Appl. Phys.20, 313-317 (1979)

Applied Physics 9 by Springer-Verlag1979

A Note on Scalar Hertz Potentials for Gyrotropic Media S. Prze~dziecki* and R. A. Hurd

Division of Electrical Engineering, National Research Council of Canada, Ottawa, Ontario, Canada KIA 0R8 Received 1 May 1979/Accepted30 July 1979 Abstract. The scalar Hertz potentials are generalized to the case of electromagnetic fields in gyrotropic media. Expressions for electromagnetic fields in terms of two potentials are presented and the system of differential equations for the potentials is derived. The results are summarized in the form of a theorem. The case of a stratified gyrotropic medium with parameters varying along the distinguished axis has been considered also. Scalar "superpotentials" satisfying a fourth-order partial differential equation are introduced. They allow expressions for electromagnetic fields to be found in terms of one scalar superpotential only. Basic results on scalar Hertz potentials in isotropic media are recalled. PACS: 41 The analysis and solution of many electromagnetic problems in isotropic media are considerably simplified by the proper use of scalar Hertz potentials. Examples start with the basic problem of radiation by a dipole [1] and comprise all areas of electromagnetics : antennas, guided waves, resonators, diffraction, etc. [2, 3]. The purpose of this paper is to generalize these potentials to the case of electromagnetic fields in gyrotropic media. Our justification for this work stems from at least one example where the solution of a diffraction problem in a gyrotropic medium [4] is made much simpler by the use of the generalized potentials. It is believed that these potentials will prove useful in many more diffraction, radiation and other problems in gyrotropic media.

1. Isotropic Medium The basic aspects of scalar Hertz potentials for an isotropic medium can be summarized in the following two theorems : * The research reported in this paper was carried out during S. Prze~dziecki'sleave of absence from the Institute of Fundamental TechnologicalResearch,Polish Academyof Sciences,Swi~tokrzyska 21, PL-00-049Warszawa, Poland.

Theorem 1. An electromagnetic field E, H generated in an arbitrary region D from two scalar functions u, v via formulas

E = V x V x ua + ico/~V x va H = - icoeV x ua + V x V x va

(1)

satisfies in D the homogeneous set of Maxwell's equations V x H = - icoeE V x E = ico#H

(2)

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

where a is an arbitrarily oriented unit vector and k 2 = co2e#. The proof follows from a simple substitution of (1) into (2). From this theorem follow two corollaries: a) The field E e, H e generated by the function u is transverse magnetic (TM) with respect to a, i.e. H e- a = 0 and the field E m,H m generated by v is transverse electric (TE) with respect to the same direction, i.e. E~.a=O.

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314

S. Prze2dzieckiand R. A. Hurd

b) Each of the fields E e, H e and E", H m satisfies the system (2). Thus it is seen that, for a fixed a, a large class of sourceless electromagnetic fields is generated in D from the set of all wave functions u, v via (1). However, a question that immediately arises is whether this class coincides with the set of all sourceless electromagnetic fields in D. An answer to this question is provided by the following representation theorem: Theorem2 (representation theorem). An arbitrary sourceless electromagnetic field determined in a region D of sufficiently simple shape can be represented in D in terms of two scalar functions u, v in the form (1) and the functions u, v can be chosen so to fulfill in D the Helmholtz equation (3). The restriction imposed on the region D to be of sufficiently simple shape means that any straight line parallel to a cannot have more than one interval in common with D. Theorem 2 can also be formulated as the following splitting theorem:

permittivity and permeability have the following forms :

~t=

0

0

~a

# 0

0 , #

(4)

where not both % and #o are zero. The z axis is called the distinguished axis of the medium. If % = #0 = 0 but not both e==e and #==# the medium is uniaxial. We denote by z 0 a unit vector directed along the z axis. An outline of analysis of electromagnetic fields in gyrotropic media was presented in [10]. We shall now concern ourselves with the generalization of Theorem i to the case of a gyrotropic medium. Assume that a region D contains a gyrotropic medium characterized by (4). Then Maxwell's equations take the form -ico~.E

V x H=

V x E =ie)l~-H.

Theorem 2' (splitting theorem). An arbitrary sourceless electromagnetic field E, H given in D can be split into a TE and a T M field with respect to the direction specified by a so that each of these constituent fields satisfies (2) and can be expressed in terms of one scalar function fulfilling (3) in the form being the respective part of (1). The restriction on the region D is the same as in Theorem 2. The proof of Theorem 2 (or 2') is given in [5] (cf. also [6]). It does not reduce to simple manipulations as is the case with Theorem 1 and involves some delicate points. The Theorems 1, 2, 2' are easily generalized to the case of a uniaxially anisotropic medium [7, 8] but with one essential restrictiofl, namely, the vector a can no longer be arbitrarily oriented but must coincide with the distinguished axis of the medium. In other words, the generalization of Theorem 2' is possible but the splitting into TE and T M fields can be carried out only with respect to the distinguished axis [9]. The restriction on the region D remains the same as above. We do not give the relevant formulas for the uniaxial case since they turn out to be particular cases of the ones to be derived for a gyrotropic medium.

2. Gyrotropic M e d i u m

A medium is said to be 9yrotropic if in an appropriate system of Cartesian coordinates x , y , z the tensors of

(5)

Let

V~=V-zo~z e = E - EzZo ,

h = H - HzZo .

For an arbitrary vector field B = b + B=zo we have Ob V x B = z o x ~zz qvV,B= x z o +zoV t -(b x Zo).

(6)

Using (6) the z components of (5) can be written as follows : V~. (h x zo) = -

icon=E=

V," (e x Zo) : iOOgaHZ.

(7)

From (6) and (7) the transverse parts of (5) can be written as 9

0e

9

Oh

-- lO)/3a~Z "-~('02/3a(~ "h) x z 0 ~ l('O#a E

Wt V t 9(h

x Zo)

(8) : (D2~a(~" e)

X z 0 ~ V t V t . (e

x Zo).

Due to the fact that E= and H= are explicitly determined by h and e, the system (8) is equivalent to (5), i.e. each solution e, h obeying (8) provides a solution to (5) if E= and H~ are determined by (7)9However, insofar as boundary conditions are concerned, system (5) has to be considered to determine equivalent boundary conditions for (8).

Hertz Potentials for Gyrotropic Media

315

Let us also record additional relations between the transverse and longitudinal fields. They follow from the divergence equations V . e . E = 0, V. ~ . H = 0 and (7). We get

By substituting (13) into (9) we get

Vt 9e =

0v +~oe~V,u)=0. V~-(h-V, Uz

G 0E~

%

8 0z c ~ #~ 0H~ + cog ~ E ~. # Oz

Vt . h -

(9)

~" e = ee + i%(z o x e)

x Zo) q- ic02ea/Aah -{- V t V t 9(h

0H~ (12)

v~ + ~- ~-U + k~ H~ = coeo~o & , where 2

#

P

k e2 ._-..=032Ca #

2 --#O ,

i~

2

2

k m = (0 Pa

2

2 E -- ~;9 e

Thus the system of Maxwell's equations has been split into two systems (11) and (12) involving transverse and longitudinal components only. Connecting formulas are given by (7) and (9). We shall now construct an electromagnetic field satisfying (5) from two scalar functions u, v. Suppose that the longitudinal components E~,H~ are given by

ea

(13)

(18)

Thus relations (16) can now be written as e q- CO/A~ Vtv q- I(.o/AVt X VZ0

One can easily check that the fields (19) satisfy (11) if u and v fulfill (14). Equations (19) can be written also in the following form ~u

io~/~

e = V , ~ + - -V Z~.V~ oe x icoe

h=

---~.Vtxuzo+V,~ P

where the tilde denotes the transpose of a matrix. Making use of (14), expressions (13) can be given the following form e

2

E ~ = ~ z 2 + - g" - ak ~ u +

0U (D ]A"Cg O Z

(2l) 021)

2V

Finally expressions (13) and (20) can be cast into the following uniform vectorial form:

G 02 + k ~ ) u = - c o # z o G #v

~ 0z

E=e-I.V•

(14)

( v ) + ~~ ~ +k:)~=~oe~o~ ~ 0~

7~-Y

P

H~ = - c o ~ z o ~zz + ~z 2 + --#, km "

If expressions (13) are substituted into (12) we see that equations (12) are satisfied if

v?+7~

(20)

Ov z,

0U

#~

(19)

0v h = Vt ~zz - mE ~ Vtu - icoeVt x uz o

~2U

- _V~u

G= _ Ev~.

(

(17)

icoeu = - tp icogv --- qo.

e

G=

P

We choose particular integrals of (17) so that

e =Vt~

T ~ - + k~ G = - copo~o Oz

e

h-V, ~ +o~ ~~ V,u=Vt x ~Zo,

+ V~V t -(e • Zo).

(V2~_~a~2)

~O _]_ -fl0 __ -,

(16)

VZ(icoeu + t~) = 0 V~Z(ico/~v- q))=0.

x Zo)

We can now easily obtain a system of equations for the longitudinal components E~, H~. Taking the divergence of (11) and using (7) and (9) we get

TO :

% =V, x ~OZo -co~TV~v

where qo and ~ are certain auxiliary functions. By substituting (16) into (7) and using (13), we get

(11)

0h ~zz = (~2e#"(e • z~ + i~

i~

Thus

(10)

and similarly for ~-h. Using (10) we can write system (8) as follows: Oe ico%~z = c~

(15)

e-Vt~

In arriving at (9) the following auxiliary relations were used :

-

0u eg Vt" ( e - V~yzz - co~ T Vtv) = 0

Oz

H=

ic~ P

~.V•

UZo +

i~# a -.V•

o

e x UZo+ g - 1 "V x ~'V x VZo.

(22)

316

S. Prze~dziecki a n d R. A. H u r d

It is now easily seen that (22) reduce to (1) for an isotropic medium and thus functions u and v can be considered as generalized scalar Hertz potentials. For % =#~ •0 but not both ~ : e and #~ = # we obtain, as predicted, formulas for an uniaxial medium. Let us add that there is a degree of freedom in defining scalar Hertz potentials, e.g. we could have no coefficients in (13). Then the system of equations satisfied by the potentials would be identical with that for E~, H~. Such a definition is employed in [6]. In a still different way the potentials are introduced in Sect. 4. We can now summarize our results concerning the potentials u, v in the form of the following theorem: Theorem 3. A transverse electromagnetic field e, h generated from two scalar functions u, v via (19) or (20) satisfies the system (8) or (11) if u and v fulfill (14). Alternatively the field E, H determined by (22) obeys the system (5) if u and v satisfy (14).

The Theorem 2 (representation theorem) can also be generalized to the gyrotropic case and an arbitrary electromagnetic field given in a sufficiently simple region D can be represented in terms of two scalar potentials satisfying (14) in the form (22). The restriction on D is that any straight line parallel to the distinguished axis cannot have more than one interval in common with D. A proof of this less straightforward matter is given in [6]. It can be seen easily, however, that the splitting theorem (Theorem 2') is not valid for a gyrotropic medium. No general TE or T M fields can exist that depend on z. Assume for example that E~=0, then from (12) it follows that 3 H J O z = O and thus no TE field dependent on z can exist other than a TEM field. Yet for a uniaxial medium the splitting into TE and T M modes is possible, but only with respect to the distinguished axis of the medium [9].

where k 2 =O.)2C, a#a . Potentials u,v given by ?U U:

(24) 8a 0 2

[(v?+

2\

satisfy the system (14). Consequently in all formulas (13), (19-22), potentials u and v can be obtained from U via (24). Thus we obtain a remarkable result that an arbitrary electromagnetic field in a gyrotropic medium can be derived from only one scalar function while in an isotropic medium two functions are necessary. This result is less surprising if we note that in a gyrotropic medium the components Ez and H~ are coupled by the system (12) while in an isotropic medium E~ and H z can be two arbitrary, independent solutions of the Helmholtz equation. Similarly an alternative potential V can be introduced with 0V

V = (DSTg ~ - Z

K(V2 ok

u:

0-'

(25)

)v

and V satisfies the same equation as U.

4. Stratified Gyrotropic Medium It is of practical interest to observe that the scalar Hertz potentials can be generalized to the case of a stratified gyrotropic medium provided the strata are perpendicular to the distinguished axis, i.e., the tensors (4) depend on z. For the components Ez, H z we now obtain the following system of equations :

Z2

3. Superpotentials

=

The idea of auxiliary functions for electromagnetic fields can be extended further in the case of gyrotropic media. The potentials u and v can be derived from one scalar "superpotential". Consider a function U which satisfies the equation

- - Co/A'Cg C~z

:

~? 1 ~ +k2)E,E z

- o)~zg + fla

8z

6o~ ~zz

0laaHz

#~Hz (26)

I~ OZ + k m

Oe,E~

It now seems convenient to introduce potentials via the following relations

+m (23)

eaEz : - V2u #~H~ = , V2 v.

(27)

Hertz Potentials for Gyrotropic Media The system of equations to be satisfied by u,v takes n o w the following form

v? +~ooV7~-f +k~

(28)

019 ) v~+~O~z~O z +k2., v ~U

317

5. C o n c l u d i n g R e m a r k s

We have shown that electromagnetic fields in gyrotropic media can be represented in terms of two auxiliary scalar functions which in the isotropic case reduce to the scalar Hertz potentials. The generalized potentials satisfy a system of two coupled second order equations. The coupling of these equations enables a new representation for electromagnetic fields to be found in terms of only one scalar function (a superpotential). The superpotential obeys a fourth-order equation. It has also been shown that the representation in terms of the generalized Hertz potentials holds for gyrotropic media stratified along the distinguished axis.

The formulas for the transverse field are as follows References 1 V 0u

ico _

e= 7 ,~+T~'V,x~*o (29) h=

ir _ __

1

Ov

~.V,xUZo+~V,~.

The uniform vectorial form for (27) and (29) is

0I; E = (~a)- 1. V x a. V x UZo - (~a)- 1 "Yz" (Zo x V x UZo)

+icon- l~;'V X VZo (30) H = - i c % - 11~- V x u z o + ( # ~ ) - 1V x ~ . V x vz o

- (~)-

&-z"(Zo x V x VZo).

1. H.Hertz: Ann. Phys. Chemie 36, 1-22 (1889) 2. J.A.Stratton: Electromagnetictheory (McGraw-Hill, New York 1941) 3. H.H/Snl, A.W.Maue, K.Westpfahl: Theorie der Beugung, in Encyclopedia of Physics, Vol. 25/1 (Springer, Berlin, GiSttingen, Heidelberg 1961) 4. S.Prze~dziecki, R.A.Hurd: Diffraction by a half-plane perpendicular to the distinguished axis ofa gyrotropic medium (oblique incidence) (a paper in preparation) 5. K.Bochenek: Methods of analysis of electromagnetic fields (PWN, Warszawa 1961) (in Polish) 6. S.Prze2dziecki, W. Laprus : "Representation of electromagnetic fields in gyrotropic media in terms of scalar Hertz potentials," Inst. Fund. Techn. Research Polish Acad. Sci. Report 17/1978 (in Polish) 7. P.C.Clemmow: Proc. IEE (London) 110, 101-106 (1963) 8. S.Prze~dziecki: J. Appl. Phys. 37, 2768 2775 (1966) 9. A. Kujawski, S.Prze~dziecki: Bull. Acad. Pol. Sci. Ser. sci. math. astr. phys. 21, 955-962 (1973) 10. P.S.Epstein: Rev. Mod. Phys. 28, 3-17 (1956)