3.1 Fourier Transform and Mutual Coupling Concepts
 Let the spectral field (in kx, ky domain) radiated by the metallic patch into the region above the periodic patch array (z > zi−1; see Figure 1) be stated as
where ap is the amplitude coefficient of the pth basis function expanding the patch electric current, with
being the two-dimensional (2D) Fourier transform of the pth basis electric current fp(x, y) existing over the metallic patch parallel to the xy plane (one printed on the top surface of the dielectric slab within each unit cell; see Figure 1), and where is the spectral Green's function (numerical type, i.e., non-analytical) for multilayer structures, i.e., the G1DMULT routine.
 Subsequently, the spatial domain field is obtained through the inverse transform that involves only a summation over spectral components, i.e., discrete spectrum, due to the periodicity of the structure along x and y. This is expressed as follows:
in which the discrete spectral coordinates are given by
where m and n are integers; θ0 and ϕ0 are angular coordinates defining the direction of the dominant Floquet modal beam; dx and dy are the periods along x and y of the 2D periodic structure, respectively; and k0 is the usual free space wave number. Under the phased array scenario, (θ0, ϕ0) defines the steered main beam direction.
 When all elements are excited for the phased array scenario beamed toward the direction defined by (kx0, ky0), the electric current over the (u, v)th patch centered at (x = udx, y = vdy) is written as
which is valid only over the range of x and y values as explicitly indicated in the argument (the current is zero everywhere else). The spatial domain vector function fp(x − udx, y − vdy) represents the pth basis current of the (0, 0)th patch that is translated by (x = udx, y = vdy) from the origin. The untranslated basis current (u = v = 0) is the generating basis current function for the periodic basis current function that spans throughout the entire array aperture. The variables kx0 and ky0 are the usual dominant Floquet harmonics pertaining to the steered main beam. The term is the solved amplitude coefficient for the (0, 0)th element under the phased array scenario steered to that particular main beam. The superscript “all” denotes the all-excited phased array scenario.
 Consequently, the current over the entire aperture [−∞ < x < ∞, − ∞ < y < ∞] can be expressed as a superposed sum of in equation (7) over all u and v element indices, i.e.,
 Now, let be the coefficient for the coupling from the (0, 0)th excited waveguide element in fundamental mode with index pin (the only one excited) to the amplitude coefficient of the pth basis function of the (u, v)th patch current. Therefore, when only the (0, 0)th element is excited (the rest passively match terminated), the current over the (u, v)th patch can be written as
being the current over the (u, v)th patch when only the (0, 0)th element is excited, with all the rest passively match terminated.
 By inversing the definition of , let be the coefficient for the coupling from the solely excited (u, v)th element in fundamental mode to the amplitude coefficient of the pth basis function of the (0, 0)th patch current. Then, the reverse case of equation (9) is written as
being the current over the (0, 0)th patch when only the (u, v)th element is excited by the (pin)th mode. It is again asserted that this current function vanishes outside the range specified, i.e., outside the (0, 0)th patch.
 Therefore, superposing infinitely many (u, v)th cases of equation (10) to obtain the current over the (0, 0)th patch under the infinite phased array scenario when beamed toward a certain direction defined by (kx0, ky0), we have (with set to unity for all u and v indices pertaining to a uniform phased array)
where is the amplitude coefficient of the pth basis function of the (0, 0)th patch current under the all-excited phased array scenario, which can also be inferred by setting u = v = 0 in equation (7). Note the very important exponential term that accounts for the phase shift between the (u, v)th element and the (0, 0)th element (at zero phase)—recall that in an infinite phased array scenario, the variation (of either fields or currents) of every element is the same in all elements, but with a phase difference determined by the phasing of the elements (phase gradient over the array) that determines the steered main beam direction. This pertains also to the dominant Floquet mode (kx0, ky0).
 Since this is scan dependent (under phased array scenario), let us write it as with expressed argument. Explicitly stating this in equation (11),
where Φx = kx0dx, Φy = ky0dy.
 Therefore, the associated forward Fourier transform is
 Placing this expression into equation (9), with in there set to unity (as we have considered uniformly excited phased array, with unit amplitude) and since the x and y coordinates are affected by neither the summation (over basis function index p) nor the integration (with respect to Φx and Φy), we are at liberty to arbitrarily define a new primed coordinate system with origin at (x = udx, y = vdy), thus yielding the following:
where fp(x′, y′) is the pth basis current of the (0, 0)th patch, being the generating basis current function of the entire periodic basis current function spanning over the whole array aperture.
 Upon using equation (7) in equation (14), we get
 Now, has already been acquired via the solution of the infinite all-excited phased array problem (as described by Ng Mou Kehn and Shafai ), and it is expressed as follows:
where is the solved amplitude coefficient of the pth RWG (after Rao, Wilton, and Glisson [Rao et al., 1982]) basis function expanding the patch electric current, as was done by Ng Mou Kehn and Shafai .
 Hence, with equation (16) placed into equation (15), we obtain
 Let us next write the far-zone electric field radiated by the patch current , under the EES, in which the primed x and y coordinates in the argument span over the entire aperture, as the following inverse transform:
in which the exponential term accounts for the phase shift from the (0, 0)th element with zero phase to the (u, v)th one, pertaining to a phase-steered beam direction corresponding to kx and ky.
 Placing equation (17) into equation (20), with an inconsequential change in the sign of the exponent for the inverse transform (discrete summation) adopting the more common convention, we have
where the primed kx0 and ky0 correspond to the observation direction under the EES.
 The k′x0 and k′y0 in the bracketed superscripts on the left sides of these two equations signify that these currents existing over the entire array aperture under the EES pertain to the situation where the field observation (under the EES) is toward that direction defined by these primed variables. Hence, there will be different current distributions over the entire array aperture for various field observation directions under the EES.
 Using the concept manifested by equation (21) and then denoting as the spectral Green's function for the Fw field (F may be E or H, and w may be x, y, or z) radiated by spectral electric current excitation (they are the pth basis functions of the patch electric currents), both under the multilayer scenario with the periodic patch array present, any general field component Fw can be expressed as
where G denotes the spectral Green's function having its usual superscript notation for the field component involved and the subscript representing the kind and location of the source (“patch” here means electric current source over the PEC patch). The bracketed subscript on the left side containing k′x0 and k′y0 denotes that this field component under the EES pertains to the situation where the field observation (under the EES) is toward that direction defined by these primed variables.
 If we take the forward transform of equation (22), i.e., take throughout, we obtain
in which Ω has been introduced to represent
which is a quantity that is scan dependent, i.e., varies as a function of θ and ϕ via and . The dependence on Φx = kx0dx and Φy = ky0dy is assumed separable, as shown.
 Note the reduction of surface integration limits to cover only the unit cell of the (uv)th element. Also, observe the incorporation of primes to the kx0 and ky0 quantities lying outside the integration with respect to Φx = kx0dx and Φy = ky0dy so as to distinguish them from the integration variables (unprimed). With further rearrangement, swapping prime and unprimed coordinates, and applying the well-known identity, we have
describing both 1 ⇌ δ and the shifting properties of the Fourier transform. With further use of another well-known result,
and using equation (24) for m = n = 0, we obtain the following:
 Hence, the strength of the Fw field component of the dominant (0, 0)th Floquet modal plane wave (where main beam is “mb”) of the infinite phased array pertaining to a certain is also the Fourier-transformed spectral domain field component under the EES, being the amplitude of the (kx, ky)th spectral plane wave component. It is equation (25) that is the ultimately computed quantity used in equations (30) and (31) for the radiated fields under the EES.
3.2 Embedded Element Radiation
 Referring to the work of Sipus  on far-field calculations using PEC equivalence involving equivalent magnetic current source aperture, we write the following PEC equivalent magnetic current located at z = zap just above the PEC patch array (see Figure 1):
 The electric field components on the right side are those of equation (22). From the work of Sipus , the far field is
where the factor 2 at the front is due to imaging. The integration spans over the entire source aperture located at z = zap with infinite extent along the transverse x-y plane. The wave numbers are
where (θ, ϕ) are the angular coordinates of the observation direction under the EES.
 Placing equation (26) in equation (27),
 Using and , the θ and ϕ components of this far-zone electric field under the EES can be expressed separately as follows:
in which the (w is x or y) is from equation (25).
 Hence, the essence here is that any certain far-field observation direction (θ, ϕ) under the EES determines a certain spectral component (kx, ky) [defined in equation (28)] of the aperture electric field (related to the PEC equivalent magnetic current source sheet), also under the EES. As asserted earlier, just after equation (25), this spectral (kx, ky) component of the E field (say, the Ew component) under the EES is simply the strength of that Ew field component of the dominant (0, 0)th Floquet modal plane wave (main beam) of the infinite phased array that is steered toward a certain main beam direction (θmb, ϕmb), with
 Subsequently, by basic Poynting's power theorem, the total power radiated out into the half-space above the periodic patch array under the EES is
where is the well-known intrinsic wave impedance of the medium above the array (typically vacuum).