Higher-order mode interaction in planarperiodic structures

N.V. Shuley. B.E., M.Eng.Sc.

Indexing terms: Dielectrics, Mathematical techniques. Periodic arrays

Abstract: A method for the inclusion of higher-order evanescent modes in planar periodic structures is present-ed. The approach utilises the wave matrix method that is extended to account for the nonpropagating orders.As an example, the case of two closely spaced singly periodic gratings is demonstrated. In order to producesuitable data concerning reflection and transmission matrices, an integral equation formulation is used todescribe the scattering behaviour of the grating. The application of the theory is intended for the design ofmultilayer dichroic surfaces.

1 Introduction

In a recent paper [1], wave matrix analysis has been usedto study the effects of cascading periodic metal screens inorder to obtain transmission/reflection characteristics. Inthe paper, the 'one-mode interaction' assumption wasmade. This means that the nonpropagating (evanescent)Floquet mode orders were not taken into account whencalculating the interaction between the periodic sheets. Ascorrectly pointed out, this approximation is valid when theintersheet spacings are large compared with the wave-length. It was also concluded that the one-mode inter-action assumption is good for suprisingly small intersheetspacings. The present paper demonstrates a methodwhereby the evanescent mode interactions can be takeninto account and gives some theoretical results for thesimple case of two free-standing singly periodic screens inclose proximity. The results obtained reduce to that whichwould be expected for the one-mode interaction case whenlarger intersheet spacings are employed. The application ofthis theory is primarily intended for thin multilayerdichroic surfaces [2]. In fact, it may also be extended tothe design of many other structures that employ the use ofplanar periodic surfaces. For example, current designs ofmeanderline sheet polarisers [3, 4] employ a spacer that isapproximately a quarter wavelength in thickness in orderto avoid coupling between the various meanderline sheets.Such designs utilise the one-mode interaction formulationto derive an equivalent reactance for each sheet which isthen cascaded so as to produce the desired phase shift. Bytaking into account the evanescent mode coupling, thespacer thickness constraint may be removed, thus resultingin polarisers that are substantially thinner and therefore,from a fabrication viewpoint, easier to construct.

2 Analysis

Consider the structure shown in Fig. 1A consisting of acascade system of M cells illuminated by a plane wave.Each cell in turn consists of a cascaded subsystem ofdielectrics enclosing a single metallic grating as indicatedin Fig. IB. In general, the M cells are different. However,each cell may only contain one metallic grating. In thisfashion, a variety of structures may be represented.

The interaction of several cascaded cells may beanalysed by extending the wave matrix approach toaccount for the nonpropagating Floquet modes. Theapproach here is essentially that which is used for periodic

Paper 3074H (E11/E12), first received 16th August 1983 and in revised form 26thJanuary 1984

The author is with the Division of Network Theory, Chalmers University of Tech-nology, S-412 96 Gothenburg, Sweden

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984

discontinuities in waveguides [5]. If we let N be thenumber of interacting modes, each grating may be rep-resented by the 2N port shown in Fig. 1C. The mode

incident planewave

z=0

Fig. 1A System of M cascaded cells illuminated by a plane wave

n th cell

metallic grating

Fig. 1 B nth cell contains a single metallic grating and s dielectric layers

metallic grating

Fig. 1C 2N-port representation of the grating

129

amplitudes for the nth grating are related by:

where the matrices Rn-i, Rn+i and Tn + i, Tn_1 are reflec-tion and transmission coefficient matrices, respectively, andcan be defined as:

("+1)= reflection coefficient of the ith mode with

theyth mode incident from the n — l(n + 1) cell

f'J , = transmission coefficient of the ith mode with("+1) thejth mode incident from the n — \(n + 1) cell

i, j — modal order 1, . . . , N

The phase of the reflection coefficient of the nth cell isreferred to the plane of the metallic grating within eachcell. This is in keeping with the convention cited in Refer-ence 1. Matrix manipulation may be applied to expresseqn. 1 in the form required for a cascade coupling:

(2)

where

T - "'On — I -

T~ in

Since the homogeneous dielectric layers do not contributeto the excitation of higher-order scattered modes, eachlayer may be represented by the generalised diagonal wavematrix. For a section of transmission line of length lr thistakes the form:

rr0

R'Ne j

T\

0

^ 0

T*"

0pr oJy\lr

1 N ^

R\

e

e Jnlr

T ^

0

jy i 'r

0

0

nrr1 N -

0

(3)

where {}',} i = 1, N are the modal propagation coefficientsfor the rth layer, the sign of which for the nonpropagatingmodes is chosen such that the exponential term decreaseswith increasing lr. The sets {R'}, {Tr

t}, r = 1, N are themodal reflection and transmission coefficients with respectto incidence from the (r — l)th dielectric layer.

The overall wave matrix for the entire structure maynow be represented by

where

(4)

where 7̂ , represents the wave matrix for the nth cell andthe grating is situated between the r'th and (r' + l)thdielectric layers, and s is the total number of dielectriclayers.

The overall wave matrix T may now be written in termsof its components:

AJ l2 (5)

130

If excitation is restricted to z < 0, d' = 0, and the overallreflection and transmission vectors may be written as:

c =

d ^

for transmission

for reflection(6)

c' and d are the transmission and reflection coefficients forthe N modes considered in the multilayered structure.

In order to consider the influence of evanescent modesin a multilayered structure, a full modal scatteringapproach may be applied to the grating. This is in contrastto a quasistatic approach where higher-order modes areneglected [6].

3 Example

We consider the particular case of two identical singlyperiodic strip gratings spaced by distance / (Fig. 2). It is

• metallic strip

E1

H^-2

I- IFig. 2 Configuration of two single periodic metallic gratings withspacing Ic = 0.745 cm, d = 1.499 cm

assumed that the strips are perfectly conducting. Further-more, we treat the case of TM (with respect to z) incidence.The geometry is thus two dimensional, and the scatteredfields produced by the grating consists of TM waves alone.In view of the symmetry of the problem, the evaluation ofthe reflection and transmission matrices in eqn. 1 isreduced by half since Rn-X = Rn+l and Tn_l = Tn+l.These reflection and transmission matrices are evaluatedby allowing one mode of unit amplitude to be incident onthe grating and calculating the corresponding reflectionand transmission coefficients. This operation is repeated inturn for the N modes considered.

In the absence of dielectric layers, the matrix TLr in eqn.3 reduces to a diagonal matrix of the form:

0

0

0

0e

0

-JVllr

0

In view of the periodic nature of the grating, the fields maybe expanded as a series of Floquet modes. The Floquetmodes are given by:

(7)±jymi

where

de"P/2mn

+ k sin d•>]= 0, ±1, ±2

1EE PROCEEDINGS, Vol. 13J, Pt. H, No. 3, JUNE 1984

is the transverse variation and

ym = } k2 — I —-—h k sin 6 I > for propagating modes

{flnm V ')1'2

= ± ; X I —r + ksmO) k2IV

+ ksmO) - k

for evanescent modes (8)

is the propagation constant, and k — 2n/LWe now consider an incident field of the form:

(9)

which may or may not be propagating according to eqn. 8.The scattered field is now expanded as a series for bothsides of the grating. This results in:

Hs(x,z) = X amVmem= oo

00

E (x z) = — Y 7 am = of'

H's(x,z) = £ bmVm

ejyfor z ^ 0

(10)

for z sc 0

Continuity of the fields at the plane of the grating, z = 0,produces

P^P- Z

t amVm=m = — JO

where am and bm are, as yet, unknown, and Zm = yjcoeo =\/Ym is the modal impedance of the Floquet modes.

We now extract the pth mode from the summations;this results in:

= t

= Z

(12a)

m — — oo

(126)

where ]T' denotes omission of the pth termThe functions ¥m(x) are orthonormal over { — d/2, d/2),

and the inner product may be defined as:

fd/2

By applying the orthogonality condition, eqn. 126 resultsin:

Zp(l-a,) =

and

E(x\ 0)T*(x') dx' m = p- c / 2

(14)c/2

- Zm am = Zm bm = E(x\ 0)V*(x') dx'l-c/2

where the region of integration has been reduced tox ^ | c/21, since the £-field is zero over the region | c/2| ^ x < |d/21

Using the above relationships in eqn. 12a produces

= Z (15)

Substituting the value of bm from eqn. 14 results in a Fred-holm integral equation of the first kind:

YmVm(x)V*(x')}E(x',O)dx'- c / 2

where p = 0, + 1 , ± 2 , . . . , N is the incident modal order.It has been assumed that integration and summation

may be interchanged. This is possible since the series isconvergent.

4 Results

Fig. 3 shows the calculated reflection coefficient for asingle grating calculated by the moment method for two

or

-5

-10

« -15(11) =

-20

-25

-308 10 12 Ufrequency, GHz

16 18 20

Fig. 3 Comparison between two methods of calculating the reflectioncoefficient magnitude for a single grating

(Dimensions as in Fig. 2)quasistatic method

+ + + moment methoda 0 = 0"b 0 = 50

angles of incidence, 0 = 0° and 50°. By way of comparison,the solution has been also calculated by a quasistaticmethod [6]. The moment method calculation used 30terms in the basis expansion and the number of Floquetmodes was 61. It can be noted from the Figure that thetwo solutions diverge somewhat around the grating lobecondition for angular incidence, indicating the reducedaccuracy for the static method in this region.

Fig. 4 demonstrates the effect of the evanescent modecoupling for two closely spaced gratings. In this Figure,comparison is made between the single-mode interactionassumption and higher-order modal interactions. Con-vergence checks show that the first two evanescent modeswere excited more significantly than the other higher-order

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984 131

modes. This also applies for very close spacing of the grat-ings. Consequently, the size of the reflection and transmis-

1.000

0.975

0.950

£ 0.925c0

0.900

0.875

0.850

N=3

0.1 0.2 0.3 0.4 0.5 0.6spacing /wave length

0.7

Fig. 4 Transmission coefficient (expanded scale) for two identical grat-ings as a function of spacing I and N mode interaction

(Dimensions as in Fig. 2)tf = 50 , / = 10 GHz

sion matrices were limited to dimension 3. From theFigure it may be seen that with spacings smaller than A/4the coupling due to the evanescent modes appreciablyalters the transmission/reflection coefficient. With largerspacings, this difference decreases and displays a slow con-vergence to coincident values at considerable spacings. In

1.0

09

0.8

0.7

S 0.6J)

§ 0.5co

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5spacing /wavelength

0.6

Fig. 5 Transmission coefficient for two identical gratings as a functionof spacing I, and N mode interaction with the system close to the gratinglobe condition

(Dimensions as in Fig. 2)0 = 0 . / = 17.6 GHz

the region beyond A/4 these small differences are not con-sidered decisive for most engineering applications.

Fig. 5 demonstrates the case for a higher frequency andnormal incidence. It is seen that the single-modeinteraction criteria are insufficient for spacings smallerthan A/2. By increasing the frequency we have moved thesystem closer to a grating lobe condition. The conditionfor onset of grating lobes is A > d{\ + sin 0). This gives,from Fig. 3, / < 2 0 GHz (0 = 0°) and / < 1 1 . 3 GHz(6 = 50°). By examination of the complex propagationconstant in eqn. 7, we note that for the evanescent modesthe attenuation factor decreases as grating lobe onset isapproached. This implies that more modes are required toobtain convergence. In two-dimensional structures caremust be exercised to include those modes that are near tothe propagation criterion. This is best accomplished byreordering the modes according to increasing wavenumber in the propagating direction.

As pointed out in the example, convergence checks haveto be carried out to ensure inclusion of sufficient evanes-cent modes for close spacings. This can result in matrixsizes that may prove unmanageable where dielectric layersadjacent to the gratings are considered. Since the approachrelies on the attenuation factors introduced via the evanes-cent mode propagation constant, it is prudent to includethe effects of the first dielectric in the scattering descrip-

T Ttion. In this case Rn Rn+i and Tn+i =/= Tn-i, and planewave incidence from both sides must be considered.

5 Conclusion

A numerical technique has been presented in order toaccount for the effects of higher-order modal interactionfor closely spaced planar periodic structures. Numericalresults confirm the trends that are expected from theexamination of evanescent mode propagation factors.Although an example has demonstrated the case for twoidentical singly periodic strip gratings, the results may beapplied to the more general case of doubly periodic grat-ings where numerical methods offer the only choice of sol-ution.

6 Acknowledgment

The author acknowledges the encouragement and supportof Prof. E. Folke Bolinder, Head of the Division ofNetwork Theory, Chalmers University of Technology,Gothenburg, Sweden.

7 References

1 LEE, S.W., ZARILLO, G., and LAW, C.L.: 'Simple formulas for trans-mission through periodic metal grids or plates', IEEE Trans., 1982,AP-30, pp. 904-909

2 HAMDY, S.M., and PARKER, E.A.: 'Comparison of modal analysisand equivalent circuit of E-plane arm of the Jerusalem cross', Electron.Lett., 1982, 18, pp. 94-95

3 LEVREL, J.R., and LAVIGE, B.: 'Scattering by an infinite two-dimensional periodic array of thin meanderline conductors on a dielec-tric sheet'. IEE Conf. on Antennas & Propagation, Norwich, April1983, pp. 258-261

4 YOUNG, L., ROBINSON, L.A., and HACKING, C.A.: 'Meander-linepolarizer', IEEE Trans., 1973, AP-21, pp. 376-378

5 COLLIN, R.E.: 'Field theory of guided waves' (McGraw-Hill, 1960),pp. 390-397

6 MARCUVITZ, N.: 'Waveguide handbook' (MIT Radiation Lab.Series, 1951), pp. 285-289

132 IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984