The electric and magnetic field of an incident light is given by

[ E i H i ] ( P ) = [ e i h i ] exp ( i k i ⋅ P ) (1)

where e i and h i are the electric- and magnetic-field amplitude; k i = [ α , β , − γ ] is the incident wave vector with α = n 0 k i sin θ cos φ , β = n 0 k i sin θ sin φ , γ = n 0 k i cos θ , k i = 2 π / λ and n 0 is the relative refractive index of region V₀; P = (X, Y, Z) is an observation point; λ is the wavelength of the incident wave; θ is the incident angle between the Z-axis and the incident wave-vector; φ is the azimuth angle between the X-axis and the plane of incidence.

The amplitude of the incident electric field can be decomposed into TE- (TM-) component, which means the electric (or magnetic) field is perpendicular to the plane of incidence. To do this, we define two unit vectors e TE and e TM that span a plane orthogonal to k i . Hence, the amplitude e i in (1) is decomposed as

e i = e TE cos δ + e TM sin δ (2)

where the symbol δ is the polarization angle between e i and e TE shown in Figure 1(b).

We seek for the diffracted fields E_l (P) and H_l (P) in each region. These should satisfy the following requirements.

(C1) The Helmholtz equations in each region;

(C2) Radiation conditions: The diffracted waves in V₀ (or V_L) should propagate or attenuate in the positive (or negative) Z-direction;

(C3) Periodicity conditions: Each component of the diffracted electric and magnetic field should satisfy

f ( X + d , Y , Z ) = exp ( i α d ) f ( X , Y , Z ) , (3)

f ( X , Y + d , Z ) = exp ( i β d ) f ( X , Y , Z ) (4)

where, α and β are the phase constants in X and Y.

(C4) Boundary conditions: The tangential components of electric and magnetic fields are continuous across the boundaries S_l.

The multilayered thin-film bi-grating which is laminating L − 1 grating layers has period d in both X- and Y-directions shown in Figure 1(a). The semi-infinite region over the multilayered bi-grating and the substrate are denoted by V₀ and V_L, respectively. Moreover, each region inside the multilayered bi-grating, which is numbered starting from the incident side, is denoted by V l ( l = 1 , 2 , ⋯ , L − 1 ) . The regions V l ( l = 0 , 1 , ⋯ , L ) are filled with isotropic and homogeneous media with refractive indices n l , and a permeability of each region is equal with that of the vacuum μ 0 . The interface between V l and V l + 1 is denoted by S l _{( l = 1 , 2 , ⋯ , L )} . The profile of S l is sinusoidal and given by

z = η l ( x , y ) = H l 2 { cos ( 2 π x d x ) + cos ( 2 π y d y ) } − ∑ i = 1 l − 1 ω i (5)

where ω l denotes an average distance between S l and S l + 1 . The value H l is regarded as a groove depth of the boundary S l

We solve the problem above using Yasuura’s method of modal expansion [10]. To do this, we first define the set of modal functions; next we construct approximate solutions in terms of finite modal expansions with unknown coefficients; and, finally we determine the coefficients applying the boundary conditions.

Modal functions: Because the diffracted waves have both TE- and TM-components, we need TE and TM vector modal functions in constructing the solutions. Here we employ the functions derived from the Floquet modes (separated solutions of the Helmholtz equations satisfying the periodicity (C3) and the radiation conditions (C2) if necessary). The modal functions for electric fields for each region are given by

φ l m n TE , TM ± ( P ) = e l m n TE , TM ± exp ( i k i ⋅ P ) (6)

where, m , n = 0 , ± 1 , ± 2 , ⋯ and l = 0 , 1 , 2 , ⋯ , L , and wave vectors in (6) are defined by

e l m n TE ± = k l m n ± × i Z | k l m n ± × i Z | ,     e l m n TM ± = e l m n TE ± × k l m n ± | e l m n TE ± × k l m n ± | (7)

and

k l m n ± = [ α m , β n , ± γ l m n ] ,     α m = α + 2 m π d x ,     β n = β + 2 n π d y , γ l m n = ( n l 2 k 2 − α m 2 − β n 2 ) 1 / 2 (8)

where Re ( γ l m n ) ≥ 0 and Im ( γ l m n ) ≥ 0 . We use the modal functions defined in equations from (6) to (8) to construct approximations of diffracted electric fields. For the accompanying magnetic fields, we employ

ψ l m n TE , TM ± ( P ) = 1 ω μ 0 k l m n ± × φ l m n TE , TM ± (9)

Approximate solutions: To satisfy the radiation condition (C2), the approximate solution in V₀ should have a form of finite linear combination of up-going modal functions with unknown coefficients. Likewise, the solution in V_L must be a linear combination of down-going modal functions. The solution in V_l, however, must have both up- and down-going waves. To show the traveling direction of a modal function, we use superscripts + and − representing up- and down-going waves. Here, we form approximate solutions for the diffracted electric and magnetic fields in V_l:

( E l N d H l N d ) ( P ) = ∑ m , n = − N N A l m n TE + ( N ) ( ϕ l m n TE + ψ l m n TE + ) ( P ) + ∑ m , n = − N N A l m n TM + ( N ) ( ϕ l m n TM + ψ l m n TM + ) ( P ) + ∑ m , n = − N N A l m n TE − ( N ) ( ϕ l m n TE − ψ l m n TE − ) ( P ) ​ ​ ​ + ∑ m , n = − N N A l m n TM − ( N ) ( ϕ l m n TM − ψ l m n TM − ) ( P ) ,   ( l = 1 , 2 , ⋯ , L ) (10)

where N denotes the truncation number.

Boundary matching: Because the approximate solutions satisfy the requirements (C1), (C2), and (C3) by definition, the unknown coefficients A l m n TE ± ( N ) and A l m n TM ± ( N ) are determined such that the solutions satisfy the boundary conditions (C4) in an approximate sense. In the Yasuura’s method, the least-squares method is employed to fit the solution to the boundary conditions [10] [11]. That is, we find the coefficients that minimize the weighted mean-square error by

I N = ∫ S ′ 1 | v × [ E 1 N d + E i − E 2 N d ] ( s 1 ) | 2 d s + | Γ 1 | 2 ∫ S ′ 1 | v × [ H 1 N d + H i − H 2 N d ] ( s 1 ) | 2 d s + ∑ l = 2 L − 1 { ∫ S ′ l | v × [ E l N d − E l + 1 N d ] ( s l ) | 2 d s + | Γ l | 2 ∫ S ′ l | v × [ H l N d − H l + 1 N d ] ( s l ) | 2 d s } (11)

where S ′ 1 denotes one-period cells of the interface S l , Γ l is the intrinsic impedance of the medium in V l and v is a unit normal vector of each boundary.

To solve the least-squares problem on a computer, we need a discretized form of the problem. We first discretize the weighted mean-square error I_N by applying a two-dimensional trapezoidal rule where the number of sampling points is chosen as 2(2N + 1) [9]. We then employ orthogonal decomposition methods [singular-value decomposition (SVD) and QR decomposition (QRD)] in solving the discretized problem [12] [13].

It is known that the solutions obtained by Yasuura’s method have proof of convergence [13] [14]. We, therefore, can employ the coefficients A l m n TE ± ( N ) and A l m n TM ± ( N ) with sufficiently large N for which the coefficients are stable in evaluating diffracted fields. The power reflection and transmission coefficient of the (m, n) order propagating mode in V₀ and V_L are given by

R m n = γ 0 m n γ | A 1 m n TE + | 2 + γ 0 m n γ | A 1 m n TM + | 2 , Re ( γ 0 m n ) ≥ 0 , (12)

T m n = γ L m n γ | A L m n TE − | 2 + γ L m n γ | A L m n TM − | 2 , Re ( γ L m n ) ≥ 0. (13)

The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power.