

C. Generation and Detection of EM beam carrying Orbital Angular Momentum

It has been shown in [5] that for a LG beam with OAM

number l, the offset in phase is given by Generation of such kind of phase offset, each element of the array antenna has to be given an offset of some phase. To obtain the properties of the electric field from an array, we can use the array factor (AF) which depends on displacement (and the

shape of the array), phase, current amplitude and number of elements. The total field for identical antennas is then obtained by

(6)

To utilize the properties of symmetry, a multiple circular grid with equal area sectors is chosen. The location of each individual element is given by

r ( ) √ ( )

is the radius vector for the

center of each subsection (7)

( ) ( )is the angle at which the antennas are separated (8)

m denotes in which ring the element is placed in and n is the location of the selected ring. M is the total number of rings and N is the total number of elements in each ring (as given in [5]).The electric field expression is given by the equation

( )

∑ ∑ ( ( ( ( )))

(9)

∑ ∑ ( ( ( ( )))

(10)

The detection of OAM and methods involved in it are also mentioned in [7]. It suggests the methods of Antenna Arrays, Direct Torque Measurement, Holograms, Triangulation, The Lighthouse effect, Radiation patterns, Rotational Doppler Effect and The Hanbury Brown and Twiss Effect. While some experiments are suitable to be conducted in a laboratory, e.g. the direct torque measurement, the triangulation of the phases of the EM field requires large detectors.

III. ORBITAL ANGULAR MOMENTUM IN RADIO

The recent discoveries on rotating helical phase fronts and OAM of laser beams are being applied to radio domain and many ideas have been proposed. A systematic study on the same has been done in [8]. As given in [8], “it is possible to unambiguously estimate the OAM in radio beams by local measurements at a single point with the use of vector field sensing electric and magnetic tri-axial antennas, provided ideal (noiseless) conditions and that the beam axis are known”. Furthermore, it has also been shown that conventional antenna pattern optimization methods can be applied to OAM generating circular arrays to enhance their directivity.

It has also been demonstrated in [8] that the basic behavior of an OAM antenna system is similar to most other conventional antenna systems. Directivity properties can be utilized for optimized OAM detection. Also, only a local measurement is needed in order to detect OAM in a radio beam, provided that the location of the beam axis is known. In the optics literature dealing with OAM, it is shown that Laguerre–Gaussian (LG) modes carry OAM and therefore have helical phase fronts. Even when the OAM antenna array system is unable to produce pure LG beams, a rotation in the phase front is clearly visible even in radio beams which carry OAM and it has its similarities with L-G beams.

A. Radio Vorticity:An experimental setup

The real radio transmission of twisted radio beams was done by Bo Thidé and his colleagues in Venice, Italy in 2011[9]. As mentioned in their paper the experiment was setup to demonstrate that when using a given radio frequency bandwidth around a fixed carrier frequency, the infinite orthogonal OAM states can ideally provide, without increasing the frequency bandwidth, a large set of independent OAM transmission channels. This new technique can be termed as topological diversity.

In the experiment, two orthogonal OAM channels: one untwisted with OAM l=0 and the other with OAM l=1, were generated and transmitted. To transmit the l=0 OAM state, a commercial Yagi-Uda antenna with the specifications mentioned in [9] was used. For l=1 OAM state, a commercial off-axis parabolic antenna was modified in such a way so as to create a spiral twist in the antenna having phase mask reflector (a helical antenna representing the same phase plates in Fig.2). The modified antenna has been shown in the Fig.3.

Figure 3: The helical parabolic antenna (Taken from [9])

By using an interferometric phase discrimination method, the two OAM modes were separated by identifying their “phase fingerprints”. The receiving station consisted of a commercial off the shelf (COTS) FM radio module receiver fed by two identical Yagi-Uda antennae (A and B). A phase shifted cable was used to connect A and B to add output at both the ends with a phase difference of between them. To differentiate between the two OAM modes, antenna A and antenna B and the field singularity are aligned along a line parallel to the horizon, and the singularity is positioned in the middle of the segment formed by antennae A and B .For perfect alignment of the setup, the twisted EM wave with l = 1 produce an exact 180 azimuthal phase difference between the two antennae, which is subsequently compensated by the cable electric delay, thereby producing an intensity maximum. The untwisted beam, with 0 azimuthal phase difference, produces an intensity minimum for the same settings.

The EM waves are transmitted with the wavelength λ, propagating along two different paths to the receiver antennas.

There is a phase difference that depends on the angle

between the incident plane wave front and the interferometer baseline, and also on relative azimuthal term between the two

receiving antennae due to beam vorticity ( when l=0

and π when l =1). Additional phase term is added due to experimental setup. Thus the total phase difference is given by

(11)

where d is the separation between the two antennae. To find the difference in signal configuration received at A and B, a λ/2 delay cable was used. Therefore we get

| | | – |

(12)

where is the voltage measured at the antenna cable end.

A maximum is obtained when kπ and k is an integer. To receive the different OAM states, antenna A can be shifted by a quantity . Thus the phase difference can be written as

(13)

The parameter n can be adjusted to improve the tuning of the receiving system. If antenna A moves towards the source, then n will be negative. When the centre of the vortex coincides with the centre of the interferometer, the two antennae experience a phase gap due to the OAM of the EM

wave lπ and maximum of signal is obtained when

– ( ) (14)

When l =1, a maximum is achieved when n = 0 and k = 0 and hence the signal intensity will be a minimum. A maximum for l=0 mode will be obtained when corresponding to the minimum.

B. OAM in Deep Space and Near Earth Communications

The OAM modes have been used as an additional degree of freedom in interplanetary communication as mentioned in [6]. The OAM modulation and multiplexing can be used to solve the high-bandwidth requirements for future deep space and near space optical communication. Although there is a lot of interference added by the atmosphere which affects the orthogonality of OAM states, it has been shown in [6] that in combination with LDPC codes, the OAM based modulation schemes can operate even under high atmospheric interference. The spectral efficiency of the scheme proposed in [6] is also times better than that of PPM.

As mentioned in [6], (n, ) (m = 1,…,K) LDPC codes are used to encode the K different bit streams coming from different information source. The outputs of the encoders are interleaved by the K x n block interleaver. The mapper determines the corresponding -ary signal constellation point by

∑

(15)

Where M is the no. of amplitude levels per OAM state and is the normalization constant. The orthogonal basis functions are given by the set { . denotes

the modulation coordinates of the ith constellation point. The codewords for N = 3 and M = 2 are {(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)}. The scheme is quite similar to OFDM (Orthogonal Frequency Division Multiplexing) technique where the data is divided into several parallel data streams or channels, one for each sub-carrier. Each sub-carrier is modulated with a conventional modulation

scheme at a low symbol rate, maintaining total data rates similar to conventional single-carrier modulation schemes in the same bandwidth. Here, instead of OFDM, the modulation technique used is OAM which is used to create the N basis functions. Therefore the latter improves SMR sensitivity by increasing the number of OAM states and spectral efficiency and use of many OAM states does not lead to bandwidth expansion.

In analogy to Space Time Coding, where transmit diversity and time diversity is utilized, the authors of [6] have discussed about a new scheme of OAM time coding which utilizes the topological diversity discussed and time diversity. The scheme of OAM time coding has been discussed in the next section.

C. OAM Time Coding

For multiple OAM states and multiple symbol times, it can be called as OAM time codes [6]. Since Space Time Block Codes enjoy the benefit of being orthogonal, so they can be used here straightforward but they are not optimal. The OAM time coding can be modeled as

Y = H X + W (16)

where H is a N x N channel matrix, Y is a N x Ns received

matrix and X = N x Ns transmit matrix and

Y = [ ,…, ] = ( )

,

X = [ ,…, ] = ( )

,

W = [ ,…, ] = ( )

(17)

where N is the number of OAM modes and Ns is the number of symbols. For an OAM time code in which the channel matrix H is known at receiver, the optimum transmit matrix is given by the minimization according to maximum-likelihood (ML) detection.

∑ || ||

(18)

The minimization is performed for all possible OAM – time input matrices.

IV. THE FUTURE OF OAM

Although OAM provides a promising scheme to transmit at the same carrier frequency at the same bandwidth, thereby leading to huge data rate transfer in a fixed bandwidth, [10] argues that for certain array configurations in free space, traditional MIMO theory leads to eigen-modes identical to OAM states. The authors in [10] argue that OAM states created orthogonal communication channel is a subset of MIMO, and therefore it fails to offer additional gain in diversity. This aspect, however, requires further research.

In order to generate radio waves with OAM properties, the authors of [5] and [7] use antenna arrays consisting of concentric uniform circular arrays (UCAs). To increment the phase by an integral multiple of 2π, each antenna elements are fed with the same input signal, but with successive delay from element to element. The schematic used is shown in Fig.4.

It is also pointed out that the different OAM states in a beam can be decomposed by integrating the complex field vector weighted with ( ) along a circle around the beam axis and the integration operation is approximated by a discrete Fourier transform (DFT) of the field in the antenna positions. In [5] and [7], it is also concluded that with the limited number of antennas N, there will also be a limit on the largest OAM number k that such that efficient retrieval can be done, namely |k| < N/2.

Figure 4: Eight element UCA with phase rotation nk

on element n to

approximate value of k (Taken from [10])

Finally, [10] shows that OAM based radio communication can be obtained in standard MIMO theory. The authors also made a system design with the “UCA’s facing each other on the same beam axis in the space, which leads to circulant channel matrices”. The matrices obtained are diagonalized by the DFT, which means that the OAM states presented in [5] and [7] are one, not necessarily unique set of eigen-modes of these channels.

For multiple Laguerre-Gaussian modes, equivalent numbers of twisted antennas, each with a different twist, are needed to transmit orthogonal OAM signals. Thus, it would require increased setup costs as compared to the MIMO systems where most of the antennas have the same shape and

configuration. A work can be done on creating twisted antennas where the twist can be controlled dynamically so that the same antenna can be used for different L-G modes. Besides this, the receiver structure of the OAM scheme is relatively less complex for two twisted transmit antennas than compared to more than two transmit antennas. Also for distance well above Rayleigh distance, only one eigen-mode/OAM state is dominant. Hence, OAM has its restrictions for its use in long space communication.

REFERENCES

[1] Gibson G, Courtial J, Padgett MJ, Vasnetsov M, Pas’ko V, Barnett SM

and Franke- Arnold S 2004 , “Free-space information transfer using light beams carrying orbital angular momentum”, Optics Express, Vol. 12,

Issue 22, pp. 5448-5456 (2004).

[2] L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light” Progress In Optics, Vol Xxxix, 39 . pp. 291-372..

[3] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, “Entanglement of the orbital

angular momentum states of photons,” Nature 412, 313-316 (19 July 2001).

[4] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-

Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).

[5] Johan Sjöholm and Kristoffer Palmer, “Angular Momentum of Electromagnetic Radiation”, UPTEC F07 056 April 2007.

[6] Ivan B. Djordjevic, “Deep-space and near-Earth optical communications

by coded orbital angular momentum (OAM) modulation”, Optics Express, Vol. 19, Issue 15, pp. 14277-14289 (2011).

[7] H.Then, B.Thide, J.T.Mendon¸ T.D.Carozzi, J.Bergman, W.A.Baan,

S.Mohammadi, and B.Eliasson, “Detecting orbital angular momentum in radio signals”, arXiv:0805.2735 (May 2008) .

[8] Siavoush M. Mohammadi, Lars K. S. Daldorff, Jan E. S. Bergman,

Roger L. Karlsson, Bo Thide, Kamyar Forozesh and Tobia D. Carozzi, “Orbital angular momentum-A system study” , IEEE Transactions on

Antennas and Propagation, Vol. 58, No. 2, February 2010.

[9] Fabrizio Tamburini, Elettra Mari, Anna Sponselli, Filippo Romanato, Bo Thidé, Antonio Bianchini, Luca Palmieri, Carlo G. Someda, “Encoding

many channels in the same frequency through radio vorticity: first experimental test”, New J. Phys. 14 033001, 2012.

[10] Ove Edfors and Anders J Johansson, “Is orbital angular momentum (OAM) based radio communication an unexploited area?”, IEEE

Transactions on Antennas and Propagation, February 2012.