The recent years have witnessed a growing interest in the possibility of enhancing the information transfer per unit bandwidth by exploiting the Orbital Angular Momentum (OAM) of light in both free-space and guided scenarios. In the proposed paper, the propagation of suitable OAM superpositions of waveguide eigenmodes in guiding structures with a 
circular symmetry is analyzed both theoretically and with the aid of the software COMSOL Multiphysics®, leading to the estimation of the power attenuation constants due to the finite conductivity of the metallic guide walls

Andrea Cagliero

Assunta De Vita

Bruno Sacco

Rossella Gaffoglio

English

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEModeling Orbital Angular Momentum (OAM) Transmission in Waveguides with the COMSOL Multiphysics® Software /Cagliero, Andrea; De Vita, Assunta; Gaffoglio, Rossella; Sacco, Bruno. - ELETTRONICO. - (2016). ((Interventopresentato al convegno COMSOL Conference 2016 tenutosi a Munich nel 12-14 October 2016.OriginalModeling Orbital Angular Momentum (OAM) Transmission in Waveguides with the COMSOLMultiphysics® SoftwarePublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2731420 since: 2019-04-23T18:35:29ZCOMSOL MultiphysicsModeling OAM Transmission in Waveguides with COMSOL  Multiphysics  A. Cagliero1*, A. De Vita2, R. Gaffoglio1 and B. Sacco2  1 Department of Physics, University of Torino, Torino, Italy 2 Centre for Research and Technological Innovation, RAI Radiotelevisione Italiana, Torino, Italy * Corresponding author:  andrea.cagliero@unito.it   Abstract: The recent years have witnessed a growing interest in the possibility of enhancing the information transfer per unit bandwidth by exploiting the Orbital Angular Momentum (OAM) of light in both free-space and guided scenarios. In the proposed paper, the propagation of suitable OAM superpositions of waveguide eigenmodes in guiding structures with a circular symmetry is analyzed both theoretically and with the aid of the software COMSOL Multiphysics®, leading to the estimation of the power attenua-tion constants due to the finite conductivity of the metallic guide walls.  Keywords: Circular waveguide; coaxial cable; COMSOL Multiphysics; Orbital Angular Momen-tum (OAM).  1. Introduction  As emphasized in literature [1], the free-space propagation of OAM beams is affected by an over-quadratic power decay with distance. However, this negative aspect does not concern guided propagation of waves carrying OAM, which is instead free from this limitation. This fact has favored the interest towards the possibility of exploiting the OAM in waveguide scenarios, where vortex beams can be thought as suitable superpositions of the standard Transverse Electric (TE) or Transverse Magnetic (TM) modes supported by the considered guiding structures [2].  In Optics, the possibility to excite OAM modes in fibers has been experimentally verified first in particular kind of fibers [3,4], with ring-shaped cores, and then in conventional optical fibers [5], paving the way to the realization of OAM-based multiplexing schemes [5,6]. The guided propagation of OAM modes has been also considered at the radio frequencies and in the microwave band, since waveguides are often part of radiating systems [7,8]; as an example, this is the case of the horn antennas, which consist of a flared metal waveguide shaped like a horn [9]. According to the considerations reported in [2], in hollow metallic waveguides with a circular symmetry, trivial combinations of the conventional eigenmodes are able to carry angular momentum, including both an OAM and a SAM (Spin Angular Momentum) contribution. In this work, the propagation of these modes in circular waveguides and coaxial cables will be investigated both theoretically and also by means of numerical simulations performed with the RF module of the software COMSOL Multiphysics. The presented analysis will explicitly prove how such OAM superpositions can be supported by the considered guiding structures at frequencies greater than the corresponding cutoff values. Then, the power attenuation law relative to the propagation of different OAM modes inside a circular waveguide with copper walls will be presented, as a result of a realistic electro-magnetic simulation.   2. Circular waveguide  Any perturbation inside a circular waveguide with metallic walls can be described in terms of TE and TM modes, properly obtained by solving the Maxwell's macroscopic equations with the boundary conditions dictated by the waveguide geometry. Therefore, these electromagnetic modes, whose mathematical expression is derived in [10,11], constitute a complete set of fields inside the circular waveguide.  In this section the TE modes for a circular waveguide with uniform cross section and perfectly conducting walls are explicitly reported. Then, it will be shown how to obtain propagating modes carrying OAM by superimposing pairwise such conventional eigenmodes. By definition, a transverse electric mode has no electric field in the direction of propagation, i.e., Ez = 0. According to [11], the transverse  COMSOL Multiphysics and COMSOL are either registered trademarks or trademarks of COMSOL AB. Microsoft is a registered trademark of Microsoft Corporation in the United States and/or other countries. electric field components of a TE mode propa-gating inside a circular waveguide of radius a can be written as:  where the harmonic time dependence has been omitted, B0 is a constant with magnetic field di-mensions, A and B are arbitrary adimensional constants, while γnj = ζnj / a, being ζn1, ζn2, ..., ζnj the non-null positive zeros of the function J'n(ζ), arranged by increasing magnitude. The indices n and j indicate a transverse electric mode propa-gating inside the waveguide with a wavenumber αnj given by:    being εr and μr the relative permittivity and per-meability of the interior medium. It should be noted that such mode can exists inside the wave-guide if and only if the operating frequency ω is greater than the corresponding cutoff frequency εµγ /nj , where ε and μ are the absolute permit-tivity and permeability. Since the lowest zero of the first derivative of the Bessel function Jn is ζ11, it follows that the TE mode which can prop-agate inside the waveguide at the lowest fre-quency is the TE11.  The two set of modes TEcnj, with A ≠ 0 and  B = 0, and TEsnj, with A = 0 and B ≠ 0, complete-ly span the orthogonal set of transverse electric modes inside the waveguide. By redefining these modes in such a way that the parameters A and B fulfill A = iB or A = -iB, the following new com-plete set of orthogonal modes can be obtained [2]:   Using the expressions for the TE modes re-ported in (1) and (2), the electric field compo-nents relative to the superpositions defined in (4) can be explicitly written as:  where K± is a new adimensional constant.  In [2] it has been demonstrated that the modes defined in (5),(6) carry a net angular mo-mentum which is, on average, ±nћ per photon. This quantity includes both an OAM and a SAM (Spin Angular Momentum) contribution. It is in-deed impossible for an electromagnetic mode to propagate inside the waveguide carrying all its angular momentum either in the form of spin or OAM. The combinations defined in (4) represent nothing but a particular basis choice in the space of the waveguide eigenmodes and the mode den-sity of the two sets is the same [2]. In order to visualize these modes and to prove their propagation in a circular waveguide, the analytic expression of the electric field (5),(6) is provided in input to a circular wave-guide modeled with the RF module of the soft-ware COMSOL Multiphysics (see Fig. 1). The ra-dius of the circular waveguide is assumed to be a = 0.02 m, the interior region is filled with air and a Perfect Electric Conductor (PEC) condition is imposed on the walls. Fig. 2 and 3 show the in-tensity profiles of two superpositions like (4), with j = 1 and n = 0, 1, at an operating frequency f = 10 GHz, sufficient to allow the propagation of these modes inside the considered waveguide. It is worth noting that the above presented approach could be extended to the TM modes, for which superpositions carrying angular mo-mentum are defined in complete analogy to (4).  ( )( ) ( )[ ] )1(sincos002zijnnjnjnenBnAJBnckiEαρϕϕργργ−+−⋅⋅=( )( ) ( )[ ] )2(cossin'00zijnnjnjnenBnAJcBkiEαϕϕϕργγ−+⋅⋅=)4(.TETE snjcnj i±( ) )5(002 ziinjnnjnjneeJKBnckiE αϕρ ργργ−±−=m( ) )6('1 00 ziinjnnnjjneeJKcBkE αϕϕ ργγ−±=mm)3(,220 njrrnj k γµεα −= Figure 1. Circular waveguide oriented along the z-axis and simulated with the software COMSOLMultiphysics.  3. Coaxial cable   A coaxial cable is made by an inner conduct-ing wire sequentially surrounded by a tubular insulating layer and an external metallic shield. This kind of structure can be viewed as a type of waveguide able to support the propagation of the Transverse Electric and Magnetic (TEM) mode (neither electric nor magnetic field in the direc-tion of propagation), by virtue of the presence of two conductors. When it exists, like in a coaxial cable,  the TEM  mode  is  the  fundamental  one,  order TE (or TM) modes, whose formulation for the coaxial cable will be introduced in the fol-lowing.  According to the approach presented in [11], the electric field components of a TE mode propagating inside a coaxial cable with outer ra-dius a and inner radius b can be written as:    having no cutoff frequency. Despite this, in order to obtain modes able to carry angular momen-tum,  it  is necessary  to resort to  the higher-  Figure 3. Normalized electric field intensity for the superposition TEc11 -  iTEs11 evaluated with the software COMSOL Multiphysics in a circular waveguide of radius a = 0.02 m.      Figure 2. Normalized electric field intensity for the superposition TEc01 - iTEs01 evaluated with the software COMSOL Multiphysics in a circular waveguide of radius a = 0.02 m. The real part of the vector field over the middle transverse cross section is also reported. ( ) ( )[ ( )( )] ( ) ( )[ ] )7(sincos''002zijnnjnnjnnjnnjnjnenBnAYaJJaYBnckiEαρϕϕρχχρχχρχ−+−⋅⋅−=     where B0 is a constant with the dimensions of a magnetic field, A, B and K are adimensional con-stants, while χn1, χn2, ... , χnj are the solutions, sorted by increasing magnitude, of the following transcendental equation:  A frequency ω greater than the cutoff value εµχ /nj  allows the propagation of the TEnj mode inside the coaxial cable, with a wavenum-ber:    Following the same approach presented in the previous section for the circular waveguide, the TE components (7), (8), relative to the coax-ial cable, can be superimposed pairwise accord-ing to the combination defined in (4), leading to:  The propagation of these modes can be veri-fied with the aid of the software COMSOL Mul-tiphysics by properly feeding the input port of a coaxial cable (Fig. 4) with the considered super-positions of TE modes. The inner and outer con-ductors of the simulated coaxial cable are mod-eled with a Perfect Electric Conductor condition imposed on their surfaces, while the inner region is filled with a Teflon dielectric layer (εr = 2.1, μr = 1). Moreover, the radii of the two conductors are assumed to be: a = 0.016 m and b = 0.005 m. Fig. 5 and 6 show the electric field intensity for the combinations TEcnj - iTEsnj, with j = 1 and n = 0, 1, propagating inside the considered coax-ial cable at an operating frequency f = 10 GHz.         ( ) ( ) ( ) ( ) )9(.0'''' =− aYbJbYaJ nnnn χχχχ)10(.220 njrrnj k χµεα −=( ) ( )[ ( )( )] ( ) ( )[ ] )8(cossin''''00zijnnjnnjnnjnnjnjnenBnAYaJJaYcBkiEαϕϕϕρχχρχχχ−+⋅⋅−=  Figure 5. Normalized electric field intensity for the n = 0 TE superposition evaluated with the software COMSOL Multiphysics in a coaxial cable filled with Teflon. The real part of the vector field over the middle transverse cross section is also reported.   Figure 4. Coaxial cable oriented along the z-axis and simulated with the software COMSOL Multiphysics.  ( ) ( )[( ) ( )] )11(''002ziinjnnjnnjnnjnnjnjneeYaJJaYKBnckiEαϕρρχχρχχρχ−±−+−=m( ) ( )[( ) ( )] )12(''''100ziinjnnjnnjnnjnnjnjneeYaJJaYKcBkEαϕϕρχχρχχχ−±−+=mm   4. Power attenuation law  The waveguides considered so far have been ideally supposed to be made of perfect conduc-tors. Inside this kind of waveguides the electro-magnetic modes can propagate above the cutoff without suffering any attenuation. However, in practice, perfect conductors do not exist, but rather there are good conducting materials, such as metals, with finite conductivities on the order of 107-108 S/m.  These great, but limited, values of the metal conductivity causes the field to penetrate a little the waveguide walls; as a con-sequence, the conduction electrons of the metal layer are set into motion, dissipating a part of the energy via the Joule effect. Hence, in real waveguides with metallic walls, the supported electromagnetic modes undergo some attenua-tion due to the conduction (ohmic) losses. More-over, if the considered waveguide is filled with a lossy dielectric material, the so-called dielectric losses introduce a further contribution to the waveguide attenuation. Anyway, restricting the present analysis to the case of waveguides filled with air, such kind of losses can be neglected.  If P0 represents the power associated to a mode with indices (n, j) at some reference point along the guiding direction (i.e., z = 0), then the power P at some other point z is related to P0 by:        where βnj is the ohmic losses attenuation coeffi-cient. In order to estimate the attenuation coeffi-cient relative to the propagation of a superposi-tion TEcnj ± iTEsnj inside a real guiding structure, a circular waveguide with copper walls is simu-lated with the software COMSOL Multiphysics. The modeled waveguide is filled with air, the radius is assumed to be a = 0.02 m and the oper-ating frequency f is fixed at 10 GHz. By means of a parametric sweep, COMSOL provides the es-timation of the S21=P(z)/P0 parameter for differ-ent values of the waveguide length z, between 2λ and 102λ, being λ the wavelength. Fig. 7 shows the S21 parameters as a function of z in logarith-mic scale, for three different TE mode superposi-tions with indices j = 1 and n = 0, 1, 2. For each of the considered cases a linear fit is performed enabling the estimation of the corresponding power attenuation constants. The same procedure is then repeated for the propagation of single TEnj modes, with j =  1 and n = 0, 1, 2. As can be seen from Table 1, the at-tenuation coefficients estimated for the TE modes coincide with those obtained for the OAM superpositions. As expected, this result shows that the waveguide mode combinations able to carry angular momentum exhibit the same behaviour in terms of power budget as the conventional modes.      Figure 6. Normalized electric field intensity for the n = 1 TE superposition evaluated with the software COMSOL Multiphysics in a coaxial cable filled with Teflon. ( ) )13(,20 zjnePzP β−=                      Finally, in order to provide a further theoreti-cal confirmation of the obtained results, a for-mula for the estimation of the attenuation coeffi-cient in a realistic circular waveguide is consid-ered. According to the approach reported in [12], for a circular waveguide of radius a, the power attenuation coefficient relative to the propagation of a TEnj mode can be evaluated theoretically via the following formula:  where σ is the metal conductivity, εµη /=  is the impedance of the inner medium, ζnj is the TE mode eigenvalue, ω = 2pi f is the operating angu-lar frequency, while fnj is the cutoff frequency associated to the considered mode and defined by:    It should be noted that, for the considered circular waveguide filled with air and surrounded by copper walls, σ = 5.7 × 107 S/m, ε = ε0,  μ = μ0                          and  η  corresponds  to   the  vacuum  impedance η0 ≈ 377 Ω. The results reported in Table 1 clearly show that a very good agreement between the theoretical and numerical results has been achieved.  5. Conclusions  In this work, the propagation of TE superposi-tions carrying angular momentum in two possi-ble guided scenarios has been investigated with the help of COMSOL Multiphysics. It is important to emphasize that the possibility to excite such modes in RF waveguides is of paramount impor-tance not only for guided transmissions, but  also      Figure 7. S21 parameter as a function of the guide length z for a circular waveguide with copper walls and radius a = 0.02 m. In the legend the results of the corresponding linear fit: S21 = -2βnj + c, with c ≈ 0, are also reported. )14(,1222222 −+−=nnffffanjccTEnj ζησωµβ)15(.2 εµpiζaf njnj =Table 1:   Ohmic losses attenuation coefficient for a circular waveguide with radius a = 0.02 m and copper walls.   βn1  (dB/m)  n = 0  n = 1  n = 2  COMSOL (OAM)  COMSOL (TE)  Th. formula (14)  0.0305  0.0305  0.0313  0.0101  0.0101  0.0103  0.0277  0.0277  0.0284  for more general applications, since most of the antennas and the RF systems are fed via waveguides and coaxial cables. Although it is known that the above defined OAM modes do not increase the waveguide mode density and present analogous propagation properties as the standard TE modes (see Table 1), they actually represent a natural basis set for circular symmet-ric geometries. Therefore, it is reasonable to ex-pect that they might bring technological simplifi-cations in all contexts were the circular symme-try is preserved.    6. References  [1] M. Tamagnone et al., “Comment on ‘Reply to Comment on “Encoding many channels on the same frequency through radio vorticity: first ex-perimental test”’”, New Journal of Physics 15 (2013). [2] E. Berglind and G. Björk, “Humblet's De-composition of the Electromagnetic Angular Moment in Metallic Waveguides”, IEEE Trans-actions on Microwave Theory and Techniques 62, 779-788 (2014). [3] N. Bozinovic et al., “Control of orbital angu-lar momentum of light with optical fibers”, Op-tics Letters 37, 2451-2453 (2012). [4] S. Ramachandran and P. Kristensen, “Optical vortices in fiber”, Nanophotonics 2, 455-474 (2013). [5] H. Huang et al., “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre”, Scientific Reports 5 (2015). [6] N. Bozinovic et al., “Terabit-Scale Orbital Angular Momentum Mode Division Multiplex-ing in Fibers”, Science 340, 1545-1548 (2013). [7] D. Zhang et al., “Generating in-plane optical orbital angular momentum beams with silicon waveguides”, IEEE Photonics Journal 5 (2013). [8] W. L. Wei et al., “Two monopole antennas for generating radio OAM waves in circular waveguide”, 10th European Conference on An-tennas and Propagation (EuCAP 2016). [9] COMSOL Multiphysics 5.1, RF Module, http://www.comsol.com (2015). [10] J. D. Jackson, Classical Electrodynamics, third edition, Wiley (1999). [11] E. Botta and T. Bressani, Elementi di elet-tromagnetismo avanzato, Aracne (2009). [12] S. J. Orfanidis, Electromagnetic Waves and Antennas, (2014) [Online]. Available: http://www.ece.rutgers.edu/~orfanidi/ewa [13] C. A. Balanis, Advanced Engineering Elec-tromagnetics, Wiley, New York (1989).    

Modeling Orbital Angular Momentum (OAM) Transmission in Waveguides with the COMSOL Multiphysics® Software

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Modeling Orbital Angular Momentum (OAM) Transmission in Waveguides with the COMSOL Multiphysics® Software

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