A reciprocity theorem relating currents and fields in the presence of a plane-stratified ionosphere

I-I Rinted

of Atmospheric and Ternstrial in Northern Ireland

Physics,

Vol. 43. No. 4, DP. 339-344,

0021-9169/81/04033W06$02.00/0

1981.

Pergamon Press Ltd.

A reclproclty theorem relating

currents and fields in

plane-straaed c. ALTMAN* and

the

presence

of

a

ionosphere

A. SCHATZBERG~ * Dept. of Physics, t Dept of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel (Receiued 20 November 1980) Abatmci-We consider the plane-wave spectrmn generated by two arbitrary current distributions, J,(r) and J,(r), in free space below, or above, a plane stratified ionosphere. We apply the 2x 2 ionospheric reflection or transmission matrices R(k) or T*(k) defined in terms of the TE or TM modal amplitudes, and perform an inverse Fourier transform on the reflected or transmitted spectral components, to obtain the resultant fields, E,(r) and E,(r), reflected from, or transmitted through the ionosphere. A scattering theorem is then applied which connects the scattering matrices in a given problem with those in a ‘conjugate’ problem, namely, R*(k) = fi*(k’), T*(k) = f’*(k’), where the propagation vector kc is the mirror image of k with respect to the vertical, magnetic east-west plane. It is shown that

I

E,(r) . J,(r) d3r = EpC(r). J,‘(r) d3r I

where Eic(r), i = 1, 2, represents the respective fields generated by the image or mirrored, current systems Ji’(r), i.e. mirrored with respect to the vertical magnetic meridian plane. It is noted that whereas the Lorentz reciprocity theorem relates currents and fields in the presence of (or inside) a given medium with those in the presence of a transposed medium (i.e. in the presence of a hypothetical ionosphere in which the geomagnetic field has been reversed), the result here derived relates instead a given and a mirrored pair of current systems, and their associated wave fields, in the same physical system. 1. LNTRODUCTION

The Lorentz reciprocity theorem is an important tool for comparing the receiving and transmitting properties of antennas. In one form it relates the electric field El(r), generated by an arbitrary cur-

rent distribution J1(r), with the field E*(r) generated by another current distribution J2(r), as an integral relation over all space,

I * E,(r)

J*(r)

d3r =

I

E,(r) *Jr(r) d3r.

present, such as an ionospheric magnetoplasma, but may be recovered if we assume that the second

current distribution Jz(r) flows in the presence of a hypothetical ‘transposed’ medium, character&d by the transposed tensors, B and L which in the case of the magnetoplasma may be achieved by reversing the direction of the external magnetic field. In this case the modified reciprocity theorem takes the form [KONG, 19751

(1) I

If, for instance, in the two cases the same current I flows through two short wires of length 81, and 61, respectively, then (1) becomes

E,(r) * &(r) d3r =

5

B,(r) * J,(r) d3r

(3)

in which the field E, has been generated by the current .I, in the given (original) medium, whereas the field E2 has been generated by the current J2 in the transposed (magnetic-field reversed) medium. El . Slz = Ez .61, In the case of the earth’s ionosphere the requirewhich states that the E.M.F. induced across 81, by a ment of a field-reversed medium to maintain recipcurrent I in 61, is equal to that induced across ~31~ rocity severely restricts the usefulness of the when the same current flows in Slz, i.e. when the theorem. roles of transmitting and receiving dipoles are reA different and seemingly unrelated type of reversed. ciprocity or scattering theorem has been found by a The Lorentz reciprocity theorem (1) is valid number of ionospheric workers (BARRON and BIDwhen all space is occupied by media characterized DEN, 1959; hlTEWAY and JESPERSEN, 1966: by symmetric electric permittivity and magnetic SUCHY and km, 1975; ALTMANand SUCHY, permeability tensors, E and CL,respectively. Recip1979) who consider the 2 x 2 reflection and transrocity breaks down when a gyrotropic medium is mission matrices, R” and p, for upgoing (+) or 339

downgoing (-) waves incident on a plane-stratified ionosphere from below or above respectively. We choose a fixed Cartesian coordinate system in which the z-axis is normal to the stratification, and the x-axis (parallel to the stratification) lies in the magnetic meridian (b, 2) plane, where 6 is the external magnetic field. (This choice of magnetic

medium for its validity, all currents here be in the presence of the medium.

and hefds ~111 same phywai

PLANEWAVE ANGULAR SPECTRUM OF AN ARBrJaARY CURIWNT DJSIRIBUTION IN FREE SPACE

2.THE

meridian plane, at y = 0, is not unique. and any other plane parallel to it, y = const., could have been used.) Let the incident propagation vector kc, in a given problem have direction cosines (I, m, n ),

All spatially dependent currents and fields, which are assumed to have an exp (iwt) time dependence. are Fourier analysed in k-space, so that typically

i.e.

Hk (Z’, m’, n’) = k,, = k,(f, m, n) = (w/c)(l. m, n)

(4)

=

whereas in another, ‘conjugate’ problem let the conjugate propagation vector k,,’ he given by k,,’ = ko(-l, m, n I.

(5)

Then the scattering theorem relates the reflection and transmission matrices in the two problems: R’(I, m) = &“(-I, m) = R.’

(6)

T”(I, m) = P(--I,

(7)

m) f _i: +

the tilde (-) denoting matrix transposition. The matrix elements in (6) and (7) give the amplitude ratios of the characteristic waves of the medium (ALTMAN and SUCHY, 1979), but if as in the present analysis the wave fields are to be in free space (below or above the ionosphere) we may use the amplitudes of the linearly polarised TE or TM wave fields. In this case (6) and (7) take the form:

IIT IIT.,+ .T,+ iTl+ Equation DEN

.T’ ,7‘:.- 1.

H(r) exp (ik . r) d-‘r H(x, y, 2 )

x exp [ik,,( l’x + m’ y + n’z )] dx dy dz H(r)

=$-$ IIIH,(Z’,

m’, n’)

xexp [-ikn(Z’x + m’y + n’z) dl’ dm’ dn’

(11)

where k = k,(l’, m’, n’) is an arbitrary point in kspace, and all points in this space are included in the Fourier integration. Maxwell’s equations VxE=-iFo&

(1%

VxH=ie,,oE+J

(13)

become k x Ek = k,Z,H,

(14)

kxH,=-k,Y,E,+&

where Z,= l/YO=(&~O)l’Z. Eliminating E;, or Hk in (14) and using the relation k . Hk = 0 (which follows from V * H(r) = 0), we obtain ikx.I, HkZ------k,,Z-k2

(9)

(8) was first given by BARFLON and BUD-

E,=-

Substitution . 2 H(r) =$

(15)

-iZ,

(16)

k

(1959).

In what follows it will be shown that a scattering theorem, such as (8) and (9), leads to a Lorentz type reciprocity relation relating a given pair of current distributions and their associated electric wave fields, with a mirrored pair of current systems (mirrored with respect to the magnetic meridian plane) and their wave fields. More specifically, reciprocity will be shown to apply to a given transmitting and a receiving antenna, below or above the earth’s ionosphere, and a second pair of antennas, mirrored with respect to the first pair, in which the role of receiver and transmitter have been interchanged. Unlike the modified Lorentz reciprocity theorem which invokes a hypothetical transposed

(10)

of (15) in (11) yields (I’, m’, n’) X Jk ~-p_m~2_n~2

xexp [-iko(l’x

+ m’y + n’z)] dl’ dm’ dn’

(17)

Now take the inverse transform of Jk and substitute in (17) to yield H(r)=%

(I’, m’, n’) X J(r’) pp_m‘2_nr*

xexp [-ikO{l’(x-x’)+ xd3r’ dl’ dm’ dn’.

m’(y-

y')+n'(z

-z’)}] (18)

341

Reciprocity theorem relating currents and fields in a plane-stratified ionosphere

generality, that the current distribution consists of an elementary dipole, p, = I&, situated at r = rl, so that in terms of S-functions

We now keep 1’ and m’ fixed, l’=l,

m’=m

and integrate over all n’. If 12+ m2> 1, there are poles on the imaginary axis at A’= fn = fi(12+m2-1)‘“. If 12+m2<1, then the poles are at n’ = fn = f(1 - 12- m2)*=. We close the integration path along the Re (n’) axis by a semi-circle at infinity in the upper or lower complex-n’ plane, according as z < z’ or z > Z’ respectively, and when I2 + m2 < 1 we indent around the poles as shown in Fig. 1. With the aid of the Cauchy residue theorem we obtain

J(f) = p18(r’- r,)

“Ptsfx’-x1)6(y’-y,)6(2’-2,)

(21)

and (20) becomes, according as z~z,,

xexp [-ik,* - (r - rl)] df dm

Pdl x exp E-i$U(x -4 + dv - Yd i n(z -z,)}] di dm. - (Plxr Ply,

y’)

xexp[-ik,{Z(x-x’)+m(yf n(z -z‘)}] d3r’ dl dm

-ko

The x and y components of E,(r) may conveniently be written in matrix form, for later use, as

; ko*xJ(r’)

=2

871’

xexp [-iko* * (r-r’)]

d3r’dl dm

(19)

where k,’ = k& m, *n), the sign of n depending on the inequality z B t’. (We note that when the negative pole, at -n, is captured, the contour is traversed in the positive sense, whereas when the pole at +n is captured it is traversed negatively, and so the result has the same sign in both cases.) The same integration procedure applied to (16) yields E(r)=3

f (k,‘[k,* xexp [-ik,’

(22)

- exp [-ik,” * (r-r,)]

dl dm

(23)

where

(23a)

* J(r’)]- k$J(r’)}

* (r-r’)] d3r’ dl dm.

(20)

Equations (19) and (20) give the required planewave angular spectrum for an arbitrary current distribution J(r’) in terms of the direction cosines, t and m, of k,,. In order to simplify the expressions in the subsequent discussion we may assume, without loss of

3. DFXIVA’HON

OF lWE

RlWIPROC!nY

‘IXDC4lREM

For a given propagation vector ko*, the electric field vector E” is fully specified by its x and y components, since in free space ko*. E’=O, and hence

(24)

Fig. 1. The complex-n’ plane integration contours.

It will be convenient to choose the TM and TE modes in such a way that the modal amplitude is positive when the electric or magnetic field vectors in the plane of incidence [TM or TE respectively) point away from the z-axis (or to be more precise point in the *ko* x (k,,” x 2) direction], as illustrated in Fig. 2. This choice conforms with the requirements of the scattering theorem #IAMAN and SUCHY, 1979). Then it is easy to pass from

342

:< T+( I, m)

In

mn

-_([) + m:

m

--1

0

C

1

PI

xexp[-iko(t(x2-xt)+~n~yYz-y,1}]dt TE modes

TM modes

Fig. 2. The electric wave vectors E,’ TM modes.

and ElIi’for TE and

fixed Cartesian to modal components:

1=

Eli” [ EL”

l

[ uny[z1”]=42*[~]

(Z2+ n12)1/2fm

drn

in which we have evaluated the matrix products C,,“(fI’))’ and AI*&“. The product p2 . E,(r,) in (29) has been called the “reaction” of the field Et on the source pz (%JMSEY~ 19%). Let us now consider the conjugate problem, and calculate the ‘reaction’ of the field Ezf, generated by a dipole pzr, placed at a point r< = (xzi, yi. z,“)_ z2( = z2> above the ionosphere, on a dipole p,’ situated at r,‘- = (x,~, yIc, z,‘), zl’ = .z,, below the ionosphere. In analogy with (28) and (29) we have

pzc exp [-iko{l”(xl’

x T (1’, m”)C&

-- xg)

+ rnC(YIL- yz’,}] dl’ dm’. s(fi*)-.’

;I:

(26)

c i I

thereby defining the ~n~o~a~on matrices a*. Consider now the integrand on the right-hand side of (23) describing the x and y components of a single plane wave in the angular spectrum generated by the dipole pt, located at rl. Suppose that rl lies below the ionosphere, and we wish to determine the overall wave field at a point r, above the ionosphere. We take the upper sign for upgoing waves in the integrand of (23), premultiply by R’ (25) to convert to TM and TE modes, and then premultiply by the transmission matrix T’(1, m) 5 r’(Z, m; z2, 2,) to pass from z1 to 2::

[‘=-I,

BlrE2C(r,‘)=-

-&&<,

JJ

C32+(fl+)-1T+(l, m)fi+D,,+p,

xexpE-iko(l(xz-x,)+m(y,-y,)}ldl

dm.

(311

ln m i--G‘ I nU2+m2) _(p+m 1JJ 0 [pIc01

x exp [-iko{l(rz”

-mn

1 -m?l -(IZ+m2)

8~’

-x,‘)-

m(y,’ - y,“)}] dl dm.

(32) (29) and (32) we see that

provided that p;=

100 1

0

-1

[0 -Zoko2

171“zr.n

in order to be able to apply the scattering theorem (7), giving

~2 * C(b)

&h) = T

(30)

We take the transpose of the right-hand side, a scalar, evaluate the matrix products and equate

Comparing

Next we premultiply by @I*)-’ to convert back to x and y Cartesian components, and finally by Cszi (24) to recover the E, component of the field E,(r,). In all,

(29)

0

0

(33)

= pIc . Ez’(rlc)

pi”Lpi*

i-l.2

(34)

1

and x;c =

(28)

Finally we premultiply (28) by the dipole vector (row matrix) &, located at rz, where p2=16i2, to

x,,

yic = -y;,

i=l,2

that is r,’ = (xi, -Yi, 2,).

(35)

Reciprocity theorem relating currents and fields in a plane-stratified ionosphere This mapping transformation, (34) and (35), gives the mirror images of the dipole vectors, p1 and pz, with respect to the magnetic meridian, (b, f), plane. With pi = ISl, and pC = ISC, (33) becomes E1(r2) * S12(r2)= Ezc(ri’) . Sl,c(rIc)

(36)

which states that the EMF induced across Sll by a current I in 61, is equal to that induced across Sl,’ by the same current I in 812’, where SIX’ and Slzc are the mirror images of SI, and 61, with respect to the vertical magnetic meridian plane, and Ei’ (i = 1,2) denotes the fields generated by the current I in Sr. This reciprocity relation is represented schematically in Fig. 3. If both dipoles lie below the ionosphere, the transmission matrix in (28) is replaced by R’(I, m), insofar as the ionospheric contribution is concerned, and C,,‘@‘)-’ by C,,-(a-)-‘. (The direct ray between p1 and pz, and that reflected from the earth’s surface, will in any case satisfy the reciprocity relation, (33) or (36), and will not be considered further). Then

B2&b2)

-nl

m

-nm

-1

-(12+m2)

0

Z&o*

=T

xR+(I, m)

In

mn

m

-1

-(lZ+mZ) 0

p1

1

I

xexp[-ikO{l(xz-x,)+m(y2-

yAd1

dm.

(37)

The same expression holds for the conjugate problem, with pl(rl), pz(r2) and 1 being respectively replaced as before by pZC(r2’), p;(r,‘) and -1 throughout. Taking the transpose of the right-hand side and applying the scattering theorem (6), we again obtain the reciprocity relations (33)-(36). 4. GENERALUA’IION

AND DISCUSSION

The

proof given in the preceding section applies without change when arbitrary current distributions py(x,y,z)

Transmitter

= Lp,(x,-y,z)

Receiver

i

/

I

,i=1,2 Transmitter

\

Receiver

Fig. 3. Reciprocity when transmitter and receiver separated by a plane-stratified ionosphere.

are

343

are used instead of simple dipole sources, starting with (20) rather than (22), except that the expressions become much more cumbersome. Equation (33), when generalized to arbitrary current distributions, becomes

I

E,(r) - J2(r) d3r =

E,‘(r). J:(r) d3r (38) I in which e(r) is the field generated by the current distribution &(r), i = 1,2 and K(r) is the field generated by the mirrored current distribution J,‘(r), where

Ji’(x, Y7 z)‘wi(X,

-Y, z,

(39)

as in (34) and (35). It will be shown elsewhere that (38) in fact represents a more general relationship than that proved here, and applies to current distributions, or antennas, both outside or inside the ionosphere. Let us consider some simple applications. Consider two linear antennas lying in a magnetic meridian plane and parallel to it, i.e. both pointing in a north-south direction. Then the two antennas are their own mirror images, and so simple Lorentzian reciprocity applies to them. If the two antennas in the meridian plane are horizontal and point in an E-W direction, then the mirroring process is equivalent to reversing the sign of the current in each of them, and so nothing has changed and Lorentzian reciprocity again applies. On the other hand, if one of the two antennas is parallel to the meridian plane and the other perpendicular to it (i.e. pointing in a horizontal E-W direction), then only one current reverses its direction on being mirrored, and we have ‘anti-reciprocity’ (Lorentz reciprocity with a change of sign). Consider next two vertical or two horizontal antennas that lie on a horizontal E-W line. Suppose that propagation is from east to west. The mirroring process, and hence the reciprocal arrangement here means interchanging the two antennas with propagation remaining from east to west. Most of these results are well known, and have been derived through other considerations BUDDEN, 1961 ; BUDDEN and JULL, 1964), In more general antenna orientations, however, one must of necessity have recourse to the mirroring process here derived,

Acknowledgement-Part of this work was done during the stay by one of the authors (C. A.) at the Institute for Theoretical Physics, University of Diisseldorf, supported by the Deutsche Forschungsgemeinschaft. Thanks are due to Prof. K. SUCHY for stimulating discussions.

344

L

AL-MAN

and A.

SCHATZFIERG

RFBJtRENCFS 34lxMAN c. and SUCHY K. 8-0~ D. W. and BUDDEN K. G BUDDEN K. G.

1979 1959

BUDDEN K. G. and Jvr.~. G. W. KONG J. A. PITIXWAY M. L. V. and JESPERSENJ. L. RUMSEY V. H. Sucm K. and km C.

1964 197s 1966 1954 1975

1961

Appt. Phys. X9, 337 Proc. R. Sot. A249, 387 Radio Waves in fhe kmosphere, Cambridge tinis Press. Can. J. Phys. 42, 113

TheoryofE&romagnetic Waves, Wiley, New York. terr. Phys. 28, 17. Phys. Rev. 94, 1483. J. PZ~ma Phys. 13, 437.

J. amos.