Appl. Phys. B27, 169--173 (1982)

Applied

phy~es

Physics B Chemistry "" 9 Springer-Verlag 1982

A Note on Debye Potentials for Spherically Gyrotropic Media J. K. Frgckowiak and S. Prze~dziecki Institute of Fundamental Technological Research, Polish Academy of Sciences, gwi~tokrzyska 21, PL-00-049 Warsaw, Poland Received 22 July 1981/Accepted 15 January 1982

Abstract. The Debye potentials are generalized to the case of electromagnetic fields in spherically gyrotropic media. A medium is called spherically gyrotropic if it is locally gyrotropic with the distinguished axis having a radial direction determined by a central point. Expressions for electromagnetic fields in terms of the generalized potentials are presented and the system of differential equations for the potentials is derived. The results are summarized in the form of a theorem. Basic facts about the Debye potentials in isotropic media are recalled. PACS: 41 The scalar Hertz potentials have been generalized recently [1] to the case of electromagnetic fields in gyrotropic media. The basic property underlying this generalization is the preservation of the rotational (axial) symmetry of the Maxwell system for the gyrotropic tensors e, IX. As is well known, for isotropic media the spherical symmetry of Maxwell's equations makes possible the introduction of another pair of auxiliary functions called the Debye potentials [2] or the spherical Hertz potentials [3]. One easily notes that this symmetry remains preserved if the constitutive tensors are locally gyrotropic with the distinguished axis having a radial direction defined by a fixed point and with the properties of the medium depending, at the most, on the distance from the considered point. This observation suggests a possibility to generalize the Debye potentials to the case of electromagnetic fields in spherically symmetric anisotropic media. This generalization is the purpose of the present paper. The general forms of spherically symmetric tensors e, Ix in an appropriate system of spherical coordinates (0, q), r) are ~= i

~ 0

00t l , :) e,

IX= i# o 0

#

0

#~

where ~ and IXare, in general, functions of r. Harmonic time dependence has been assumed with the time factor exp(-ie)t) which will be suppressed throughout. A medium that is described in an appropriate spherical system (0, q), r) by the constitutive tensors of forms (1) will be called spherically gyrotropic provided not both % = 0 and #0 = 0. If % = #9 = 0 but e :~ ~r and/or # :I: #r the medium will be said to be radially uniaxial. The physical relevance of spherically gyrotropic media is rather very limited. Nevertheless the generalized Debye potentials could be used advantageously in the analysis of propagation in a plasma in the vicinity of the poles of a magnetic dipole. A magneto-ionic model of the ionosphere for high and medium latitudes, widely exploited by Krasnushkin [4, 5] for investigations of propagation of very long waves, is based on the tensor e of form (1), # being a scalar. Also impedance boundary conditions employed by Wait [6] would follow from such a model of the ionosphere. The generalization of the Debye potentials to the case of radially uniaxial media was presented in [-7]. 1. Notation

(1)

We shall denote by r the distance from a fixed point and by r o the outward unit normal to the spheres

0721-7269/82/0027/0169/$01.00

170

J. K, Fr~ckowiak and S. Prze~dziecki

r = const. We shall refer to the direction r o as radial or longitudinal and to the directions tangent to the spheres r = const as transverse. We introduce the following notation for the transverse parts of the operators grad, div, and Laplacian V~= V

-

0 r o ~or

1~ 2 V , . = V . - ~ r r ( r to. )

E m . r 0 = 0.

10fraL 1 v~=v~vt=v2- r~V\ 0r/"

The operator curl can now be written as V x =r o x -

7~(r )

r o x V,(r o. ) - roV,(r o x ).

2. Isotropic

(2)

Medium

For isotropic media the basic facts characterizing the Debye potentials can be summarized in the following two theorems : T h e o r e m i. An electromagnetic field E, H generated in a region D from two scalar functions u, v via formulas E = V x V x rur o + icopV x rvr o

H=

--

ico~V rur o + V x V x rvr o

(3)

x

satisfies in D the homogeneous set of Maxwell's equations V x H = - io)eE

V x E = icgpH

(4)

if the functions u and v fulfill in D the Helmholtz equation

where k z =o2e#. With the aid of the transverse operators, (3) can be split into their transverse and longitudinal parts

~ru

e = V~ ~-r + ic~

x ro (6) ~3rv

h = - icoeVtru x r o + V~ 3r ' E~ = - V{ru

(7)

H r = - V~rv,

where e = E - E , . r o,

h=H-Hrro,

E , . = E . r o,

Let us also add that the choice of the central point r = 0 is completely arbitrary. The proof of Theorem 1 follows from a simple substitution of (3) into (4). The functions u and v are called Debye potentials [2] or spherical Hertz potentials [3]. Their relation to the Hertz vectors is clarified in [2]. F r o m Theorem 1 follow two corollaries: a) The field E ~, H e generated by the function u is transverse magnetic (TM) with respect to the radial direction r o i.e. He.r0 = 0 and the field E m, H m generated by v is transverse electric (TE) with respect to ro, i.e.

H r = H . r o.

b) Each of the fields E e, H e and E 'n, H m satisfies the system (4). The formulas (3) show that with their aid we can generate a large class of sourceless electromagnetic fields from the set of all wave functions u, v determined in D. A question that immediately arises is whether this class coincides with the set of all sourceless electromagnetic fields in D. (The point r = 0 is considered to be fixed.) An answer to this question is provided by the following representation theorem. Theorem 2 (Representation T h e o r e m ) . An arbitrary sourceless electromagnetic field given in a region D of sufficiently simple shape can be represented in D in terms of two scalar functions u, v in the form (3) with the functions u, v satisfying in D the Helmholtz equation. The restriction on the region D to be of sufficiently simple shape means that any straight half-line of radial direction must not have more than one interval in common with D. An alternative formulation of Theorem 2 can be given as the following splitting theorem: Theorem 2' (Splitting Theorem). An arbitrary sourceless electromagnetic field E, H given in D can be split into a T M and a TE field with respect to r o so that each of these constituent field satisfies (4) and can be expressed in terms of one scalar function in the form being the respective part of (3) with the scalar function satisfying (5). The restriction on D is the same as in Theorem 2. In both Theorems 2 and 2' the point r = 0 can be chosen arbitrarily as long as the restriction imposed on the region D is fulfilled. A proof of Theorem 2 (or 2') for the interior or the exterior of a sphere r = const is given in [3]. For more general regions a proof could be conducted along the lines similar to those adopted in [8]. Theorems 1 and 2 (or 2') can be considered as mutually inverse provided Theorem 1 is confined to regions for which Theorem 2 (or 2') holds. Theorems 1, 2, 2' are easily generalized to the case of a radially uniaxial medium [7] but with one essential

Debye Potentials for SphericallyGyrotropicMedia

171

restriction that the point r = 0 is no longer arbitrary but is uniquely determined by the medium. The restriction on the region D remains as for the isotropic case. We do not give the relevant formulas for the radially uniaxial case since they turn out to be particular cases of the ones to be derived for a spherically gyrotropic medium.

and similarly for It.h. With the aid of (12) we can rewrite the system (10) as follows ico#, 0 (rh) r Or = co2e#,(e x ro) + io)2%Ge + VrV~.(e x ro) (13)

icoe~ 0

r 0r (re) = o)%,#(h x to) + ieo%dxgh+ V,V,- (h x ro).

3. S p h e r i c a l l y Gyrotropic M e d i u m

We can now easily obtain a system of equations for the longitudinal components Er, H~. Taking the divergence of (13) and using (9) and (11) we get

Consider the system of Maxwell's equations V x H = - ime. E

V x E = icolt. H

(8) 0 1 0 /-ke2) r2GE~

with e, tt given by (1) in a spherical system defined by the central point r = 0. We shall now split the system (8) into its longitudinal and transverse parts. As for the isotropic medium we denote

/#o 3

(14) V

E = e + E~r o

H = h + H~r o

a 1 a +#'U~ # Or

Vt. (r o x e) -- - ico#~H,.

+kZ=)r2/~'H~

/e~ O

With the aid of (2) we obtain from (8) V,. (r o x h) = icoGE,

O %\ 2

(9) where

Similarly using (2) we get from (8)

2

io)#~ O (rh) = ~o2#~(a9e) x r o + VtVt. (e x ro) r Or

2

k e = o3 gr

2 2 /L/ - - ]Ao

g2 2

km = (~

2 - - gO

(10) icoG 0 r Or (re) = cozq(g 9h) x r 0 + VtV,. (h x r0). Due to the fact that E~ and H~ are explicitly determined by h and e the system (10) is equivalent to (8) i.e. each solution e, h obeying (10) provides a solution to (8) if E~ and H, are determined by (9). Only when boundary conditions are involved system (8) has to be considered to find equivalent boundary conditions for (10). The second pair of relations between the transverse and longitudinal fields follows from the divergence equations V-~.E=0,

Thus the system of Maxwell's equations has been split into two systems (13) and (14) involving transverse and longitudinal components only. Connecting formulas are given by (9) and (11). We shall now construct an electromagnetic field satisfying (8) from two scalar functions u, v. Suppose that the longitudinal components Er, H, are given by q E r = _ VZru

Vt.h

(15)

If expressions (15) are substituted into (14) we see that (14) are satisfied if

V.It.H=0. Vg+er~reOr+k

We obtain Vt.e =

#rH~ = - VZrv.

1 1 0 (r2grEr) - (.oeg e r 2 Or -7-GHr

toe, \ • ar + ar e ) rv

(11)

/~1 r21 0r0 (r2/~H,) + ~ G E ~ "

In arriving at (11) the following auxiliary relations were used a. e = ~e + i%(r o x e)

ru

(12)

(

910

)

V2+~,~0r +k 2 =~~

(16) rv

+ ar ~/ru.

We shall now find the expressions for e and h in terms

172

J . K . Frockowiak and S. Prze•

of u and v. Let us substitute (15) into (11), we get

Finally expressions (15) and (21) can be cast into the following uniform vectorial form

Oru V,. e - ~ V t Or

E = (~,)- 1. V x ~,-V x rur o 1 Or, _(g~)- 1 . . . . (r o x V x ruro) +icoe- l~.V x rvr o r Or

o0% Vtrv ) = 0 e

(17) Vt'(h - t ~ v , ~r-& v + ~ 1 7 6

H = - i c o # - l ft. V x rur o

Similarly as for planar vector fields we can represent the solenoidal fields from relations (17) in the following way e-

_s vtl &Uor

~%e V,rv

= V x ~or o = Vtq~ x r 0

(18) 1 Orv co#~ h - ~ V, ~ + # V~ru = V x ~oro = Vttp x ro,

where ~o and tp are certain auxiliary functions. By substituting e and h determined from (18) and E , H, given by (15) into (9) we get V~(q~- i~rv) = 0

V~(~p+ icoru) = 0.

(19)

We choose particular integrals of (19) so that q~= icorv

(20)

tp = - icoru.

Thus relations (18) can now be written as

1 Oru (.o%

e = 7 V,~

+

e Vtrv + iooV~rv x r o

h = - iooVtru x r o

co#gVtru + #

1

arv v,g.

(21)

One can easily check that the fields (21) satisfy (13) if u and v fulfill (16). Formulas (21) can also be written in the following way 1 &u ico e = ~ V, ~ + --~ ~" V, x rvro

(22) ico h=---ft'Vtxrur #

1 Oru o+ Vt ~ Or '

where the tilde denotes the transpose of a matrix. Making use of (16) Eq. (15) can be given the following form e~E~= e, Or e Or + k + c~

ru

+ Or e ] rv

(23)

;,,nr= -o

,17 a + L Or #4ru #/

+ #'~-r#&

(24)

1 011 + (pit)- 1. V x Ix. V x rvr o - (pit)- t. r Or

9(r 0 x V x rvro). It is now easily seen that (24) for an isotropic medium reduce, in essence, to (3)9 A minor discrepancy arises from the different position of constant coefficients. This results from the difference between the basic equations (15) and (7). There is a degree of freedom in defining the scalar potentials and the definition adopted here seems more convenient for the case under consideration. Equations for the potentials (16) reduce exactly to (5). Thus the scalar functions u and v can be considered as generalized Debye potentials. For % = # 0 = 0 but er4=e and/or # r + # we obtain, as predicted, formulas for a radially uniaxial medium. We can now summarize our results concerning the generalized Debye potentials in the form of the following theorem : Theorem 3. An electromagnetic field E, H generated in a region D of a spherically gyrotropic medium from two scalar functions u, v via (24) satisfies in D the Maxwell set (8) if the functions u and v fulfill (16) in D. Alternatively the transverse field e, h determined by (22) obeys (10) if u and v satisfy (16)9 Obviously the central point r = 0 is determined by the medium whose constitutive tensors are of the form (1) in a spherical system defined by the point r = 0. Theorem 2 (representation theorem) can also be generalized to the spherically gyrotropic medium and an arbitrary electromagnetic field given in a sufficiently simple region of a spherically gyrotropic medium can be represented in terms of two scalar potentials satisfying (16) in the form (24)9 The restriction on D is the same as for the isotropic case9 It can easily be seen, however, that the splitting theorem (Theorem 2') is not valid for a spherically gyrotropic medium9 No general TE or TM field with respect to r o can exist that depends on r. Assume, for example, that E r = 0 then Hr has to fulfill simultaneously two different equations which in general are not compatible and thus no TE field depending on r can exist other than a TEM field (i.e., a field for which e~=u,=O). Yet for a radially uniaxial medium the splitting into TE and T M modes is possible with respect to r o determined by the medium.

Debye Potentials for Spherically Gyrotropic Media

173

Finally as a curiosity we may mention that there exists a peculiar medium that might be called generalized radially uniaxial medium. It is determined by the relation % / e = - / ~ j # = c o n s t [9]. Electromagnetic fields in such a medium behave similarly as in radially uniaxial media.

Similarly a dual superpotential V can be introduced starting from the relation v = r-~V Let us add that for ordinary gyrotropic media the auxiliary functions called in [1] superpotentials were introduced by Gurevich [10] without resorting to the intermediate step involving the scalar Hertz potentials. We thank Dr. A. J. Turski for pointing this out to us.

4. Superpotentials In analogy with the situation for ordinary gyrotropic media [1] the question may be posed as to whether an analysis of the system (16) [or (14)] can be reduced to that of one equation of higher order. In the same context we may ask whether the Debye potentials u, v can be generated from one scalar "superpotential". In general, the variable coefficients in (16) make this impossible. Yet there exists a peculiar case where the potentials u, v can be derived from one superpotential. This occurs for gg __

~8/g

In this case we have u=r-iU

L{v2

la

)

(25)

v=o)e~vr \ ~ + e , . ~ z cgr + k 2 U

and the potentials u, v will satisfy (16) if U fulfills the equation # 01"

/ qv

" ( V 2 + s r ~ 1 0g"~r -t-k2)+C02l~rr] U =O '

(26)

where

v:_ 0_(< Or \el" Consequently in all formulas (15), (21)-(24) potentials u, v can be obtained from U via (25).

5. Concluding Remarks We have shown that electromagnetic fields in spherically gyrotropic media can be represented in terms of two auxiliary scalar functions which in the isotropic case reduce to the Debye potentials. The generalized potentials satisfy a system of two coupled second-order equations. The basic fact underlying this generalization is the spherical symmetry of the Maxwell equations with spherically gyrotropic constitutive tensors. Analysis of the system of equations for the generalized potentials cannot be reduced, in general, to an analysis of one differential equation of a higher order.

References I. S.Przekdziecki, R.A.Hurd: Appl. Phys. 20, 313-317 (1979) 2. L.A.Vainstein: Electromagnetic Waves (Soviet radio, Moskva 1957) (in Russian) 3. K.Bochenek: Methods of Analysis of Electromagnetic Fields (PWN, Warszawa 1961) (in Polish) pp. 103 106 4. P.E.Krasnushkin: Suppl. Nuovo Cimento 26, Ser. 10, 50~112 (1962) 5. P.E.Krasnushkin: Theory of Propagation of Very Lon9 Waves (Vychyslyt. Centr AN SSSR, Moskva 1963) (in Russian) 6. J.R.Wait: Can. J. Phys. 41, 299-315 (1963) 7. B.Friedman: "Propagation in a Non-Homogeneous Medium", In: Electromagnetic Waves,ed. by R.E. Langer (The University of Wisconsin Press, Madison 1962) 8. S.Prze~dziecki, W.Laprus: On the representation of electromagnetic fields in gyrotropic media in terms of scalar Hertz potentials. J. Math. Phys. (in press) 9. A.Kujawski, S.Prze2dziecki: Bull. Acad. Pol. Sci. Ser. Sci. Math. Astr. Phys. 21, 955-962 (1973) 10. A.G.Gurevich: Ferrites at Super-High Frequencies. (Gos. Izd. Fiz.-Mat. Lit., Moskva 1960) (in Russian)