Propagating and evanescent edge diffracted waves for a semi-infinite periodic dipole array

In order to understand and quantify the high-frequency wave processes associated with Floquet wave (FW) edge diffraction, this paper deals with the canonical Green's function of a semiinfinite phased array of dipoles in free space. The diffracted field contributions due to each incident FW field are cast in a dyadic form that highlights the behavior of TE and TM waves. Special attention is given to evanescent diffracted fields, which are associated with high order FWs.


INTRODUCTION
The electromagnetic modeling of large finite array antennas may be carried out by using a Floquet Wave (FW) representation of the infinite array Green's function. To account for array edge effects, this representation needs to be modified as described in 11; in particular, the Green's function of a finite array is collectively tepresente d as the radiation from a superposition of continuous truncated FW distributions over the array aperture. Since the FW series exhibits excellent convergence properties when the observation point is located away from the array surface, this representation is found to be more efficient than the direct summation of the spatial contributions from each element of the array, especially wh& each FW aperture distribution is treated asymptotically. Furthermore, the collective approach provides basic physical insight into the relevant scattering mechanisms for this class of problems. By invokin the locality of high-frequency phenomena as formalized in the Geometrical T%eory of Diffraction (GTD), an actual rectangular array may be treated by accounting for local canonical edge and corner effects 121.
In order to understand and quantify the high-fre uency wave processes associated with FW edge diffraction, this paper deals wit% the canonical Green's function of a semi-infinite phased array of dipoles in free space. The diffracted field contributions due to each incident FW field are cast in a dyadic form that highlights the behavior of TE and TM waves. Special attention is given to evanescent diffracted fields, which are associated with high order FWs.

FORMULATION
The geometry of a phased array of dipoles oriented along the direction J,=J,, ?+J,,t is shown in Fig. 1, with definition of both Cartesian and cylindrical coordinates; here and in the following a caret denotes a unit vector and an overbar a general vector. Referring to Fig. 1, d , and d, are the interelement periods in the z and z directions, respectively. The dipoles are linearly phased, with y= and v1 denoting the element-to-element phase shift along the z and z coordinates, respectively. With a suppressed time dependence exp j u t ) , the dipole currents (1) j = j, , -1 3 4 , can 6' e represented as nm is the position -of (n;m)-th dipole. TM. and TE. fields are cal-cul&ed via (he magnetic and with the branches of the square roots defined so that Sm(k )<0 for I k,, I >k, and Sm(yk,,,)<O when k~, + k~, > k * . Furthermore, Q dllotes the number of poles extracted and F represents the standard Fresnel-type transition function of the Uniform Theory of Diffraction (UTD) with argument 6pq* =*sin[(+fdpp)/2]. The two contributions defined in (5) and (6) are discussed next.

FWn contributio~
Except for the Heaviside unit step function Lr that bounds their domain of existence at the shadow boundary planes qi = #g , the residue contributions ( 5 ) represent the FWs of the doubly infinite array of dipoles. In particular, it can be seen that #B=92e(dw)tan-'(sinh(%n(+ )) for Ik 1 < k while qisB=r 2 for Ik I > k, ?here # -co~-~(k,,/k,,,) spec!& the dir&iTn of the dfmuthal comGnent of the Fwwavevector. . Poles with k z , t k~ < k2 are associated with propagatin Floquet waves (PFWs but all t i e otters with that propagate in the zpp direction with the speed of fight. For them the shadow boundary an le coincides with the angle of propagation in the azimuthal plane (ds$f=# ). The EFWs are inhomogeneous plane waves that propagate slowly (v?r.t, %e speed of light) in the direction k i+k i and decay exponentially in the i$ direction away from the arraTpldE. The transition between the homogeneous and evanescent FWs is defined by the cut-off condition k =O. Owing to the EFWs exponential decay along I y I, the convergence of%e first series in (4) is very rapid when the observation point is located sufficiently far away from the array surface.

9.g. Ewnwcent and propagating difiacted wawa
The uniform saddle point evaluation of 3) provides diffracted FW field discontinuous when its com lex argument crosses the sitive imaginary axis, which occurs exactly at tEe shadow boundary a n g f defined earlier. The diffracted field contributions smoothly compensate for the discontinuity of the FWs at the shadow boundary. Each pq-th FW diffracts at a point Q on the array edge according to a generalized Fermat principle that ma; be expressed as where Z denotes the pindependent location of Q Therefore, the diffracted field co&ribution has been tagged with only a Zngle summation index,g.
Diffracted rays produced by FWs with different axial component k arise from diatinct diffraction points Q, (~n s tor each 7). Fer lLzg!<Lr -11 d:fP-led rays emanatin from these points lie on a diffraction cone with aperture semiangle /3 = cos'(k /k) which becomes more acute with decreasing FW phase ve1ocit;along z. #he: Ik 1 4 , the diffraction cone collapses onto the z-axis; for Ik I>k, there is no Yeal point Q on the edge satisfyin ( 8 ) , and 6, beco$& complex, as do the diffracted'rays. The resulting digacted field is evanescent along the p direction, with exponential decay term exp(-lk Jp). This yields rapid conver ence for diffracted ray series in (4), sufficient6 far from the edge. The di&acted FWs which contribute substantially to the scattered field are generated by all PFWs, and those EFWs for which Ik,,l<k.

ILLUSTRATIVE EXAMPLE
Numerical calculations have been carried out to test the accuracy and effectiveness of the asymptotic solution (3), as well as to highlight the effects of the cut-off transition. A reference aolution for a strip array along z is constructed via element-by-element summation of individual source contributions over a square array with m 5 M=2000 and n 5 N=100. The mdimension permits neglectin the truncation effects along z. The dipoles are tilted 45. with respect to z. 1, Fig. 2, with A being the wavelength, the near field scan is at a radial distance p=2.2A from the z-axis in the 2-0 (i.e. m=O) plane; dr=0.5A, d,=l.lA, 7='---0.945A-' and 7,'=0.5X-'. The i and 4 electric ield components are shown along the scan. The solid curve is obtained by the high-frequency solution (4), including diffracted fields from both edges. These curves coincide with the reference solutions (circles). Additionally, dashed curves are presented, that are obtained by neglecting the diffracted field contributions. The eometrical configuration is such that Et:, are propagating, while is evanescent hut close to its cut-off condition. The three corresponding dikracted fields E ! , , Et, and E : all propa ate away from the edge. Higher order FW and their pertinent diffracted fie12 are neglected due to their strong exponential decay along and p, respectively. It is evident that the diffracted field contributions strongly affect the accuracy of the prediction, especially that relevant to the close-to-cut-off EFW. It is worth noting that even in this critical re ime, the field predicted by 4 is in excellent agreement with respect to t%e reference solution, and tbe didracted field (6) well compensates for the discontinuity of the EFW. PEFERENCES cattering by Weakly aperiodic Truncated thin Wire Gratings", Joum. Opt.