Reflection dyadic for the soft and hard surface with application to the depolarising corner reflector

I.V. Lindell P.P. Puska

Indexing terms: Reflection dyadic, Depolarising comer reflector, Plane-wave Jields

Abstract: Applying scalarisation of the plane- wave fields, an expression for the reflection dyadic corresponding to the soft and hard surface (SHS) is derived, applicable for both the electric and magnetic fields of the plane wave. Eigenvectors and eigenvalues of the reflection dyadic allow one to express the dyadic in a particularly simple form in terms of circularly polarised unit vectors. The expression can be effectively applied to problems involving consecutive reflections from many SHS surfaces. As an example, reflection of a plane wave from a corner reflector involving three SHS faces is studied. The total reflection dyadic has a simple form showing that the handedness of a circularly polarised wave is not changed in the reflection. The reflection phase angle of rays after different reflection paths is given a simple formula readily applicable for a physical optics analysis of the totally corrugated corner reflector.

1 Introduction

The soft and hard surface (SHS) has been introduced as a convenient idealised concept for the tuned corru- gated surface [ 11, and other corresponding structures [2]. Such a surface has found numerous applications in microwave engineering. Its unique property is that it gives the same boundary condition for both the electric and the magnetic field. If the unit vector v denotes the direction of conductivity on the SHS, we can write the boundary conditions in the form:

One of the interesting qualities of the SHS due to the conditions (eqn. 1) is that, if it is being illuminated with a circularly polarised wave, the reflected circularly polarised wave retains its handedness [ 11. This makes the SHS an attractive construction material in radar polarimetric reflectors [3-51. If all the faces of the cor- ner reflector are made of a perfect electric conductor, the handedness of a circularly polarised wave is changed in every reflection, eventually leading to an 0 IEE, 1996 IEE Proceedings online no. 19960585 Paper first received 13th November 1995 and in revised form 1st May 1996 The authors are with the Electromagnetics Laboratory, Helsinki University of Technology, Otakaari 5A, Espoo 02150, Finland

v * E = O v . H = O (1)

outcoming wave of opposite handedness. To have a returning wave of the same handedness as the incident wave after three reflections, it was suggested that one of the faces in the corner reflector be replaced by a SHS [5]. To have the same property after any number of reflections, all three faces should have SHS covering, Fig. 1. The analysis of such a reflector is apparently more complicated and calls for a simplified method.

Fig. 1 SHS corner reflector Lines are parallel to conducting direction v on each of faces

In the present paper, applying six-vector notation [7] and expansions in circularly polarised vectors, together with dyadic formalism, a systematic method of analys- ing consecutive reflections is proposed. As an applica- tion, the total reflection dyadic for the SHS corner reflector is derived showing a simple form for the final result.

2 Theory

2. I Scalarisation of the plane wave Let us consider a plane wave propagating in a homoge- neous isotropic medium. The direction of propagation is defined by the unit vector U. The electric and mag- netic fields can be conveniently written in terms of six vectors [6, 71. A six-vector field, or six-field, is denoted by a sans-serif symbol:

The plane-wave six-field obeys exponential function dependence on the position vector r:

( 3 )

(4)

IEE Proc.-Microw. Antennas Propag., Vol. 143, No. 5, October 1996 417

From the Maxwell equations, the amplitude field vec- tors satisfy:

(5) 1

rl U x H = --E

which in six-vector form is:

(6)

The antisymmetric matrix J satisfies J2 = -I, where the I is the unit matrix. The six-field f of the plane wave satisfies U . f = 0.

Now it turns out to be possible to express the electric and magnetic fields of a plane wave in terms of their components along a direction defined by any unit vec- tor v which does not coincide with U. In this case, the triple {U, v, w} with w = U x v forms a basis of vectors. Let us denote the reciprocal basis by U’, v’ and w’ [8, 91:

w I = p - - - K K K K (71

( 8 )

(9)

(10)

(11)

u x v w , v x w I w x u U = - v =-

K = U ’ V x w = ju x VIZ = w . w

a = (u’u + v’v + w’w) . a

f = v’(v * f ) - w’(v. U x f )

f = [v’l- w’J] (v . f j

Any vector or six-vector a can be written in terms of the two sets of basis vectors as

Because of U . f = 0, the field f has only two compo- nents,

Inserting eqn. 6, we have

This is the required result: the six-field f of a plane wave is now expressed in terms of its scalar component v . f - More explicitly, we can write

-1 [U x (U x v j l + (U x v j J ] (v . f ) (12) f=m

or, in terms of the electric and magnetic field vectors, -1 u x ( u x v ) u x v ) = ( -uxv u x ( u x v j ) (;.EH)

(131 There is no other restriction to the unit vector v than U x v 9 0. In particular, the unit vectors U and v need not be real if only they satisfy U . U = v . v 1. How- ever, in the following, v is assumed real.

.2 Incident plane-wave six-vector J, is reflected from soft and Jaavd surface (SHS) whose conducting direction is along unit vector v Reflected fieldf, is obtained through reflection dyadic

2.2 Reflection from SHS Let us consider a plane wave with the six-field f, inci- dent along the unit vector U, on a planar SHS with the unit normal n. The conducting direction of the SHS is assumed to be along the unit vector v satisfying v . n = 0 (Fig. 2).

Let us denote the reflected plane wave by the six-field vector f , and its direction of propagation by the unit

418

vector U, satisfying:

The total six-field f =J1 +f, satisfies the boundary con- dition (eqn. 1) implying

From eqns. 12 and 15, the reflected six-field can be written in terms of the v component of the incident six- field,

v . f = O =+ v - f , = --v ’ f z (15)

Applying eqn. 6 we can write

which inserted in eqn. 16 gives us the following expres- sion for the reflected six-field:

J(v. f . ) = v . J f , = v . U, x fi = (V x U,) . fi (17)

The boundary condition can be immediately checked by substituting eqn. 18 in v . (f; + f,) which after arranging terms is seen to vanish.

Becausef, is orthogonal to U, we can writeJ1 = -U, x (U, x f,). Applying this, a more symmetric expression is obtained. The relation can be written as:

- f, = E * f i

-1 - - R=- [(U, xvj (ut XV)+(U, x (ur x V ) j (ut x (ut xv))]

IurxVl2 - (20)

E, = R . E , H, = R * H , (21)

It is seen that the same reflection dyadic R operates on both the electric and the magnetic field:

which is due to the similarity in the boundary condi- tions (eqn. 1). It must be noted that this property is not shared by the perfect electric conductor (PEC) or per- fect magnetic conductor (PMC) surfaces, for which the reflection properties are:

with upper signs referring to PEC and, lower signs, to PMC. It is seen that, for PEC and PMC surfaces, the reflection dyadic does not depend on the direction of incidence.

- - -

- - E, = F E . E, a, = G.H, (22)

2.3 Properties of the reflection dyadic The reflection dyadic satisfies the obvious conditions:

- - -

u , . R = O R . u , = O ( 2 3 ) The first one simply states that the reflected field is normal to the propagating direction U,. The second one has no special meaning because the incident fields are orthogonal to up The reflection dyadic is a planar dyadic whose three-dimensional determinant function is zero.

2.3. I Power reflection: The Poynting vector of the plane wave transforms in the reflection as

S - -E, x I€: = - (R.E,) x (R.IiC:) = R . S , (24)

where is the double-cross square of the dyadic [9]. It can be expanded after some simple algebraic steps as

=(2) - 1 1 = ‘ - 2 2

IEE Proc.-Micuow. Antennas Puopag., Vol. 143, No. 5, October I996

Thus, in a lossless medium, the Poynting vector of a plane wave preserves its magnitude but the direction is changed in reflection from ui to U,. Here, we have assumed real unit vectors ui, U , so that R* = R.

2.3.2 Eigenpolarisations: It is of interest to find the eigenpolarisations of the incident field defined as those which do not change in the reflection. It is well known that TE and TM polarisations with respect to the normal direction n are invariant in the reflection from an isotropic planar surface and, thus, form the eigenpolarisations in that case.

Let us define the invariance of polarisation with respect to the direction of propagation. The incident fieldf, is defined to have the same polarisation as the reflected field f , iff, and the mirror image off , are sca- lar multiples of one another:

-

C . f r = Rfz ( 2 9

C * R * f = R f (27)

This leads to the eigenvalue problem of the dyadic C R: - - -

where R is the eigenvalue and f the eigen six-vector. The problem can be visualised as reflection changed to transmission through the plane so that U , becomes ui (Fig. 3). The eigenfields are then changed only in mag- nitude by the scalar factor R.

\ - c . ur

Fig.3 through mirror dyadic C square root of unit dyadic 7

Direction of LeJection U, and direction of incidence U , are related

Applying a property of the mirror imaging dyadic

- - - 191,

C . (a x b) = -@.a) x (E. b) - - (28) valid for any vectors, and noting t h d C. v = v, C . U, = U ; and C . ui = U, the dyadic C . R is seen to be symmetric:

1 _ _ _ _ _ C.R= ~ [(uixv) ( U ~ X V ) - (uix(uixv)) (uix(uzxv))]

(29) IUi x VI2

VI2 = IU, x v12.) (Note that, because of ui . v = U, . v, we also have Iu; x

The eigenvalues R and eigenvectors a can be directly identified from eqn. 29. They are without normalisa- tion:

- - Ro = 0 a 0 = U, C . a 0 = U, = bo (30)

- - R+ = +1

R_ = -1

a+ = ui x v

a- = uix(uixv)

C.a+ = -U, x v = -b+ (31)

C.a- = urx (u rxv) = b-

Here we also denote by b, the reflected polarisations corresponding to the eigenvectors of the incident wave.

- -

(32)

IEE Proc.-Microw. Antennas Propag., Vol. 143, No. 5, October 1556

The three eigenvectors ao, a+, a- are seen to be orthog- onal and form a basis unless ui = v. Any incident polar- isation is orthogonal to ui = a. and thus can be expanded in terms of a+ and a-.

2.3.3 Circularly polarised vectors: The incident field can more conveniently be expressed in terms of right-hand ( r ) and left-hand ( I ) circularly polarised vec- tors, both orthogonal to U;

ar = a+ - ja- = ui x v - jui x (ui x v)

a' = a+ +jab = ui x v + j u i x (ui x v) (33 ) They satisfy:

a r . a r = a e . a e = 0 ar.ae=2a*.a* =2(uixvI2 (34) whence the unit dyadic can be expanded as

(35) 1

ar . at

- - I = u;u; + - (arae + sear)

The dyadic (eqn. 29) can now be written in the form

which implies the simple relations:

Denoting the reflected right-hand and left-hand circu- larly polarised vectors by:

_ _ - - _ - - - C . R . a r = a ' C . x . a ' = a ' (37 )

b ' = b + - j b - = u , X V - - U , x ( U ~ X V )

b' = b+ + jb- = U, x v + jur x (ur x V) (38) we can also write:

- - - - - - - R . a ' = C . a l = - b ' R . a t = C . a ' = - b ' (39)

which means that the handedness of the wave is not changed in reflection from a SHS surface. The opposite is known to be true for reflections from PEC (R, = -1) or PMC (Ri = +1) surfaces. The SHS reflection dyadic has thus the simple form:

1 ar 3 ae

- - R = --(bear +bra')

3 SHS corner reflector

As an application of the reflection dyadic expression, let us study plane-wave reflection from a corner reflec- tor made of SHS material. A similar idea with just one of the faces made of SHS material was recently pub- lished [5 ] . The advantage of such a structure was to obtain a reflected circularly polarised signal with the same handedness as the incident signal, which is not the case for a PEC corner. For an all-SHS corner, the required operation is obviously independent of the number of reflections from the three faces, although the practical advantage from this seems to be only mar- ginal.

3.1 Basic notation The corner is defined by the inequalities x, y , z 2 0. The conducting directions v on the different planes are assumed orthogonal to one another. Such a structure can be defined in two ways resulting in what may be called right-hand and left-hand SHS corners, as defined in Table 1.

Table 1

plane normal right-hand corner left-hand corner

x = 0 U, v,= uy v,= U,

y = 0 UY v y = U, v y = U,

z = 0 U, v, = U, v,= uy

419

The corresponding unit normal vectors are n, = ux, ny = U?, and n, = U,. Consider an incident plane wave with the direction ul, = au, + Buy + yu, with positive a , 8, y values, entering the face 1 of the corner reflector. The wave is reflected from all three faces in the order 1, 2, 3, which may be any permutation of x, y , z . Denote the three incident and reflected directions, respectively, by U,, andz,,,=m = I, 2, 3 and the corresponding dyadics by C,, R,.

It is easy to show that the value of (U,, . v,)~ is the same for all indices m and coincides with that of (U,, . vJ2. This is because the vectors U,, and U,, are all of the form ct au, ct Buy, ct yuz, with different sign combi- nations. For a similar reason, all quantities Iu,, x v,I2 and /U,, x v,I2 have the same value for fixed n.

eflection dyadic The reflection dyadic corresponding to three reflections from the corner without the phase factor due to the path delay is

with the three reflection dyadics of the form (eqn. 40)

Since the reflected direction on the face m equals the incident direction of the face m + 1: U,, = u ( ~ + ~ ) , , (with

the vectors b;,l to a$&. . a$+l = 0 and bA. a i + s

= 0, which means that b; and a;+l are scalar multiples of one another and so are bA and a i + l .

The following inner products of circularly polarised vectors are needed in the analysis:

= -uli), we can rela te easy to show that

(43)

(44)

= J(ag . ai)(a; * af)eF3p1 (45) The upper and lower of the double signs refer to the respective first and second supetcripts on the left sides of the equations. The last expressions can be checked by taking squared magnitudes and noting (e.g. that since u2, = ulU) we have /U1, x vll = / U l r x vll = Iu2i x V I / .

The phase angles cp, can be shown to equal the geo- metric angle between the projections of the v, and v,-1 vectors in the plane orthogonal to the vector u,~.

The direction of the wave reflected from the face 3 is easily checked to be u3U = u1, = -US!, which is opposite to the direction of the original inci- dent wave. Also, the vectors bl and bj are related to the respective vectors ai and ai through:

(47)

The total reflection dyadic can finally be written in the simple form

with the coefficients:

Applying eqns. 43-45, the coefficients can be simply written as:

A = -e3((Pl+(PZ+'P3) -e3'P (51)

(52) B =.= -e-J('Pl+'P2+(P3) = -e-?'P

After some algebraic manipulations with eqns. 4 3 4 5 , the following simple expression can be derived for the total reflection phase angle cp corresponding to a ray incident in the direction u1 on the face with corruga- tions aligned parallel to the vector vl:

(53) From eqn. 53 we can see how the phase varies for rays reflecting from the faces of the corner in different order. All rays incident on the corner can be divided in six ray groups which make six sectors in the aperture plane. Rays entering on any of the faces may reflect from the two other faces in two different orders. Because the factor v1 x v2 . v3 is either +1 or -1, depending on the handedness of the vector triple, these two ray groups have the opposite sign in their total reflection phase cp. Also, all of the total phase angles change the sign when the handedness of the corner reflector, or of the circularly polarised incident wave, is changed. Because the reflected polarisation is the same for a circularly polarised incident field, the aperture field of the corner reflector constitutes of six sectors of circularly polarised field with different phases in each sector. The field in the backscattering direction be found by applying the physical optics approximation. In a more exact analysis, diffraction from the edges must also be taken into account. Its relative contribu- tion, however, diminishes with increasing size of the reflector.

For example, a right-handed circularly polarised wave ai arriving symmetrically along the diagonal u1 = -(ux + uv + uZ)h/3, satisfies u1 . vl = -1143 for a l l j = 1, 2, 3, whence the phase angle cp alternates between cp = n13 and cp = 4 3 in the consecutive sectors. By making a composite reflector of consecutive PEC and SHS cor- ners, one can obtain reflected circular polarisation of opposite handedness for a wave incident from certain sectors. By noting the line where the handedness changes, it is possible to determine a fixed direction from the reflector with great accuracy. This may have application in navigation because it does not require active responders [IO].

Conclusion

An analysis method applicable to structures involving many soft and hard surface (SHS) faces h a been devel- oped in this paper. The reflection dyadic R was given a

IEE Proc -Microw Antennas Propag , Vol 143, No 5, October 1996 420

simple and efficient form in terms of eigenvectors and circularly polarised unit vectors. As an example, a cor- ner reflector with three SHS faces was analysed and a simple form for the reflection dyadic was obtained, which shows how the handedness of a circularly polar- ised plane wave is not changed in the total reflection from the corner. The phase angle of different rays reflected from the corner was given a particularly simple form. The result can be applied for a physical optics analysis of the totally corrugated corner reflec- tor.

5 Acknowledgments

Thanks are due to Professor E. Jull, who demonstrated the application of a corrugated surface in a corner reflector during a visit of I. Lindell at the University of British Columbia, Vancouver, in July 1994. The authors also wish to thank the anonymous reviewer for finding a sign error which impaired the first version of the manuscript. The present study was partly funded by the Academy of Finland.

6 References

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2 CHEN, S., ANDO, M., and GOTO, N.: ‘Frequency characteris- tics and bandwith enhancement of artificially soft and hard sur- faces’, ZEE Proc. Microw. Ant. Propag., 1995, 142, (4), pp. 285- 294 SHEEN, D.R.: ‘The gridded trihedral: a new polarimetric SAR calibration reflector’, IEEE Trans., 1992, GC30 , (6), pp. 1149- 1153

4 MACIKUNAS, A., and HAYKIN, S.: ‘Trihedral twist-grid polarimetric reflector’, ZEE Pvoc. F, 1993, 140, (lo), pp. 216-222

5 MICHELSON, D.G., and JULL, E.V.: ‘Depolarizing trihedral corner reflectors for radar navigation and remote sensing’, ZEEE Trans., 1995, AP-43, ( S ) , pp. 513-518

6 ALTMAN, C., and SUCHY, K.: ‘Reciprocity, spatial mapping and time reversal in electromagnetics’ (Kluwer, Dordrecht, 1991)

7 LINDELL, I.V., SIHVOLA, A.H., and SUCHY, K.: ‘Six-vector formalism in electromagnetics of bianisotropic media’, J. Electr. Waves Appl., 1995, 9, (7/8), pp. 887-903

8 BRAND, L.: ‘Vector and tensor analysis’ (Wiley, New York, 1947)

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