EM Scattering by an Array of Perfectly Conducting Strips by a Physical Optics Approximation

The scattering of electromagnetic waves by planar arrays of perfectly conducting strips is analyzed by a simple method based on physical optics. The induced current as determined by physical optics is used in simple hand computation to obtain the amplitudes of various propagating space harmonics. Results are compared against some exact results available in the literature to show the accuracy of the proposed approximate method.


I. INTRODUCTION
The diffraction of electromagnetic waves by a planar array of perfectly conducting strips has been of great interest, and a number of rigorous, semirigorous, and numerical solution techniques have been developed for the analysis of the problem [ 11, [ 21.These methods are normally mathematically involved and require an extensive programming effort and the use of a digital computer.The object of this communication is to present a simple approximate method based on physical optics which only requires very little hand computation.In this method a periodic Green's function is first found for the problem [2].The induced surface current density is taken as twice the magnetic field intensity of the incident wave.A simple integration operation which is easily performed by hand computation yields the amplitudes of various waves.Some typical results obtained are compared against rigorous results available in the literature [ 11, which indicate the accuracy of the present approximate technique.

METHOD OF ANALYSIS AND RESULTS
The structure under consideration is a periodic array of perfectly conducting strips.The period of the structure is d, the spacing between the strips is a, and the width of each strip is equal to b .The incident wave is E-polarized as shown in Fig. 1.The electric field of the incident wave after suppression of the exp G u t ) time variation takes the form where Po = k sin Oi, yo = k cos Bi, k is the wavenumber, and 0 is the incidence angle.
The slectric fields of the reflected and the transmitted waves will have a y component only and the problem is, therefore, treated as a scalar problem (hereafter we write E , as $).To solve the problem a periodic Green's function, which is the electric field due to an infinite periodic array of line currents on the x axis, is sought.The currents flow along the y axis causing a y-directed electric field.The distance between any two adjacent line currents is d.The scattered electric field $scat at every point in space is calculated using this Green's function and the induced surface current density J , on the reference strip (the strip at The surface current flows along the y axis, and its density is approximated using physical optics as twice the tangential component of the incident magnetic field: Substituting (3) into (4) we obtain In view of the periodic nature of the problem, the scattered field can be alternatively expressed in terms of space harmonics, namely, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL.AP-28, NO.    (7 1 Notice the simplicity of the fiial solution presented in ( 6) and (7).
Formulation for the H-polarized problem is obtained by a similar mathematical procedure.Instead of an infinite array of electric line currents, we take in this case an infinite array of magnetic line currents as the source of the Green's function.The surface magnetic current in the opening of the reference cell (-a/2 < x < a / 2 with the z axis shifted to the center o f the opening) is taken as twice the tangential component of the incident electric field intensity.The fiial result for the magnetic field amplitudes of the transmitted waves Bn are as follows: 27~ exP (7Pna/2) -exP ( i P n a / 2 ) B =- cos ei.A self-complementary strip grating (a = b = d / 2 ) was considered at normal incidence for different periods.This structure was previously analyzed exactly by Baldwin and   Heins [ 11. Their results for the amplitudes of the waves of orders zero and one, together with the results based on the present method, are shown in Table I.Both magnitudes and phases in degrees are compared and indicate good agreement.The exact magnitude of the order zero wave has an oscillatory behavior which settles to the physical optics value of 0.5 as the structure period is made larger.The phase of the order zero wave goes similarly through oscillations with peaks occuring at periods which are whole multiples of the wavelength at which strong resonances take place.The magnitude and phase of the order one wave agree much more favorably with the physical optics values as the structure period is in- creased.
and (8)  indicates that for the self-complementary structure (a = b = d / 2 ) the solutions are identical.

TABLE I COMPARISON
O F WAVE AMPLITUDES OBTAINED BY PHYSICAL OPTICS AND THE EXACT METHOD O F BALDWIN AND HEINS FOR SELFCOMPLEMENTARY ARRAYS O F DIFFERENT PERIODS FOR AN E-POLARIZED NORMALLY INCIDENT PLANE WAVE Relative Power Physical Optics Relative Power Baldwin and Heins