A New Mathematical Formulation for Studying the Scattering Properties of

Perfectly Conducting Periodic Structures.

*Antonio Roberto Panicali

ABSTRACT: Previous methods for studying the scattering properties of perfectly conducting periodic structures were derived with basis

on the boundary conditions the fields must satisfy at the structure surface; in this work it is shown that those scattering properties plusthe induced current along the structure surface can be studied through adifferent mathematical formulation exclusively derived with basis on thescattered far-fields characteristics. Complementary periodic structures,periodic structures with dielectric layers and corrugated waveguides we-re also studied as applications of the method. Numerical results obtainedwith the new method are shown to be in good agreement with the resultsobtained by other methods and also with experimental data.

1. MATHEMATICAL FORMULATION: a perfectly conducting Planar Periodic Structure (PPS) with profile described by

Z= g(y')= g(y' + nd) n = 0, ±1, ±2,..., max g(y') = 0 ( )is illuminated by a linearly polarised plane wave with electric field -xE parallel to the PPS grooves (TE-case), Fig.la; let denote theangle between the incident wave vector g and the negative Z-axis. -Letthe current-density at a point p' on the PPS surface be described by

J(p') = x f(y') , pl = y'f + g(y') z (2)Based on Flochet's theorem one can interpret (2) as describing an infinite periodic array uniformely excited with a progressive phase shift; therefore the scattered field Es(p) , can be expressed as an infinite set

E0 ZE- ZE(~~-) 0h

g(y (a) (b) (c)Fig.l: a-basic geometry;b-complementary PPS;c-PPS with dielec

tric layer.of Space Harmonics CSH) with amplitudes proportional to the array ele-

ment pattern I11- - 00 + -+ -

E (p) = x-E y exp i ($. p), + for z>O, - for z<min g(y') (3)

+ = (2 K) 1/2 (K+ d)-l U - KKk 2T/X (4)p vi 1]

V = K (y sin V + z cos V) (5)

sin =p X d + sin So (6)

k- = K cos CV sgn (z) = positive real or positive imaginary (7)

*Depto. de Engenharia de Eletricidade da EPUSPUniversidade de Sgo Paulo - Sao Paulo - Brasil

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Uj(J,f(y')| = array element pattern at direction $=1/2 d i i2 1/2

= -n0 (k/B 'r) fK f(y') 1 +ID g(y') exp-i fBp'(y')dy' (8)0 ~~~~0

with f(y') and p'(y') defined as in (2). Since the total field (incident+ scattered) must vanish for all z < 1min g(y')| one concludes from -

(3) thaty- --E , y=0 1 $0 (9)To 0 1.

Thus, from (4) and (9)1/2U ,f(y')j = -2 E d(k/n) and U(F) = 0 p $ 0 (10)

0 ~~~0 11It can be shown that the constraints imposed by (10) on the array element pattern plus the fact that the reactive energy stored along each -structure cycle must be finite are sufficient to uniquely specify theinduced current on the PPS. As a first step in solving (10) one assumesthat f(y') can be written as a linear combination of M = 2N+l base functions f f (Y')} i.e.:

f(Y') X _ an fn (Y') (11)

Substituting (11) into (8) and imposing (10) for -N < p < N the unknowncoefficients {an I can be obtained, formaly at least, as the solution -of a set of independent linear equations:

al = A 1 (12)

a=< a, a2 ..., an> JY = YNY_+1*- so N >(13)

where {y-} are given by (9) and A is the square matrix {amn- ~~1/2 - -(YI mn

amn = (2nrd) (K d) un.f (y')j, m,n=l,...,M,p=m - (N+1) (14)

Taking {a I from (12) into (11) and substituiting into (8) and then -

into (4) nthe amplitudes (y'} , -N < p < N, can be written asp

y A+ IA-1 y= R yo (15)where now r+ = < Y+N Y+ - YN >T and A is obtained as in(14) substituting the (- ) supersorits by (+). If additional planes wa -

yes, others than the zero-th order one, are simultaneously incident onthe PPS, the total amplitudes of the "reflected" HS for z > o can stillbe obtained from (15) provided the new incident wave-vectors coincide -with B- in (5) and that the components of (y-I in (13) are replaced bythe amplitudes of the corresponding incidentpwaves. Clearly R in (15)can7be interpreted as a generalized reflection matrix; in the particularcase of a perfectly conducting plane coincident with z = o, R reduces -to the identity matrix E.

2. REFLECTING PROPERTIES OF COMPLENENTARY PPS: two PPS (labeled 1 and2) will be said to be complementary (or inverse) to each other, Fig. lb,if their profiles are described by

z-g=(Y') and z'=g (y') = -g (y') (16)

Eq.'s ( 2 ) ( 5 ) and ( 7 ) imply that for each y' and p

r*Pi (y') = q*P2 (y') (17)p ~~ ~~~~~~~~++ -Therefore, from (14), ( 15) and (17), A = -A and A 2-A1 From (15)

R = A+ IA-I-1= - 1 218R A+IA2IU-A I-All =A (A )I =R (18)2 21 1 1

3. PPS COVERED BY A DIELECTRIC LAYER:let the space above a PPS be filled

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with homogeneous dielectric material, up to a height z = h > 0, and letthe whole system be illuminated by a TE plane wave incident from thefree space, Fig.1c. Let Edo and 6- denote respectively the amplitude -and the wave-vector of the transmY2ed wave within the dielectric, compu

ted as iif the dielectric extended from z=h to z = -X; let y- be definedas in ( 13) and C 9 ) with E0 replaced by Ed. Again, by force of Flo -chet's theorem one can decompose the field within the dielectric material into two sets of SH's: let y- and {S } denote respectively the am -plitudes and the wave vectors of the SHO incident on the PPS with{S }obtained from ( 5) making 5 = 5 . Similarly , one can define y+ ald{8d"}for the HS's reflected by tte PPS.

Clearly y+ = R y- with R defined by C 1S) and (14) with{V }d v

replaced by (V I and with k and n replaced by the corresponding values for the dieYHctric material. On the other hand one can also relater4 to y by considering the "reflection" at the free-space/dielectric-interface. These two relations can be combined resulting the followingconsistency equation:

y- ODSRdy + y (19)

where the matrix S takes into account the scattering of the SH's at theinterface with the free-space while the diagonal matrix DO defined as

d = exp - i du . hz = exp i d . hi, p = m - (N + 1) (20)

takes into account the phase shift or attenuation experienced by the -SH's propagating between the PPS and the free-space dielectric interface. From ( 1)O ry and y+ can be easelyott-iained1and from that one cancompute the amplitudes of the SH's scattered back into free-space

4. PARALLEL PLATE WAVEGUIDES WITH PERIODICALLY CORRUGATED WALLS: cpnsider the waveguide formed by tws PPS with periods d and d2I and let m,n be the smallest positive integers such that md, nd2 d , Fig. 2d us therefore the waveguide period. Let h denote 2I-the distance between the two PPS. Any field con-figuration satisfying Flochet's theorem within -the guide can therefore be decomposed into two E SIsets of SH's; following a similar procedure asiin item-3, let the column vectors y and Y+ reS2present the amplitudes of the SH's respectively- d

incident and reflected by PPS-1. As before y andrl must satisfy tho consistency relation (13> - Fig.2: corrugatedexcept that now y - 0 , Rd = R 1, S =R2, with R guand R2 denoting tte corresponding reflection ma-trices for the two PPS's. With D defined as in (20) onecan write:

= 0 R2D R1-y7 = N (21)yl 2 1X1 Y 21

For a non-trivial solution of ( 21), it is required that

det (M - E) = 0 (22)

thus stablishing a dispersion relation for the waveguide. For each pairIfrequency x V_ c or ])f satisfying ( 22), the possible field configu-rations can beOobtaines as the eigenvectors of M in ( 21).

5. NUMERICAL RESULTS: figures 3a and 3b show respectively the magnitude.and phase of the current density induced along sinusoidal structures -with varying sizes of the period, when illuminated by a vertically in-cident TE-polarised planewave. As expected, as the period increases interms of wave-lengths the induced current penetrates deeper into the

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grooves and in the limit it approaches the values predicted by the physical optics approximation. Fig. 4a shows the energy of the scattered propagating modes, computed by the present method and also by Zaky and Neureuther | 2f ; despite some minor discrepancies the two methods can be saidto be generally coincident. Fig. 4h shows the percentual difference be -tween the incident and reflected power, which in all case agreed within5%. Figure 5& compares the power in the scattered propagation modes as,-computed by the present method with the results obtained by Ikuno and -

Yassuura | 3 for the case of a sinusoidal structure with unitary periodand a total depth equals to 0,30 when illuminated by a TE-polarised inci-dence: as indicated the results coinced extremely well. Finally, Fig. 6a,6bshows a comparison between some experimental and theoreoritical resultsobtained with the present method, regarding the phase of the TEI0 rmodereflection coefficient, which occurs in a section of WR-187 waveguide -terminated by two types of metallic plugs as shown in Fig 6b I and II'or frequencies varying between 4.0 GHz and 6.0 GHz; once more the results coincide quite well.

6. REMARKS REGARDING SOME COMPUTATIONAL DIFFICULTIES: as implied in iten-l, additional constraints on the boundness (oftype of singularities) of the induced current are necessary -in order to uniquely specify a solution to (10). However inorder to obtain simpler relations between incident and scattered fields as (12) and (15), these constraints have beencompletely ignored in deriving (12) ;the price one had to payfor this simplicity, is that now f(y') as computed by ( 12 )does not necessarely satisfy the required boundness conditi-ons as M + o; in the present work this type of computional -difficulty has been solved through the proper choice (mostlyby trial and error) of the base functions in (11);triangularbase function, approximated by rectangular pul1ses-, were usedfor computing the results displayed in Fig. 3 to 6. The useof more elegant methods such as regularisation or constrai-ned minimization procedures are now beeing investigated.

LIST OF REFERENCES

1 - H.Y.Yee, L.B.Felsen and J.B.Keller, "Ray Theory of Re-flection from the Open End of a Waveguide", SIAM J.Appl.Math, vol.16, n9 2, pp.268-300, Mar. 1968.

2 - K.A.Zaki and R.Neureuther, "Scattering from a perfectlyConducting Surface With a Sinusoidal Profiles: TE Polarization", I-EEE Trans.Antennas Propag.Vol. AP-19,pp.208-216,Mar. 1971.

3 - H.Ikuno and K.Yasuura, "Improved Point-Matching MethodWith Application to Scattering from a Periodic Surface',IEEE Trans. Antennas Propag., Vol.AP-21, pp.657-662,Sep . 19 73.

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