The study structure is a planar structure, consisting of three layers of different thicknesses. Between these layers are planes of discontinuity Ω₁ and Ω₂. The excitation circuit “ Plane Ω₁” constituted by an excitation source of dimensions “c₁ × d₁” strictly less than the wavelength and a metal strip of dimensions “c₁ × l₁”, negligible thickness and “open line“. The “Plane Ω₂” contains a metal surface incorporating a radiating aperture dimensions “a₁ × b₁”. Between the metal surface and the top cover we include/layer water. The assembly is placed in a metal case. The lower cover of the housing forms the ground plane. This structure is depicted in Figure 1.

The theoretical formulation of the structures for the iterative method is based on determining the relationship between the incident and reflected waves [4] in the spectral domains and spatial domains shown in Figure 2.

The evolution of iterations through the spectral domain to the space domain is done using the Fourier transform modal “FMT” which considerably reduces the calculation time. Modal Fourier transform [5] requires the fragmentation of

discontinuity planes (Ω₁ and Ω₂) in pixels and this so that the electromagnetic behavior of the overall circuit will be summarized by writing the boundary conditions and continuity of the tangential fields on each pixel. The iterative process stops when it reaches the convergence of results.

The evolution of iterations through the spectral domain to the space domain is done using the Fourier modal transform “FMT” [2] [5] [6] which considerably reduces the calculation time. Fourier modal transform requires the fragmentation of discontinuity planes (Ω₁ and Ω₂) in pixels and this so that the electromagnetic behavior of the overall circuit will be summarized by writing the boundary conditions and continuity of the tangential fields on each pixel. The iterative process stops when it reaches the convergence of results.

The terms below link the incident waves “ A → 1 , A → 21 , A → 22 and A → 3 ” to the reflected waves “ B → 1 , B → 21 , B → 22 and B → 3 ” when they pass the space domain to the spectral domain:

( A 1 A 21 ) = Γ ^ Ω 1 ( B 1 B 21 ) + ( A 1 0 A 2 0 )

( B 21 B 22 ) = Γ ^ Q ( A 21 A 22 )

( A 22 A 3 ) = Γ ^ Ω 2 ( B 22 B 3 )

( J → 21 J → 22 ) = ( Y ^ 11 Y ^ 12 Y ^ 21 Y ^ 22 ) ( E → 21 E → 22 ) (1)

Γ ^ k : Reflection Operator ensuring the link between the incident waves and the reflected waves. It is defined in the spectral domain. It contains information on the housing walls and the relative permittivity of the different mediums of the structure, k Î {medium 1, medium 3}.

Γ ^ Q : Diffraction Operator at each interface (Ω₁ and Ω₂).

The operators of diffraction Γ ^ Ω 1 and Γ ^ Ω 2 contain the images of the circuits that are in the Ω₁ and Ω₂ planes, defines these two planes of discontinuities.

The evolution of the iterative method for a planar structure with three layers of different mediums is very detailed in our articles [7] [8].

According to the diagram in Figure 3 we can write:

( B 21 B 22 ) = Γ ^ Q ( A 21 A 22 ) (2)

● Parameters Y ^ i j du quadruple Q:

{ J 21 = Y 11 E 21 + Y 12 E 22 J 22 = Y 21 E 21 + Y 22 E 22 (3)

{ J → 0 = J → 1 + J → 21 J → 2 = J → 22 + J → 3 J → 1 = Y ^ 1 ⋅ E → 1 J → 3 = Y ^ 3 ⋅ E → 3 E → 1 = E → 21 E → 22 = E → 3

The symmetry of the structure allows us to write:

{ Y ^ 11 = Y ^ 22 Y ^ 12 = Y ^ 21 (4)

After some mathematical manipulation, it is possible to determine the matrix:

Γ ^ Q = 1 C [ 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 − 2 Y 12 Z 02 − 2 Y 12 Z 02 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 ] (5)

with: C = ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2

Γ Q = ∑ m , n , α [ | f m n α 〉 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 〈 f m n α | | f m n α 〉 − 2 Y 12 Z 02 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 〈 f m n α | | f m n α 〉 − 2 Υ 12 Z 02 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 〈 f m n α | | f m n α 〉 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 〈 f m n α | ] (6)

f m n α : Bases function of the box modes.

{ B 21 α = ∑ m , n , α | f m n α 〉 ( 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 ) α 〈 f m n α | A 21 α + ∑ m , n , α | f m n α 〉 ( − 2 Y 12 Z 02 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 ) α 〈 f m n α | A 22 α B 22 α = ∑ m , n , α | f m n α 〉 ( − 2 Y 12 Z 02 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 ) α 〈 f m n α | A 21 α + ∑ m , n , α | f m n α 〉 ( 1 − ( Y 11 Z 02 ) 2 + ( Y 12 Z 02 ) 2 ( 1 + Y 11 Z 02 ) 2 − ( Y 12 Z 02 ) 2 ) α 〈 f m n α | A 21 α (7)

with α ∈ { TE , TM }