Appl. Phys. 19, 337-343 (1979)

Applied Physics 9 by Springer-Verlag 1979

Generalization of a Scattering Theorem for Plane-Stratified Gyrotropic Media C. Altman Department of Physics, Technion, Haifa, Israel

K. Suchy Institut fiir Theoretische Physik, Universit~it, D-4000 Diisseldorf, Fed. Rep. Germany Received 8 September 1978/Accepted 31 December 1978

Abstract. A previously derived eigenmode scattering theorem for plane-stratified gyrotropic media established that the scattering matrices S and S', defined, respectively, in terms of a given and a conjugate set of eigenmode amplitudes, are mutually transposed, i.e. S = S', the theorem being valid specifically for the incoming and outgoing eigenmodes of gyrotropic bounding media. In this paper the theorem is extended to include linearly and circularly polarized base-modes in the isotropic media which may bound the multilayered, gyrotropic structure. Transformation of the eigenmode scattering matrix leads to a generalization of the scattering theorem to include base modes which are not necessarily eigenmodes of the medium. In the important special case in which the ambient magnetic field is parallel to the plane of the stratification, the scattering matrix is shown to have a special symmetry whose form depends on the base modes chosen (eigenmodes, linear modes or circular modes). PACS: 42.10 In an earlier paper [1] we considered normalized eigenmodes (characteristic waves), gs (s= __+1, +2) in the form of 4-component eigenvectors which propagated in the positive or negative z-directions of a plane-stratified gyrotropic structure, and which varied as exp[i(cot-koSx)J where ko=cO/c, and S is a constant of the propagation; the z-axis is normal to the stratification, and the x - z axes define the plane of incidence which is at an azimuthal angIe r with respect to the magnetic meridian plane (;~,b0), where b 0 is the external magnetic field vector; g, is composed of the normalized wave-field components parallel to the stratification

U=U_t =

0

1

1

0

0

0

"

(3)

By means of the modal matrices G=[gl

g2 g-1

g-2]

(4)

the biorthogonality condition (2) may be expressed as (~UG = J

(5)

or

~s=[Ex

-Ey

~x

~'~y3,

(1)

where ~ = (#o/~o)1/2H, the tilda (~) denoting the transpose. The normalized adjoint eigenvectors gs, obtained by reversing the direction of the external magnetic field, were shown to be biorthogonal with respect to the given eigenvectors gs with ~sUg~= 6s, sign (s)

(2)

(] = U(~- ~J,

(6)

where j=

1

0

0 0

-1 0

(7) 1

0340-3793/79/0019/0337/$01.40

338

C. Altman and K. Suchy

Now whereas in [1] a prescription was provided for constructing the normalized adjoint eigenmodes, we shall use (6) as a definition and means of calculating t], the modal matrix adjoint to the given matrix G, which is composed of suitably normalized base modes g,. A given wave field vector e composed of the tangential electric and magnetic field components, as in (1), may then be decomposed into the base modes gs, according to

with respect to the magnetic meridian plane (~, b0). The incidence plane for the given (original) set of wave fields is, by contrast, at an azimuthal angle ~b with respect to the magnetic meridian plane. The z'= z axis is normal to the stratification as before. The normalized conjugate eigenvectors g's in the conjugate system (x', y', z') and their adjoints g's were seen to be simply related to the given normalized eigenvectors in the original system (x, y, z), with

e=Ga

g;=L~_s,

(8)

where

~; =Lg_~

(14a)

and in terms of modal matrices (9)

G' = L(]C,

(] = LG'C

(14b)

represents the modal amplitudes in the given wave field. At an interface between two media, v and v + 1, the scattering matrix S relates ingoing and outgoing modal amplitudes

(]' = LGC,

G:L(]'C

(14c)

=[-a 1

a2

a_l

a_23,

where

- 1 00

~176176 1: 0 0

(10a)

aou t = Saln

which, written out in full, reads

1 0

(15)

1

-0 lay_+ 1 ] ,

I2:~+1] = [::

:~1 [a~_ ]

(10b)

where r -+ and t • are 2 x 2 reflection and transmission matrices for incident modes in the positive or negative z-direction ; a + and a _ are condensed notation for the modal amplitudes :

a+ = [a~ 1,

a_ = [a - l ] .

La2]

(11)

a_ 2

C=C_I=

V=V_l=

where Girl=[g, ~ g~ Gout=[-g~-i

_gV+l -g~-2

_g~+213 g~+l g[+Z].

(13)

Equation (10a) defines the scattering matrix S also for a multilayer slab, for which the ingoing and outgoing modal amplitudes are specified in the lower and upper (with respect to z) gyrotropic bounding media. In this case the reflection and transmission matrices composing S are denoted R-- and T-- to distinguish them from the 2 x 2 interface matrices in (10b). The conjugate eigenmodes were then considered, with field quantities varying as exp [i(cot- koSx')], with the conjugate x ' - z ' axes specifying the conjugate incidence plane which is at an azimuthal angle 0 z - r

(16) "

Gin = V G i n ,

G'ou t = VGou t

(17)

where

(12)

GoutS = Gin ,

0 0 0 0 1 0

In the important special case of the external magnetic field in the plane of the stratification, the relationship between the given and conjugate modal matrices, (4) and (13), simplifies to G ' = VG,

It was shown in [1] that the interface scattering matrix relates the normalized base modes gs, i.e. the characteristic polarizations, and not only the modal amplitudes, with

0

--1

0

0 0

-1 0

(18)

and the eigenvalues qs (i.e. the z-components of the modal refractive index vectors) are equal in the two systems :

q; =qs.

(19)

Finally, application of (5) to the two sides of an interface, after the adjoint mode's are transformed by means of (14b), yields with the aid of (12), the eigenmode scattering theorem for an interface S=S'

(20a)

which, in terms of the 2 x 2 interface reflection and transmission matrices, reads r • = F -+ ,

t • =t'~

(20b)

Scattering Theorem for Plane-StratifiedGyrotropic Media Application of a set of recursion equations then extends this results from an interface to any gyrotropic multilayered slab bounded by gyrotropic media, to read R_+ =1~,+ ,

T + = ~[,, -v-

(21)

1. Isotropic Bounding M e d i a

The eigenmode scattering theorem, (20) and (21), was defined in terms of the eigenmodes of gyrotropic media [1]. We need first to extend the theorem to include gyrotropic multilayer slabs, which are bounded at one or both ends by isotropic media. Although the generalization of the scattering theorem to base modes which are not necessarily eigenmodes is deferred to the following sections, we note that any set of "modes" in the isotropic medium which satisfy (5), the biorthogonality condition, and (14), the mapping transformation from the adjoint to conjugate systems, will obey the scattering theorem, for these were the only equations used in its derivation aside from (12), whose validity is not restricted to eigenmodes. The wave fields of an acceptable "mode" i.e. satisfying Maxwell's equations and propagating in the positive or negative z-direction with an x, t dependence given by exp [i(cot-koSx)] in an isotropic medium, will have the general form, in an auxiliary cartesian system (~,r/,{) tied to the wave normal [1]:

Es=[1,G,O]Er

Jfs=[-G,l,O]nEr

(22)

where n is the refractive index of the medium, G is the transverse polarization of mode s, with the ~-axis in the wave-normal direction and the ~-axis lying in the plane which contains the wave normal and the external magnetic field directions. If we wish to simulate gyrotropic modes in the isotropic bounding media, then reversal of the external magnetic field should lead to a change of sign of the transverse wave polarization [1,2], and hence the adjoint fields should have the form : 1~ = [ 1, - G, 0]/~r

. ~ = [G, 1, 0]n/~r

(23)

The modal product (2) is just the z-component of the Poynting-type vector product

= 2[0, 0, 1 -

Get]nEr162

(24)

339

the case of normal incidence on the stratification. This condition is too restrictive however. Consider the case of linearly polarized modes with the electric field in the plane of incidence or perpendicular to it, with normalized amplitudes 1/(2cos0) 1/2 where 0 is the angle of incidence. Let us form the (same) modal matrices in the given and conjugate systems (i.e. with incident wave normal vectors at azimuthal angles ~b and ( n - q S), respectively), G=[gl

g2

g-1

g-2]

~zosO _

1

0

(2COS0)1/2

0

0

cos0

-1

0

1

0

--COS6

--COS0

1

0

0

- 1

=G'.

0

(26)

It can easily be verified that the matrix G is self-adjoint in the sense of (5): (lUG--J,

(]=G.

(27)

Furthermore it may be verified that ~s =gs=Lg'_s

(28)

and (27) and (28) are just the sufficient conditions for validity of the scattering theorem. Hence, for a multilayer slab bounded by isotropic media

["+ l-I T+

R-

iT'-

*:+1 R'-J

where R • = [IIRI~

flR[]

(30)

are the "conventional" reflection coefficients in terms of linear modes [4], and T • are the transmission matrices similarly constructed. The relation (29) between the reflection matrices (R • =11 '• is the "reciprocity theorem" of Barron and Budden [5], and the relation between the transmission matrices in (29) is the appropriate generalisation of that theorem. It is often convenient to use circularly polarised base modes in the isotropic bounding media. Typically the normalised eigenmode matrices in the given and conjugate systems will have the form:

cos0

G= G'= - 2c0sl/20

cosO

cos0 cos0

I770-i o 0-icos0 9

-i

i

-i

ico7

and the modes will be biorthogonal provided that (1 -

GOt)= (1 -

0 • ~0 • 2) = 0

(31)

(25)

which will hold, for instance, for circularly polarized waves with 0 = -+ i. Equation (25) has been shown [3] to be a sufficient condition for the validity of the scattering theorem in

It may be verified that the transformed matrix LG'C, cf. (14b), is biorthogonal with respect to G, and is thus the adjoint of G: (]=LG'C,

~UG=J

(32)

340

C. Altman and K. Suchy

and in terms of right-handed (r) and left-handed (I) circular base modes in the isotropic bounding media, the scattering theorem (29) is again valid with

and the scattering theorem will apply to the base modes at an interface provided that, in analogy with

R-- = r~R~ [,R+

F+ = L F ' ~

~R~] zRz-+]

T -+ = [~T'• ~Tl-+] [tT~-+ ,T~-+]'

(33)

(14) (41)

or

We note that the modal matrix (31) is not self-adjoint as in the case of the linear modes, the polarisations of the base modes s = - + l and s=_+2 having interchanged in the adjoint modal matrix (~.

G• f2 • = LG'; ~'=.

(42)

But since, from (14), (]_+ =LG'_7, this means that ~)+ =g?'~

(43)

which, with (39), yields

2. The Interface Scattering Theorem in Terms of Base Modes which are not Eigenmodes

~)'~ ~-+ = i .

We construct positive-going base modes ~,1 and ~2 from linear combinations of the positive-going eigenmodes 01 and g2, and similarly for negative-going modes. For convenience, we form 2 x 4 matrices, F+ and F , from the positive- and negative-going base modes. Then

Equations (39) and (44) are thus the sufficient conditions for the interface scattering theorem to apply to general base modes. To determine the modal amplitudes c%1, 0~+_2 which are related by the scattering matrix (as are the eigenmode amplitudes a+l, a+2), the total field e may be expressed as

F + = [~1

3~2]= [01

FO~ f2~-2] = G+f2+ ~2+d

023 L~I

(34)

(35)

where ~ + and $2- are the 2 x 2 transformation matrices. We combine the base modes into a modal matrix F so that

rd:EG+

G_3

01

a--Gf~

(36)

~+ =

, ~2

~_ =

i l]

in analogy with a• (11), this may be written

[G+ G-][a+]La_j=EF+F lie+ ]e_ + 2[::]

/~UF = a

(37)

as in (5) and (6). The relationship between r and the adjoint eigenmode matrix (] then determines the adjoint transformation matrix g] through Y=GO.

(38)

Substitution of (36) and (38) in (37) yields

~dUGa=a:dUG ~+g2 + = i

(39)

=i

where i is the 2 x 2 unit matrix. In the conjugate system there is a corresponding modal matrix r ' = Er'+

r ' _ 3 = EG'+

(47)

with the aid of (36). This implies that g2+~+ = a + ,

g2-~_ = a _

(48)

and we see that the modal amplitudes are related by the same transformation matrices, (34) and (35), as the modal vectors (although the form of the transformation is of course different).

3. Generalization to a Multilayer Slab

whence

h-a-

(46)

0~-2

and form the adjoint modal matrix /~ = U/~- 1a,

(45)

s

or, in the condensed notation

F_ = G _ g 2 - ,

r=Er+

e = ~, asg s = ~ ~s~s (s = _+1, _+2) s

and similarly

(44)

G'_]

0 ~G'I2' ~,_

(40)

Consider next the transformation of the scattering matrix S for a multilayer slab, containing ( # - 1) layers and bounded by media v and v+#. In terms of eigenmode amplitudes, (49)

aou t = Sain

which, written out in full, reads

a%]= ,+

L+,

raaV+/Z +l 9 L-J

(50)

ScatteringTheoremfor Plane-StratifiedGyrotropicMedia The corresponding scattering matrix ~ in terms of base-mode amplitudes a_+ satisfies

,ut IS)

[

=.9 ~ a~++

(51)

the linear modes (26): c! 0 F - (2cos 0)1/2

R-j[o

L~+q (52)

Hence

2cos1/201icos 0 L 1

where +

Q~+.]

Apply now the scattering theorem S = S' to the slab to obtain i

~, =(~i.)

-

i

~~,o ~~,t

(56)

which again yields ow=~'

(57)

provided that ~'i n O - - o u t =~)'o u rO '-in =I.

(58)

Decomposing (58) into the component transformations (54), in each bounding medium, we find $]'-+f2 ~ =i

- - COS I

0

--1

0

cos0

--i

i

- i cos 0

-icos

1

c~

ic_~

0

--1

I1 01)

01 11 _i 21/2

(54)

and we have the well known scattering matrix transformation [6] in both the given and conjugate systems

f2out~12 ~

0

(60)

-

.Q~+~]' f/~

1

- - COS 0

COS0

0

=Gf2

(53)

= f~outlm~in~in

.f/i-=[~~

0

'

whence

~out = f ~ i n

cos 0

Icos0 1

T+

0 -1

1

Now transform (50) with (48):

~Vq[~d"J

341

0

i

1

-_o,

I1

Evidently, ~'-+12; =i as expected, confirming that the linear base modes in free space are valid for use with the scattering theorem.

4. External Magnetic Field in the Plane of the Stratification

a) When the gyrotropic symmetry axis is in the plane of the stratification, as is usually the case in problems involving ferrite sheets, or in ionospheric propagation at the equator, there is a simple relationship (17) between the normalised eigenmodes in the given and conjugate systems: gs=Vg'~,

Gi. =VG'in,

Gout= VG'out

(62)

where

(59)

which is the same sufficient condition (44), derived earlier, for the interface scattering theorem to apply to general base modes. We consider a simple example of a slab bounded on either side by free space. Suppose the free space "eigenmodes" to be circularly polarized. (We ignore for this purpose our earlier discussion in Sect. 1 on isotropic bounding media, and consider free space, as for instance at the base of the ionosphere, to be a magnetoplasma in the limit of infinite electron collision frequency and zero electron density. In this "quasi-longitudinal" limit the magnetoplasma eigenmodes are circularly polarized [7], and the scattering theorem applies to them.) Now consider the transformation to linearly polarized base modes. The same transformation matrices f2 = g2' in the given and conjugate systems connect the modal matrices G ( = G') for the circular modes (31) and F ( = F') for

V=V_I=

0

-1 0 0

0 -1 0

'

Thus the given and conjugate eigenmode electric field t components, (Ex,- Ey) and (E'x,- Ey) are mirror images in the (s x bo) plane (the magnetic East-West plane), and the magnetic field components are mirror images in the magnetic meridian plane (2, bo). We have seen (12) that the interface scattering matrix S is determined by the modal matrices, i.e. by the characteristic polarisations, on both sides of an interface : Gouts = Gi.,

GtoutSt = Gin .

(63)

Premultiply by V- I(=V) and with (62):

G'outS=%

(64)

342

C. Altman and K. Suchy

and hence the interface scattering matrices are equal S=S'

(65)

(although defined in terms of different, mirror modes). In conjunction with the scattering theorem, S = S', {65) yields the important result that the interface scattering matrices are symmetric S=S.

(66)

Equation (66) may be extended to an arbitrary multilayer, gyrotropic slab by means of the recursion relations [Ref. 1, Eq. (60)], and following the same arguments of that paper. If Sv is the scattering matrix for a multilayer slab whose uppermost layer is v, and Sv + 1 is the scattering matrix after an additional elementary layer has been added, then it is easily shown that if S~ = Sv, then S v+ 1 = Sv + 1. The proof by induction now commences with the slab which is just the lowest elementary layer, in which the matrices R + and T + are replaced by r + and t + for the first interface. But for these it has been shown (66) that S = S, and hence the result applies to the slab as a whole, i.e. S=S,

R -+ =1~--,

T • =~T

(67)

giving the special symmetric structure of the eigenmode reflection and transmission matrices in this system. b) An important extension is to linear, mirror basemodes in isotropic media bounding the multilayer system. Typically, if the modal matrix in the given system is, as in (26), -[cos 0

[ 0

1 G~

(2COS 0) 1/2

0

cos 0

0

- 1

0

1

(68)

0

--COS0

0

--COS0

1

0

-1

0

then the corresponding mirror-mode matrix in the' conjugate system is

G~nlr =

VG=

(2cos 0) 1/2

0

cos0

COS 0

0

0

-1

C

G' = G'~rL,

(71) Ii

- 01 00

L=L-1 =

00

Let the interface scattering matrix S relate linear base modes in an isotropic medium v and eigenmodes in a gyrotropic medium v + 1 on the other side of the interface; and let S'~r relate the corresponding mirror modes in the conjugate system. Now (63)-(65) still apply, i.e.

00

(72)

01 _01

and hence the scattering matrix S' in terms of the linear modes G', may be transformed as in (55), with S'mir-- L S--' L--

(73)

= L S L (because of the scattering theorem, S ' = S) =S

[because of (70)],

and so S=LSL.

(74)

This result was first derived by Heading [8], and gives the specific symmetry of the scattering matrix in terms of linear base modes when the ambient magnetic field is parallel to the stratification. In terms of the "conventional" reflection and transmission matrix elements [Ref. 4, Ch. 7], this reads

=iV.

'

(69)

S = S~air.

which linear modes are chosen which are mirror images of each other in the two systems. These may again be derived by means of the set of iterative equations [Ref. 1, Eq. (60)]. With q'~=qs (19) in all layers, it is easily shown as before that if S~ = S'v(mio then S~+ 1 =S'~+l(mir) , and since this equality holds for the first interface (70), it holds for the multilayer system as a whole. The scattering theorem on the other hand does not apply to these mirror modes (i.e. S :0FSmir) ~' for even though the modes G and G~i ~ (68) and (69), are self-adjoint, the mapping transformation (14) from adjoint to conjugate systems does not hold. However, there is a simple transformation between the linear base modes G' (26), for which the scattering theorem, S = S', has been shown to be valid (Sect. 1), and G~i ~9

A completely analogous analysis can be made for circular base modes in the isotropic bounding media. The important difference is the interchange of modes g; (s= _+1 interchanging with s = _+2) in the conjugate mirror system with respect to the original circular base modes (31): Gm~r = G ' F

(76)

with

(70)

Consider now the scattering matrices S and S~, for a multilayer structure bounded by isotropic media in

F=F-I=

0 0 0

0 0 1

"

(77)

Scattering Theorem for Plane-Stratified Gyrotropic Media

This leads to S =FSF

(78)

in analogy with (74). In terms of right-handed (r) and left-handed (l) modes, this may be written as: jR+ =,.R? ,R? =jR?

iT,-+=,Tz TjTt-+ = ~T,.~

343

else directly derivable from them. Their derivation will be considered elsewhere. Acknowledgements. Much of this work was done during a stay by one of the authors (K.S.) at the Physics Department of the Technion. This was made possible by financial support from that department and from the "Deutsche Forschungsgemeinschafi".

(79)

giving the specific symmetry of the scattering matrix in terms of circular base modes in isotropic bounding media. Concluding Remarks The methods adopted here to derive scattering relations in gyrotropic multilayer systems are applicable to anisotropic (crystalline) media too, with due regard to the different type of symmetry involved [9]. All the "reciprocity" (scattering) theorems for plane stratified systems in the radio or optical domains which were cited in [1], can be shown to be either special cases of the generalised scattering relations here established, or

References 1. 2. 3. 4. 5. 6.

7. 8. 9.

C.Altman, K.Suchy: Appl. Phys. (to be published, 1979) K.G.Budden, G.W.Jull: Can. J. Phys. 42, 113-130 (1964) C.Altman, A.Postan: Radio Sci. 6, 483-487 (1971) K.G.Budden: Radio Waves in the Ionosphere (Cambridge Univ. Press, London, 1966) Ch. 7.4 D.W.Barron, K.G.Budden: Proc. Roy. Soc. A249, 387-401 (1959) K. Rawer, K.Suchy: "Radio Observations of the Ionosphere", in Encyclopedia of Physics, Vol. 49/2, Geophysics III/2, ed. by J. Bartels (Springer, Berlin, Heidelberg, New York 1967) Sect. 13, pp. 1-546 J.A.Ratcliffe: The Magneto-Ionic Theory and its Applications to the Ionosphere (Cambridge Univ. Press, London 1959) Ch. 8 J.Heading: J. Plasma Phys. 6, 257-270 (1971) K.Suchy, C. Altman : J. Plasma Phys. 13, 437-449 (1975)