A Periodicity-Induced Generalized Fourier Transform Pair

— The field radiated by an infinite periodic structure can be ex- pressed in terms of Floquet waves (FWs), both in the frequency domain (FD) and time domain (TD) [1]. A new periodicity-induced generalized Fourier transform (FT) pair is derived relating FD-FWs to TD-FWs and vice versa, based on tabulated transforms and physical conditions at in- finity. The new FTs are directly related to the simple canonical problem of a line array of sequentially excited dipoles that is a basic building block for more general phased periodic structures.


I. INTRODUCTION
Floquet waves (FWs) generated by one-dimensional (1-D) phased periodicity along a rectilinear coordinate z are parameterized by the dispersion relation k zq (!) = ! z + q ; q = 2q=d; q = 0; 61; 62; . . . (1) where ! is the radian frequency, k zq is the z-domain wavenumber, z is the interelement phase gradient, d is the interelement spacing, and q is the FW index [1]. The dispersion relation for q 6 = 0 differs from the nondispersive case q = 0, i.e., k z0 = ! z , only through the constant term q. Closed form relations between frequency-domain (FD) and time-domain (TD) FWs can be established by conventional tabulated Fourier transforms (FT) when q = 0 [2, pp. 277]. However, no corresponding direct tabulations seem to exist for q 6 = 0. This has motivated the study of a generalized FT pair for a class of functions that differs from those listed in the mathematical tables by involving Hankel functions with an ! dependence of the form kq(!) = k 2 0 k 2 zq instead of k0(!) = k 2 0 k 2 z0 , with k = !=c (c being the ambient wave speed) and k zq (!) given in (1). The periodicity-induced FT will establish direct relations between FD-FW and TD-FW with q 6 = 0.
The important nondimensional parameter = z c (2) permits distinction between two cases depending on jj 1, in which the phase velocity v p z = 01 z = c= of the excitation of the periodic structure along z can be larger (jj < 1) or smaller (jj > 1) than the ambient wavespeed c.
This periodicity-induced FT is directly related to simple radiating systems such as the sequentially excited periodic line array of dipoles that has been studied in detail in [1]. There, TD-FWs have been defined and found via various analytic methods. Here, we prove this generalized periodicity-induced Fourier transform pair going from TD to FD and vice versa, in a direct manner. This constitutes the basic building block for more complicated periodic structures with sequentially excited periodicity cells. Manuscript obeys the definitions (3).
On the FD side of (4), the top Riemann sheet of the radial wavenumber k q (!) in (5) is chosen to render =mk q 0, consistent with the radiation condition at = 1 ( is the cylindrical coordinate perpendicular to z). Furthermore, <ekq 0 or 0 for ! > 0 or < 0, respectively, in order to satisfy the radiation condition for positive and negative real frequencies. The conditions jk zq j jkj determine an exponentially decaying or oscillating function along , respectively. On the TD side of (4), U() = 1 or 0 if > 0 or < 0, respectively. In (4)-(7) 0 is positive real for jj < 1, 2 0 is negative for jj > 1, and it is convenient to define the branch of the root in (7) as 0 = 0jj 0 j (this is in accord with the root of k q , since k 2 (1 0 2 ) = k q j q=0 ).
Case jj > 1: We recall that in this case we have 2 0 = 0j 0 j 2 , with j 0 j = 2 0 1=c, in (6) in which the square root is defined as =m ! 02 0 ! 2 q < 0 and <e ! 02 0 ! 2 q 0 or 0 for ! > 0 or ! < 0, respectively, in accord with that for kq in the text after (7). Separation of positive and negative ! occurs at ! 0 = 0 ! q between the two branch points.
In order to Fourier-invert the FD function that satisfies the radiation condition at 1 [see text after (7)] for any !, the integration path from 01 to +1 is shifted below the branch cuts [see Fig. 1(a)] where the sign of the square root is in accord with the radiation condition at 1.
This choice is also in agreement with [4, pp. 35] where to ensure the existence of the Fourier pair in (3) the ! 0 variable in (3) and, therefore, the ! 0 contour of integration in (18), is shifted slightly from the real ! 0 axis into =m! 0 < 0.
Using the large argument asymptotic approximation H (2) 0 () (2=()) 1=2 exp(0j 0 j=4), it is easy to see that for t < 0 the integrand decays exponentially for =m! 0 < 0. Therefore, for t < 0, the integration contour can be deformed onto P 01 where the integral vanishes by Jordan's lemma, and since no singularities are included in the deformation, the integral in (18) vanishes by Cauchy's theorem.
For t > 0 , the integration contour is deformed onto P 1 + P 2 + P 1 , with the integral on P1 vanishing. The integration of the even part of the integrand on the symmetric integration path P 1 + P 2 vanishes.
The integration of the odd part on P1 is equal to the contribution from P 2 , andÎ(t) in (18) can be evaluated as twice the integral on P 2 Since on the upper and lower side of the branch cut the square root assumes opposite negative/positive values, then using the relation H (2) 0 (e 0j ) = 0H (1) 0 (), reversing the integration path above the real ! 0 axis, and combining H (2) 0 () + H (1) 0 () = 2J0(), leads tô in which the square root assumes positive values. This sine transform is given in [2, pp. 113], yielding directly the right-hand side of (4).
Case jj > 1: Using the definition j0j = 2 0 1=c, (17) is rewritten aŝ Since j 0 j > 0, the square root in (21) is still defined as =m ! 2 q 0 ! 02 < 0, and <e ! 2 q 0 ! 02 0 or 0 for ! > 0 or ! < 0, respectively. Branch points are still located at ! 0 = 6 !q, with the only difference that the branch cuts are now located as in Fig. 1(b). Note that now the separation of positive and negative ! frequencies at ! 0 = 0 !q occurs outside the branch point region, as shown in Fig. 1(b), for the case !q > 0; similarly, at ! 0 = j ! q j for the case ! q < 0. Moreover, from Fig. 1(b), the whole region 0j ! q j < ! 0 < j ! q j corresponds to ! > 0. Therefore, in order to Fourier-invert the FD function that satisfies the radiation condition at 1 [see text after (7)] for any !, the integration path from 01 to +1 is indented between the cuts as in Fig. 1(b), where <e ! 2 q 0 ! 02 0 (for !q > 0). In the case of !q < 0, the whole region 0j ! q j < ! 0 < j ! q j corresponds to ! < 0, and the integration path is still defined in between the cuts, where now <e ! 2 q 0 ! 02 0. Since for any deformation a branch point singularity is included, the integral is nonvanishing for any value of t j 0 j, and its evaluation is given in [4, pp. 481] which leads directly to the right-hand side of (4).

ACKNOWLEDGMENT
The author would like to thank Prof. L. B. Felsen for his suggestions and encouragement.