We analyze the behaviour of TE, TM electromagnetic fields in a toroidal space through Maxwell and wave equations. Their solutions are discussed in a space endowed with a refractive index making separable the wave equations.
TE TM Fields; Toroidal Space; Wave Equation; Laplace Equation1. Introduction
Toroidal functions have been known for a long time and an important bibliography is given in [1,2] with some applications to diffraction of acoustic and electromagntic plane waves by a torus. These functions have played a major role in the analysis of plasma confinement in a torus configuration [3], where further references can be found] and, more recently, important works have been devoted to their application in particular situations [4-6].
We are interested here in TE, TM electromagnetic fields in a toroidal medium which becomes of importance for the so-called transformation media: a tool used to tackle invisibility problems [7]. We start with a presentation of the main properties of the toroidal coordinates x, q, f. Then, we discuss the Maxwell equations satisfied by the components Ef, Hx, Hq of TE modes (it is easy to transpose these results to TM modes Hf, Ex, Eq) and, we finally get the wave equation satisfied by Ef. Approximate solutions of this equation are obtained when the index of refraction in the toroidal space makes separable the wave equation.
2. Toroidal Coordinates2.1. Geometric Parameters
In terms of the Cartesian coordinates x, y, z, the toroidal coordinates , , are defined by the relations [8-10]
in which with the inverse relations
The surfaces of constant correspond to spheres with different radii
the surfaces of constant x are non intersecting tori of different radii (anchor rings)
the normal derivative to these surfaces is
Now, from the toroidal metric
we get the scale functions
so that the volume element is
and the surface elements
Remark: the metric (5) may be written with (i,j) = 1, 2, 3 and summation on the repeated indices
and, denoting the matrix with components, it comes g = diag. a2h2 (1, 1, sinh2x) where.
2.2. Differential Operators
Using the general definition of the differential operators in orthogonal coordinates [10,11], we get gradient:
divergence:
curl:
and, for the laplacian of a scalar field
3. TE Field
The components of the TE field are not function of the coordinate and, using (8c), it is easy to get, in a homogeneous isotropic medium with permittivity and permeability the Maxwell equations they satisfy
Substituting (10(a) and (b)) into (10(c)) gives the wave equation satisfied by
Multiplying (11) by and using (6(a)), we get with
and, since
, this equation becomes
To get a more tractable wave equation, we introduce [8,10] fhe variables and the function
,
and we write (13) A + B + C = 0.
Then, the expression (6(b)) of is written with the parameter a deleted
and is changed into implying
so that
Then, taking into account (14), (15b), (16), the first term of (13) becomes
and a simple calculation gives since the terms cancel
Similarly with
the second term of (13) becomes
and, a simple calculation gives since the terms cancel
Taking into account (17a) and (19a), we get
which reduces as easy to prove to the simple expression
Now, using (14) and the expression ofthe last term of (13) becomes
So, according to (20) (20a), (21), (22), the equation (13) takes the simple form justifying the choice of the function (14) made in [8,10] where calculations for A, B are also made
Deleting the last term in Equation (23) reduces the wave equation to the Laplace equation with the variables separated so that looking for in the form where m is an integer
gives since
with as elementary solutions [8,10] of this Laplace equation, the half order Legendre functions of first and second kind.
Now, the wave (23) for TE fields may be generalized to a medium with a constant permitivity while the permeability, consequently the refractive index, depends on and. So, to make (23) separable, we assume that the refractive index n is with constant
Then, the last term of (23) becomes where and with
in which m is no more assumed to be an integer, the wave equation satisfied by is
and with we get
whose solutions are the conical (Mehler) harmonics [10, 12] and.
Then, according to (11a), (14), (27), the component of the TE field is
in which and solution of (23). Substituting (30) into (10(a) and (b)) gives the magnetic components of the TE field.
TM field: These results are easy to transpose to TM fields: just change, and . into and . with a constant permeability while the permittivity is defined so that the refractive index has still the expression (26).
4. Discussion
This work is an illustration of the importance of the function (14) in toroidal coordinates either as in [8,10] to make manageable the 3D Laplace equation which separates into three equations. or as here to get a similar result with the 2D Helmholtz equation. It is remarkable that for TE, TM propagation in a toroidal medium this choice of a particular form for the component of these fields works so efficiently that we get an exact equation whatever is the refractive index. In some sense, the function (14) is consubstantial to toroidal and bispherical coordinates.
The results obtained here are only illutrative because to get analytical solutions of the exact Equation (23), satisfied by, we had to impose rather drastic conditions on the refractive index with a constant permittivity (TE) or constant permeability (TM) , in order to get a separable equation. In practice, to solve (23), requires numerical calculations with algorithms to tackle 2D partial differential equations which is not a real difficulty in particular for propagation in isotropic homogeneous media where the refractive index is constant.
The toroidal geometry has made a comeback in two different domains: first in the string theory of elementary particles [13,14] and second in cosmology. The quest of cosmic origin [15] has suggested [16] that the Universe could have the shape of a drough nut that is of a 3D torus.
Clearly, the present work on electromagnetic wave propagation in a toroidal space could be of some interest to toroidal cosmology [17-19].
REFERENCESReferencesV. H. Weston, “Solutions of the Toroidal Wave Equation and Their Applications,” Ph.D. Thesis, University of Toronto, Toronto, 1954.P. Laurin, “Scattering by a Torus, Ph.D. Thesis, University of Michigan, Ann Arbor, 1967.W. N.-C. Sy, “Magnetic Field Due to Helical Currents on a Torus,” Journal of Physics A: Mathematical and General, Vol. 14, No. 8, 1981, pp. 2095-2112.
doi:10.1088/0305-4470/14/8/031G. Venkov, “Low Frequency Electromagnetic Scattering by a Perfect Conducting Torus, the Rayleigh Approximation,” International Journal of Applied Electromagnetism and Mechanics, Vol. 24, 2007, pp. 96-103.J. A. Hernandez and A. K. T. Assis, “Electric Potential for a Resistive Toroidal Conductor Carrying a Steady Azimuthal Current,” Physical Review E, Vol. 68, No. 4, 2003, 10 p.J. A. Hernandez and A. K. T. Assis, “Surface Charges and External Electric Field in a Toroid Carrying a Steady Current,” Brazilian Journal of Physics, Vol. 34, No. 4b, 2004, pp. 1738-1744.U. Lehonardt and T. Tyc, “Broadband Invisibility by Non-Euclidean Cloaking,” Science, Vol. 313, No. 5910, 2009, pp. 110-112.
http://www.sciencemag.org/cgi/data/116632/DC1/1P. M. Morse and H. Feshbach, “Methods of Theoretical Physics,” Mac-Graw Hill, New York, 1953.H. Margenau and E. M. Murphy, “The Mathematics of Physics and Chemistry,” Van Nostrand, New York, 1955.H. Bateman, “Partial Differential Equations of Mathematical Physics,” University Press, Cambridge, 1959.P. Hillion, “Rayleigh Acoustic Scattering from a Torus,” Journal of Applied Mathematics, Vol. 1, No. 1, 2011, pp. 1-11.F. W. J. Olver, “Asymptotic and Special Functions,” Academic Press, New York, 1974.P. C. Fereira, I. Kagan and B. Tekir, “Toroidal Compactification in String Theory from Chern-Simons Theory,” Nuclear Physics B, Vol. 689, No. 1-2, 2000, pp. 167-195.
doi:10.1016/S0550-3213(00)00407-7M. Bianchi, G. Pradisi and A. Sagnotti, “Toroidal Compactification and Symmetry Breaking in Open String Theories,” Nuclear Physics B, Vol. 376, No. 2, 1992, pp. 369-386. doi:10.1016/0550-3213(92)90129-YA. Guth, “The Inflationary Universe. The Quest for a New Theory of Cosmic Origin,” Perseus Books, New York, 1997.A. A. Starobinski, “A New Type of Isotropic Cosmological Models without Singularities,” Physics Letters B, Vol. 91, No. 1, 1980, pp. 99-102.
doi:10.1016/0370-2693(80)90670-XM. Mc. Guigan, “Constraints for Toroidal Cosmology,” Physical Review E, Vol. 41, No. 10, 1990, pp. 3090-3100.
doi:10.1103/PhysRevD.41.3090M. Bastero-Gill, P. H. Frampton and L. Mersini, “Modified Dispersion Relations from Closed Strings in Toroidal Cosmology,” Physical Review D, Vol. 65, No. 10, 2002, 12 p. doi:10.1103/PhysRevD.65.106002J. Smulevici, “On the Area of the Symmetric Orbits of Cosmological Space-Time with Toroidal or Hyperbolic Symmetry,” 2009.
, 
, 
are defined by the relations [8-10]
with the inverse relations
correspond to spheres with different radii
and, 
denoting the matrix with components
, it comes g = diag. a2h2 (1, 1, sinh2x) where
.
are not function of the 
coordinate and, using (8c), it is easy to get, in a homogeneous isotropic medium with permittivity 
and permeability 
the Maxwell equations they satisfy
and using (6(a)), we get with
, this equation becomes
and the function 
,
is written with the parameter a deleted
is changed into 
implying
of (13) becomes
terms cancel
of (13) becomes
cancel
of
the last term 
of (13) becomes
takes the simple form justifying the choice of the function (14) made in [8,10] where calculations for A, B are also made
separated so that looking for 
in the form where m is an integer
of first and second kind.
(23) for TE fields may be generalized to a medium with a constant permitivity while the permeability, consequently the refractive index, depends on 
and
. So, to make 
(23) separable, we assume that the refractive index n is with 
constant
where 
and with
is
we get
and
.
of the TE field is
and 
solution of (23). Substituting (30) into (10(a) and (b)) gives the magnetic components
of the TE field.
, and 
. 
into 
and 
. 
with a constant permeability while the permittivity is defined so that the refractive index has still the expression (26).
component of these fields works so efficiently that we get an exact equation whatever is the refractive index. In some sense, the function (14) is consubstantial to toroidal and bispherical coordinates.
, we had to impose rather drastic conditions on the refractive index with a constant permittivity (TE) or constant permeability (TM) , in order to get a separable equation. In practice, to solve (23), requires numerical calculations with algorithms to tackle 2D partial differential equations which is not a real difficulty in particular for propagation in isotropic homogeneous media where the refractive index is constant.