(7 * Int. Conf. on Mathematical Methods in Electromagnetic Theory M M , d 04 Sept. 14-17, 2004. Oniepropetrovsk, Ukraine

POLARIZATION TRANSFORMATION BY THE PERIODIC ARRAYS OF COMPLEX-SHAPED ELEMENTS

Sergey L. Prosvirnin, Dmitrv 0. Tvrnov

Institute of Radio Astronomy of National Academy of Sciences of Ukraine Krasnoznamennaya str. 4,61002 Kharkov, Ukraine

E-mail: tymov@rian.ira.kharkov.ua

Abstract - Periodic layered structures are considered. Each layer of the structure is an array of plane metal strips of a complex shape placed on a dielectric substrate. Diagonalization of complex reflection and transmission matrices of single array in real eigenvector space is demonstrated. I t yields to existence of two "principal" directions for this array. If incident wave polarized in the one of this directions, it's polarization is not transformed by the structure. The frequency dependencies of the eigenwave propagation constants and polarization angles are established.

Let each layer be constructed of the curvilinear plane metal elements oriented in same direction (see Fig. la). The layers are oriented perpendicularly to the OZ axis (see Fig. Ib). In the case when on the array period small compared with the incident wavelength, the elements may show the first or even the second current resonances.

i ;

z

L

(a) (b) Fig. 1. (a) Array of the complex-shaped elements;@) Artificial medium constructed of such arrays.

The field near this array may be presented as the sum of the incident field and the fields reflected or transmitted by the array. The reflected and transmitted fields can be expanded in the series of spatial harmonics (see [I]), as given by

(la) j'j = Z i Z - i i i + Czve-;kipi

for z > 0, q . P

4 .P - - - - where k b = k, + I, + Fzyq, kd, = kL +I, - Fzyqp are the wavevectors of spatial harmonic with

number qp for reflected and transmitted field; zqp is the vector amplitude of this harmonic; k, is the parallel to

XOY component of f , wavevector of incident wave; g, = Z x ( 2 q I d,) + Zy(2rq, I d,) ;

y, = ,/-, d, = d, = d are the array periods along OX and OY axis.

The magnetic field may be expressed as follows: H = --[V,E]

-

- i

W O

0-7803-8441 -51041$20.00 0 2004 IEEE 192

10'" Int. Conf. on Mathematical Methods in Electromagnetic Theov MM&T*04 Sept. 14-17. 2004. oniepropetrovsk, Ukraine

Equations (la,b) are written for the case when the metal elements are placed in free space. For the case when the metal elements are placed on a dielectric substrate, the results are similar but much more complex. When the boundary conditions that the solutions for the electric and magnetic fields must satisfy are specified at the metal elements and in free space, the result is:

where 5 - vector in XOY plane that points to the array element, equation (Za) applies to the metal surfaces and (2b) does to the areas without metal. On performing the vector multiplication ofequations (2a,b) and integrating the result over the array cell, we obtain:

. . Consider the incident wave at normal incidence and the frequency range where only one spatial harmonic

with a propagation constant k is propagated. Then, (3) can be recast in the following form:

q.!J The real part of this equation can be written as

where the sum with the *' I' superscript indicates only the propagating harmonics. The matrix elements of the reflection and transmission operators (R and T, respectively) for one layer can be obtained by using the moment method. When only one harmonic and two orthogonal polarizations are taken into account, the reflection and transmission operators are square 2 x 2 matrices.

Let the z' be real, and R = R e R , then Re;, = R'Z'. Using' the reciprocity theorem, the conclusion can be reached that R' is a symmetrical matrix, and

consequently it's eigenvalues and eigenvectors are real. Let the z' be eigenvectors of R', then R'Z' =E' . 12 , ; ' 2

x'yqp\Ziqpl =-kIjZ 1 ; thus A,,4 < O . qP

Therefore, the real parts of reflection (and transmission) operators are negatively defined and both the real and imaginary parts of these operators are symmetrical. Hence, these complex reflection and transmission operator matrices can be diagonalized in the real eigenvector space ([2]).

Thus, the array under consideration has two "principal" directions, When the incident field is polarized along one of these directions, the anay does not transform the polarisation of the reflected and transmitted waves.

Consider the artificial medium presented in Fig. Ib. The operator equations governing the eigenwaves in such a medium are possible to derive in the same way as in [3]. They are as follows:

(I-e-'PLT'e)jt -R-ej; = O , ( 4 4

R'e2f - ( I - dFT-e)2; = 0, (4h) -+

where e is the operator governing the changes in a field propagated from any layer to the adjacent one, A; are

the amplitudes of the eigenwave partial constituents in the interval between j and j+l layers (hereinafier the "+" and "-" supersrripts refer to the propagation of the wave from left to right and from right to left, respectively), L is the period of the structure, p is the propagation constant of the eigenwave.

The dispersion relations for the eigenwaves are solved numerically, and the results are presented in Fig. 2a (eigenwave is polarized in the OX plane) and Fig. 2b (eigenwave is polarized in the OY plane). Frequency dependencies of the matrix elements IR,( and lRyyl are presented in these figures too. Compare dependencies of

0-7803-&141-5/041$20.00 0 2004 IEEE 193

I O t h Int. Conf. on Mathematical Methods in Eiectromagnetic Theory Sept. 14-17, 2004. Dniepropetrovsk. Ukraine MMkp04 . . ...

Im(PL) and lRxxl, lRwl in Fig. 2. I t can be seen that in those areas where the reflection factor for one layer is close to unity. a sharp increase is observed in the imaginary part of magnitude of the propagation constant. This increase can he explained by the total reflection of the incident field from each layer, and therefore the eigenwave cannot be excited as a propagating wave. Thus, the imaginary part of its wave number tends to infinity. Other cutoff zones explains by the layered structure of artificial medium.

Fig. 2.

3ff

27r

0.0

0.0 0.2 0.4 0.6 0,8 1.0 0.0 0.2 0.4 0.6 0.8

(a) (b)

Dispersion relations of eigenwaves in the layered StNCNre.

/Yy/

1 .o

0.5

0.0

The frequency relations of "principal" angles for considered array are presented on the Fig. 3. This angles totally coincided with polarization angles of eigenwaves in artificial medium. The strong frequency dependence in the range d/h explains by the excitation of higher spatial harmonics in the dielectric substrate. These harmonics can't propagate in the free space between arrays, so they have no influence on the dispersion relations of eigenwaves in the layered medium, hut they have an influence on matrix elements of R and T.

8o &gr--r

.i23 Y-polanza~on

.',Ea .2W ~~~~ io Fig. 3. medium. They are totally coincided.

Frequency relations of "principal" angles for one array and angles of eigenwave polarisation in the layered

The main result of this report is the diagonalization of complex reflection and transmission matrices of array of complex-shaped elements in the real eigenvector space. It yields to existence of two '"principal" directions for this array. If incident wave polarized in the one of this directions, it's polarization is not transformed by the structure. This effect explains mathematically and investigates numerically.

REFERENCES

[l] S. L. Prosvimin and S. Zouhdi, "Multi-layered arrays ofconducting strips: switchable photonic hand gap StmCNrCS". Inr.

[2] A.D. Myshkis, "Mathematics for technical colleges. Special courses", Moscow, 1971. (in Russian). [3] S . L. Prosvimin and T. D. Vasilyeva, "Eigenwaves of periodic layered ShucNre of complex arrays," 8-rh In1 Conf on

Electromagnerics ofcompler Media, Lisbon, Portugal, Proceedings, p. 24 I , September 27 - 29,2000.

. Jorrmal ofElecrronics and Communications, Special Issue: Bianisorropics 2000, vol. 55, no. 4, p. 260,2001.