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## Mathematical modeling of multi-layer polarizers on meander lines

Published: 10/29/2003
Original: Radio engineering (Moscow), 1994, №9, p.p.71...75

When considering the problem of transformation of electromagnetic field of the linear polarization within the field of elliptic polarization, and vice versa, the multi-layer meander polarizers (MMP) in the wide band are of practical interest.  shows the solution of the boundary problem of electrodynamics as well as defines the scattering signatures of a flat wave on one polarizer layer.

For a number of layers however an approximate approach based on neglect of interaction between the layers as per the highest type of spatial harmonics has been used. This allowed to find the scattering matrix of a multi-layer polarizer via the scattering matrix of one of its layers. The problem related to estimate of applicability boundaries of such approach has not been con-sidered.

The objective of this work is the solution of the boundary problem for MMP in the strict electrodynamic setting, taking into account the full interaction between the layers of the polarizers.

Let’s consider several dielectric layers in some of which the meander lines (ML) made of some infinitely fine ideal conductive material are periodically located (Fig.1). Let be one of the sides of the i-th ML surface. The boundary problem of electrodynamics can be formulated as follows: find the second electromagnetic field, the electric vector of which meets the following condition (1)

it meets the Maxwell’s homogeneous equation outside the location area of its sources; in case of infinity it meets the radiation conditions providing for no-wave coming from the infinity and having no source over there, on ridges — its meets the Meixner conditions , and in the corners — the integrability condition. In (1) — is the tangent outside electrical field, — is the tangent component of the secondary electrical field excited by the electric current induced at the MMP conductors taking into account the medium interface.  Fig.1

As is well known , the formulated problem has the unique solution. The outside electromagnetic field will be considered to have the form of a monochrome ( ) flat wave. Then by virtue of the MMP geometry the Floquet’s theorem  can be used.

Let’s introduce the following symbols: — is the carryover factor of the i-th Floquet’s harmonics (i- is the generalized index ) from the l-st into the s-th layer ( ); — is the “reflection” factor of the i-th Floquet’s harmonics from the upper bound of the l-st layer; — is the “reflection” factor of the i-th Floquet’s harmonics from the bottom bound of the l-st layer.

The and factors (where l<s) can be found when exciting the l-st layer of the multi-layer structure (while all ML are absent) by the Floquet's harmonics going in the positive direction of the OZ axis, and and factors (where l>s) — by the Floquet’s harmonics going in the negative direction of the OZ axis.

By applying at the appropriate layer the Lorentz lemma towards the ancillary field and the required field in the integrated form  and using the orthogonality condition of the subwaves , it is possible to get the expressions combining the field development factors with the electric surface current , which presents the total of currents flowing on both sides of the k-th ML .

If using the boundary condition (1) we can get the system of the first genus operator equations (as to terminology of ) as respects the currents : (2)

where — is the result of the solution for the problem of the flat excitation wave on the multi-layer structure for the m-th layer, — is the total number of ML, — ist the Green’s electric tensor function of the Maxwell’s equations  — is here — the symbol of dyadic product of vectors, — is the point of observation, — is the source point, , — is the norm of the i-th eigenwave of the l-th layer .

For the convenience a flat wave can be recorded in the spherical coordinate frame (3)

where — is the wave number.

To solve (2) Galerkin’s method can be applied. The conductors cross configuration shown on Fig.2,a and proposed in  can be selected as a ML model, considering that and that the period lengths and of ML ( — is the wave length in dielectric in which ML is located), and the transverse currents are neglected. The dependency of longitudinal currents upon the transverse coordinates can be selected in a way to provide for the required singularity on the ridges accordingly to the Meixner conditions . Let’s consider the specific element ML (Fig.2, b), oriented at an angle to ML axis. According to the notes stated above the current within the local coordinate frame  , where , and is to be defined by the mode of the basis set chosen along the axis is the actual series development by this basis with unknown factors.

After finding of current on ML it is possible to calculate the scattering matrix elements ( ) of the polarizer. For determinacy reasons let’s think that i,j = 1,3 conform to the flat wave polarized at an angle of to ML axis, and i,j = 2,4 — correspond to the flat wave polarized at an angle of . Moreover, the j indexes equal to 1 and 2, relate to the wave incident in positive direction of the OZ axis, whereas the j indexes equal to 3 and 4, — in the negative one. In this case the element represents for instance the conversion coefficient of the wave polarized at an angle of incident in negative direction into the wave which has passed through the MMP and polarized at an angle of : . The transmission factor of the wave polarized at an angle of is , where , — are the electric field components of the wave which has passed through the MMP in the spherical coordinate frame.

Based on the formulas received a PC calculation program has been made up. Here the full orthogonal system is used as basis function, where , , , , (please refer to Fig.2,а), and and define the direction where the flat wave come from (3).  Fig.2

The is introduced to eliminate the phase jumps in the points of ML conductors connection .

To check the algorithm and program efficiency the calculations for one MMP layer from  were performed. The results received for the differential phase shift between the factors and are congruent with the results of [1 ] having graphic accuracy.

The calculation results for the four-layer MMP, where the layers are divided by air space =1, 0,1667 wide, having the parameters as follows: =0,0067 , =5,4 (i=1… 4); =0,09067 ; =0,32 ; = =0,0107 ; = 0,056 ; = 0,099 ( — is the wave length at the initial frequency ) within the frequency band at , are given on Fig. 3 (continued and dashed-line curves). This figure also represents the results of experimental investigations on a model where foamed plastic instead of air space was used. The difference does not exceed 8… to 10% and is there due to the lack of fit between the experimental specimen and the mathematical model which is especially noticeable by a large number of MMP layers. This lack of fit lies first of all in the finite size of the real MMP and in use of foamed plastic instead of air space having >1 ( =1,05…1,15). Fig.3

If the space between the ML layers is less than (0,18…0,2) it is necessary to consider the interaction as per higher spatial harmonics. Use of MMP as compared with one-layer ML allows to essentially expand the wave lengths effective range (up to one octave and more).

The author thanks to L. I. Sidorenko for the experimental investigation results provided as well as to V. V. Koryshev for the discussion of received theoretical results and his attention towards this work.

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