Reflection of obliquely incident plane wave from chiral slab backed by soft and hard surface

A.J.Viitanen and RRPuska

Abstract: The general case of a plane wave reflection from a chiral slab backed by a soft and hard surface is considered. Fields in the chiral slab are represented in right-hand and left-hand circularly polarised basis. The same basis is used to derive the reflection dyadic. Expanding fields in circularly polarised vectors are advantageous, because circularly polarised waves are the eigenwaves of the chiral medium; moreover they are not coupled by the boundary condition of the soft and hard surface. However, there is some coupling at the interface of the chiral medium and free space, which complicates the analysis.

1 Introduction

Chiral materials have been intensively studied because it is widely believed that they can be used in producing novel microwave devices and structures [ N I . Applications for chiral materials are, for instance, polarisation transformers, phase shifters and devices that correct the crosspolarisation in lens antennas.

Properties of soft and hard surfaces (SHS) have been considered in [SI. These surfaces can he manufactured by machining corrugations into a conducting surface and pos- sibly flling them with dielectric. The depth of the comga- tions is dictated by the Wavelength used.

The most remarkable feature of an SHS is that it does not couple right-hand (RCP) and left-hand circularly (LCP) polarised waves in reflection, whereas one of the characteristic properties of a chiral medium is that the RCP and LCP waves are propagating there independently, with- out coupling. In other words, the SHS houndav condi- tions and the eigenwaves of the chiral medium are in some respects compatible. Thus it would be interesting to study the possibility of using soft and hard surfaces to guide or confine electromagnetic fields into or within a chiral mate rial. SHSs would then form the walls of chiral material fded cavities or reflectors. However, the complete analysis of such structures requires a solution to the problem of electromagnetic waves impinging obliquely on an SHS ter- minated chiral slab, which has been previously unavailable. In an earlier paper the authors solved a simpler problem of normal incidence [6,7], but the present paper completes the analysis with the solution to the more general oblique inci- dence problem.

In this study, we analyse a structure consisting of a chiral slab and a soft and hard surface forming a terminating back wall of the structure. We begin by writing incident

fields in terms of circularly polarised vectors. In free space RCP and LCP waves have equal propagation factors but in the chiral medium circularly polarised waves of opposite handedness are propagating with different velocities. Even though there is no coupling at the SHS boundary, nor in the chiral slab itself, the eigenwaves are coupled at the free spacuchjral material interface. The coupling has an effect on the polarisation of the reflected total field, and this cou- pling as well all other effects can be compactly represented in the form of a reflection dyadic we derive in Section 3. As the general form of the dyadic may not reveal all intricacies of the field behaviour so easily, we study in detail some spe- cial cases in the final part of the paper, Sections 4 . 1 4 3 .

!46 IC, @o'Eo;>i S I ~

U

c__

d Fig. 1 Plane wave refeclionfrom SHS matedwith chiralskzb

0 IEE, 1999 IEE Prace&&~ onhe no. 19990565 Dot lO.l049/ipmap:19990565 Paper fist received 26th October 1998 and in revised form 14th May 1999 'Ik authors are with the ElecvOmagnetk Laboratory, Dqmtment of Elem- cal and Communications Engineering, Helsinki University of Technalogy, Po Box 3w0, Fwaz015 HUT, Finland

IEE Proc.-Microw. Anlennas Propag.. Vol. 146, No. 4, Augusl 1999

Fig.2 Geomeny ofproblan

2 Theory

The geometly of the problem is depicted in Figs. 1 and 2. Fields in free space written in terms of RCP and LCP vectors are

27 I

Ui x w-jug x (U' x w) J2[1 - (Ui ' w)2]

a, =

U T x 2) - j d x (U' x w ) J 2 [ l - (UT ' w)2]

b, =

(4) u ~ x w + j u ~ x ( U ~ x w )

J 2 [ l - (U' ' w)2] bl =

are RCP and LCP unit vectors for the incident and reflected waves, respectively. The denominators of eqns. 3 and 4 are equal in free space. The unit vectors

1 1 ut= -[K+pu,] u'=-[K-pU,] ( 5 )

A, A0

indicate the propagation directions of the incident and reflected plane waves, respectively.

Vector Kis the transverse part of the propagation vector, 14 = K = k, sin 0 and p = \/(k,2 - P) = k, cos 0. The propagation direction of the reflected plane wave car-be determ>ed yith the aid of the reflection operator d= Cui where C = I - 2uzu, [[lo], p. 431.

The unit vector v indicates the direction of the soft and hard surface's anisotropy. The circularly polarised unit vec- tors obey the following orthogonality conditions:

a,'a,=O a , . q = l (6)

b , . b , = O b , . b l = l (7)

D = E E - ~ K S H ( 8 )

B = pH +j.-E (9)

In a chiral medium the constitutive relations are

where E, fi and K are the permittivity, permeability and chirality parameters, respectively [4]. The fields inside the chiral slab are a sum of the transmitted and reflected parts:

Ez(r) = E+a,+e-jk+" + E-ui-e-jk-"

+E' + b rf , - j k ; + + Eybl-e-ik'.T (10)

Hz(r) = - j [E+a,+e-jk+'r - E-al-e-jk-" 17

1 +E' b e-jk$.7' -E' - bl-e-3k'.T f r+

(11) where 11 = \/NE. The circularly polarised unit vectors for the transmitted partial waves are

U$ x w-ju; x (U$ x w) a,+ =

\/2[1- ( U 5 . w)2]

and for the reflected partial waves UT, x 2) - jUT, x (U; x w)

UT x W + j U T x (U: x 2)) J 2 [ l - (U: ' w)Z]

b ~ + = J \ / ~ [ I - (U?. wj~]

bl- (13)

In a chiral medium the RCP (+) and LCP (-) waves have different propagation factors. On the other hand, hell's law states that the transverse part of the propagation vector is equal in both media K = k, sin 0 = k+ sin 0, = k. sin 0L Hence the unit vectors indicating the propagation direc- tions of the transmitted and reflected fields are

1 1 ~g = -[K+4*uz] U; = ~ [ K - / 3 ~ t u Z ] (14) hi where p+ = \/(k> - e) = k, cos e,, IC, = k,(n T K ) and n = \/prcr The propagation vectors in the chiral medium are k, = k,u,' and k: = k*ul,

The boundary condition for an SHS located at z = 0 is [5,91

W . E , = O w . H z = O (15) Since v . a,, = v . br+ and Y . U,- = v bi_, the amplitudes of the reflected field can be written in terms of the amplitudes of the field incident on the SHS

E' + - - -E+ EL =-E- (16) indicating that + and - fields in the chiral medium are not coupled by the SHS boundary condition - clearly a conse- quence of the isotropy of the medium. With eqn. 16 the total electric and magnetic fields inside the chiral slab become

1 E ~ ( T ) = E + ar+e jk+" - b,+e-jkt'T [ ' - '

+ E - [uL-e-jk-" - b l-e-jkP"] (17)

3 Reflection dyadic

The refleckd fields can be derived by matching the tangen- tial fields at the interface of free space and the chiral slab. Manipulations are straightforward if a little laborious. Some simplification can be achieved by noticing that the phase factors can be separated into axial and transveme components of which the transverse component is invariant across the interface. Hence the electric and magnetic fields at the interface z = -din free space just outside the slab are

El(p , -d ) = e C j K ' P E' e3OduV + Eiejpdul I + ' +E$eCipdbY + ETe-jSdbi I

11

(19)

H,(p, -d) = J e - 3 K . P F i p 3 p d a 7 - ELe3Odq II.

+Eqe-jOdb, - Ere-3Pdb

(20) whereas the fields inside the chiral slab take the form

IEE Proc.-Microw. Antenna Propag.. Val 146, No. 4, Augu,~, 1999 212

~ ~ ( p , -d) = e--jK.p[z+[ej@+dar+ - e - j p + d b ? + ]

+E-[ejp-dal- - e-jp-dbl-]]

(21)

H2(p, -d) = J e - j K . P [ E + [ & + d a,+ - e -3p+dbr+] 71

-E-[ejp-dal- - e-j@-dbi-]]

Let us then take two perpendicular Unit vectors KIK and U, x KIK, both tangential to the interface, and consider the field continuity in these directions. The unit vector v written with these basis vectors has the form Y = KIK cos q + (uz x KIK) sin q (Fig. 2). The continuity of the KIK component of the electric field results in the following equality:

Etdo'+ [- cos 0 sin 'p + j cos2 6' cos $91 + Elejpd [- cos 0 sin 'p - j c03' 0 cos 'p]

+ E;e-jpd [cos 0 sin (0 + j cos2 6' cos 'p]

+ ~ L e - j o ~ [cos e sin 'p - j cos2 e cos 'p]

(22)

J 1 - sin2 e cos2 'p

J 1 - sin2 e+ cos2 'p 2E+

x I- cos e+ cos P+dsin 'p - cos2 e+ sin P+d cos (01

- -

J 1 - si? B cos2 'p 2E- J 1 - sin2 e- cos2 'p

+ x [ - c o s ~ ~ c o s ~ ~ d s i n ' p + c o s 2 e ~ s i n ~ ~ d c o s ' p ]

and the continuity of the KIK component of the magnetic field gives

2 [EieJod [- cos B sin 'p + j cos2 e cos 'p] 70 - E t e 3od [- cos 0 sin 'p - j cos2 e cos 491 + E;leKJod [cos 0 sin 'p + j cos2 Ocos'p]

- EYe-30' [cos e sin 'p - j cos2 B cos 'p] ]

(23)

JI - sin2 0 cos2 'p

J1- sin2 e+ cos2 'p 2E+

x 1- cos e+ cos P+dsin p - cos2 e+ sin P+d cos p]

- -

J I - sin2 e cos2 'p

J I - sin2 e- cos2 'p - 2E-

x [ - c o s e ~ c o s ~ ~ d s i n 9 + c o s 2 ~ ~ s i n ~ ~ d c o s ' p ]

Similarly, the continuity of the U, x KIK component of the electric and magnetic fields are as follows:

(24)

[cos 6' cos 'p + j sin 'p]

+ EcejPd [cos 0 cos 'p - j sin 'p]

+ Epe-jpd [- cos 0 cos 'p + j sin 'p]

+ E7'e-jpd [- cos e cos 'p - j sin 'p]

41 - sin2 e cos2 9 J 1 - sin2 e+ cos2 'p

2E+ - -

x [cos~+cos~+dcos 'p - sinP+dsin'p]

1 - sin2 0 cos2 'p + J 2E-

J 1 - sin2 0- cos2 'p

x [cos& cosp-dcos'p + sinfl-dsinq] (25 )

and

22. [Eiejgd [cos e cos 9 + j sin 91 70 - Elejod [cos 6' cos 'p - j sin 'p]

+ g;e- jPd [- cosecos 'p + j sin 'p]

- EZe-@d[-cosOcos'p - js in 'p]]

dl - sin2 6 cos2 'p

J 1 - sin2 e, cos2 'p 2Et - -

x [cosO+ cosf++dcos'p - sinp+dsin'p]

- 2E-

x [cos 0- cos 0- d cos 'p + sin P-d sin 'p]

J1- sin2 e cos2 'p

41 - sin2 0- cos2 'p

(26 ) We add and subtract eqns. 23 and 2-3 in t u n to obtain the expressions for the coeficients E+ and E-. These expres- sions are substituted into eqns. 25 and 26, thus eliminating two equations and leaving us with a pair of equations con- taining only the amplitudes of the incident waves E,' and the amplitudes of the reflected waves E,'. The equation pair in a matrix form is

where

A+ = [cos6'cos'p+jsin'p] x [cos 0* (cos sin /3*d cos 'p * cos P*d sin 'p)

kjcosO(cos6'~ cos&icosp T sinp+dsin'p)]

(29 ) and where the asterisk * denotes the complex conjugate. The parameters A, contain the angles 0 and q, the refrac- tion indices of both media and the thickness of the slab.

The reflection dyadic can now he written, and in the RCPLCP basis it has the form

R = R++b,al + R+-b,a, + R-+blal + R--bla,

the reflection coefficients being obtained from eqn. 27. Explicitly, the reflection coefficients read

- -

(29)

4$A+A-ej20d R++ = 2

( 1 + $) A;A- - (1 - $) 'A+A1 (30)

IEE Proc.-Mierow. Anlennm Propog., Val. 146. No. 4, Augusl 1999 273

41A* A* @ R d or. + -

2 R_- = (I+ ~ ) ' A ; A - - (1 - 2) A+A:

(32) The absolute values of the copolarised reflection coeff- cients are equal, but the phases are different. The cross- polarised reflection coefficients are equal due to the reci- procity of the materials.

1.0-

0.5

P

chiral slab air

,.-.\ ,' ,' I' ;

- i ; i

I i

(elliptically POI.) t z

'. '. \.

'. 1. \

_ - - - - - - - - - - ' ! - - -

\. '. \

'. ,; '.

-1.0 10 50 100 150

a, deg. Fi ,4 f 8 . i wrth dr nr went chiralslab t h i c h s s u 0 = 3 0 " , m = 4 5 " , r = O . b , n = 2 . 0 , d q , = 0 . 3 kod= - 0.01, ~~~ 0.1, ~. 1.0

Am littide of helicitj wto? as afimclion of direction of LP incident

When the medium becomes more chiral the phase shift between the two copolarised reflection coefficients increases, while the effect on the crosspolarised reflection coefficient remains small. This can be demonstrated easily, by f ~ n g the incidence angle to a certain value, say 0 = 30", rp = 45", with kod = 2, 0, n = 2, 0, q/q, = 0, 3, and varying K: when K = 0.02, the values of the coefficients are R,, = 0.5521-57.9", R+- = 0.8341-158.4" and R- = 0.552/-78.9". When the medium is strongly chiral, e.g. K = 0.4, the corre-

274

sponding values are R,, = 0.550/29.9", R+- = 0.8351-161.1" and R- = 0.550/-172.1". Another apparent consequence of the chirality is that the ellipticity and the direction of the main axis of the field ellipse change in the reflection. Figs. 3-5 illustrate this change. The incident field is linearly polarised (LP) and is characterised by a tilt angle a, the angle being measured in the plane perpendicular to the propagation direction, Fig. 3. The magnitude of the helic ity vectorp [[SI, p. 101

(33)

can be seen in Fig. 4. p describes the ellipticity and handed- ness of the field. The magnitude p of the helicity vector ranges from -1 to 1: when p = -1 the field is LCP, p = 1 corresponds to RCP, andp = 0 results when the field is LP.

loor

-1001 0 50 100 150

a. deg. Fig. 5 Field eii@ m i n arir direction as afirnclion of direction of LP iBci. dm1 fild Direction is measured in angles wiih respen lo some arbitrary reference angle. Three different slab thicknesses an considered R = 30'. rp= 45'. I( = 0.6. n = 2.0, q/q,, = 0.3 i<"d = - 0.01, ~~~ 0.1, - - . ~. 1.0

In the last graph, Fig. 5, we can see how the angle y of the main axis of the reflected elliptically polarised field var- ies as a function of a. Three dierent slab thicknesses are considered and the direction of the incident plane wave is fmed to 9 = 30", r~ = 45". The results do not greatly change if some other angle rp is chosen, but the effect of varying 8 is stronger, as discussed below.

Apparently the effect on polarisation is proportional to the distances the RCP and LCP eigenwaves travel in the slab, and indeed this is the case, for the longer the dis- tances, the larger the phase difference between the eigen- waves and consequently also the effect on polarisation. The distances are determined by the thickness of the slab and the incidence angle 8, but of these the thickness is probably more suited for controlling the effect in applications, because it is usually practical to keep the incidence angle fixed. A very thin slab is a limiting case for the amount of control that can be exercised by altering only the slab thick- ness. In this case, the eigeuwaves are not affected by chiral- ity, and the reflection appears to be very much like a reflection from an SHS. In more precise terms, the value of the helicity veclor does not change and the main axis angle follows linearly the tilt angle, In [lo] contraptions with these characteristics were classified as retrodirectors. On the

LEE Proc.-Micraw Anlcnnas Prqag.. Voi. 146. No. 4, August 1999

other hand, when the slab is made sufkiently thick, all the polarisation states can be obtained from an LP field just by rotating the incident field, and the structure can be classi- fied as a polarisation transformer. The curve k,d = 1.0 in Fig. 4 depicts exactly this kind of a situation.

4 Special cases

4.1 Thin chiral laver with small chirality parameter When the thickness of the chiral slab reduces to zero, the coefficients A+A_= A+*A-* = -A+'A and R,, - -1, R_ -f -1 and RL- = R, - 0. which is the boundan condition for the SHS [9].

small ( K << n). the parameters A, reduce to the form For a thin chiral slab when the chirality parameter K is

A+ M * j (cos' 6' cos' 'p + sin2 'p)

x (cos 6'* =F j cos 6'/3*d) + (cos 6'cos p + j sin 'p)

The last term is small compared to the first term when the incident angle is not vely large. Denoting cos 8, - cos 0, T (sinz 0 ~ ~ 0 s 0,) Kln, where cos 0, = pik, we get

A+A- M (cos' 6' cos' 'p + sin' p)' x cos'Bt(l +j2cos6'k0d/€) (35)

ATA- M - (cos' 6' cos' 'p + sin2 p)'

x cos'Ot(l +j2cosBk,dn) (36) and substituting these into the reflection coefficients it is seen that the chirality parameter K affects principally the copolarised reflection coeffcients R,, and R-, while the effect on the crosspolarised reflection coefficients is small, i.e. proportional to s. Even in a case of a thick slab the reflection coefficients behave as in the thin slab case just discussed, but the difference between CO- and crosspolarised coefficients is not as large.

4.2 Linear polarisation and normal incidence Quite expectedly, chirality alters the polarisation of the incoming LP field by having a stronger influence on the component of the reflected field perpendicular to the incoming field than on the one aligned with the incoming field. If the reflected field is decomposed into two orthogo- nal LP components, CO- and crosspolarised (note that there is 90" phase shift between the components), the leading term within the expression of the crosspolarised component is

which is proportional to the chirality parameter K , whereas the copolarised part is to 9. In the case of the normal inci- dence the terms 'alignedicopolarised' and 'perpendicular/ crosspolarised' become exact, and we see the dependence on K explicitly: now 0 = 0, = 0, then p* + k, and, p + k,. The coeffcients A, reduce to A, = *je7Jkid. The directions of incoming and reflected waves are now ul = U, and U' = +r Using orthogonal LP unit basis vectors v, w = U, x v instead of the CP basis, the reflection dyadic becomes

R++ -E- - N Im{A+A-} (37)

+jR++ - R-- [vw + wv] 2

R+++R--+R+-+R-+ 2

- ww (38) -

The term in the middle is the already familiar eqn. 37, responsible for the orthogonal component of the reflected field.

The equation above can also be used to check the valid- ity of our results, for after some algebra the reflection coef- ficients of eqn. 38 assume the form

j[l-($)'] s in (k++kL)d+Z$ c a s ( k + - k - ) d & 2 k o d R,, = - 311+($)21 s in (k++k- )d+2$ c o s ( k + + k - ) d

(39)

which coincide with the results given in [6, 7 ]

4.3 lsoimpedance case When the wave impedance of the chiral slab is equal to the wave impedance of the free space, or q = rlo, henceforth called isoimpedance, the crosspolarised reflection coeffi- cients vanish. Equivalently we can say that there is no cou- pling between RCP and LCP waves. The absence of the coupling allows some exotic effects to take place: in [6, 71 it was shown that if one chooses the chirality parameter and the thickness of the slab properly, the normally incident LP wave is rotated 90" in the reflection. What if the field is obliquely incident? Let us investigate the general case.

An elliptically polarised incident field can be written in the RCPLCP basis as

E" = E+u, + E - u ~ (42) where the coeffcients E, are complex numbers. The helicity vector, eqn. 33, is now

The direction of the main axis of the field ellipse can be recovered from the real part of the complex vector [@I, p. 101,

where 6 = 4 arg(E+E_). In the orthogonal LP basis (d x v, ui x (U' x v)), the direction of the main axis becomes

x [Re{e-j"(E+ + E-)}ui x v +Im{e-j'((E+ - E-)}ui x (ui x U)]

(45) The copolarised reflection coeffcients reduce to phase fac- tors

IEE Proe-Mieroiv. Anrennas Propag., Vol 146, No. 4, Augusf I999

(46)

275

which cause a phase shift for the respective eigenpolarisa- tions. The reflected field

G = e?‘+ E+ b, + e?’- E- bl (47) has the helicity vector

Neither the ellipticity nor the axial ratio is changed in the reflection, because the phase factors, eqn. 46, do not appear in the expression of the helicity vector, eqn. 48. The hand- edness remains unchanged too. However, the main axis of the reflected field is rotated. From

and denoting y = (8, - 8_)/2, we get for the main axis direc- tion of the reflected field

x [cosyuT x w + sinyu’ x (U‘ x w)]

+ Im{e-jff(E+ - E - ) ]

x [- sin yur x w + cos yu’ x (u’ x w)]] (50)

in the (U‘ x v, U‘ x (U‘ x v)) basis. One can readily see that the ellipse is rotated by y. To summarise, in the isoimped- ance case the ellipse is rotated by an angle depending on the difference of the phase shifts for + and - waves, while the ellipticity remains untouched.

5 Conclusion

In this paper the reflection dyadic of a chiral slab backed by a soft and hard surface in the case of an oblique inci- dence was given, and the polarisation properties of the reflected field were considered. The fields and the reflection dyadic were written in terms of circularly polarised vectors,

for a CP basis appears to be natural for the problem con- sidered. The use of the word natural can be understood if one recognises that the CP eigenfields of the chiral medium are not coupled by the soft and hard surface boundary condition. However, some coupling occurs at the interface of free space and the chiral medium, resulting in a change of polarisation. Even so, we can construct certain special cases where the the coupling is absent. In these cases the ellipticity of the field does not change in the reflection, but the orientation of the ellipse is altered. The results show that the structure made of a chiral slab and a soft and hard surface acting as a termination may have many applica- tions as a polarisation transforming surface.

6 References

I BASSIRI, S., ENGHETA, N., and PAPAS, C.H.: ‘Dyadic Green’s function and dipole radiation in chiral media’, Alfa Freq., 1986, LV, (2), pp. 83-88

2 JAGGARD, D.L., SUN, X., and ENGHETA, N.: ‘Canonical sowces and duality in chiral media’, IEEE Trans. Anlennm Propag., 1988.36, (7), pp. 1007-1013

3 MONZON, J.C.: ‘Radiation and scattering in homogeneous general bi-isotropic regions’, IEEE Trans. Antennas Propag., 1990, 38, (2), pp. 227-235

4 LINDELL, LV., SIHVOLA, A.H., TRETYAKOV, S.A., and VIITANEN, A.J.: ‘Electromagnetic waves in chiral and bi-isotropic media’ (Artech House, Nomood, NY, 1994)

5 KILDAL, P.-S.: ‘Artificially soft and hard surfaces in electromagnet- ics’, IEEE Danx. Antennas Propag., 1990, 38, (IO), pp. 1537-1544

6 VIITANEN, A.J., and PUSKA, P.P.: ‘Plane wave reflection from chi- ral slab backed by soft and hard surface with application to polarisa- tion transformers’, IEE PYOC. Microw. Antennay Propag.. 1998, 145, (4), pp. 299-302 PUSKA, P.P., and VIITANEN, A.J.: ‘Polarization transformer con- structed from chiral slab and soft and hard surface’. Proceedmgs of 27th EuroDean Microwave conference. Jerusalem. Israel. 8-12 Seuiem-

I

ber 1997, bp. 512-517 LINDELL. I.V.: ‘Methods for electromametic field analysis’ (Claren- 8 . . don Press, Oxford, 1992) LINDELL, LV., and PUSKA, P.P.: ‘Reflection dyadic for the soft and hard surface with application to depolarising comer reflector’, IEE Proc Microw. Anlennm Propag., 1996, 143, (5 ) , pp. 417421

10 CENSOR, D., and FOX, M.D.: ‘Polarimetry in the presence of vari- ous external reflection and retrodirection mirroring mechanism, for chiral and gyrotropic media’, .I Eleclromagn. Waves Appl,, 1997, 11, (3), pp. 297-313

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