Leaky Bloch-like surface waves in the radiation-continuum for sensitivity enhanced biosensors via azimuthal interrogation

Dielectric multilayer structures with a grating profile on the top-most layer adds an additional degree of freedom to the phase matching conditions for Bloch surface wave excitation. The conditions for Bloch surface wave coupling can be achieved by rotating both polar and azimuthal angles. The generation of Bloch surface waves as a function of azimuthal angle has similar characteristics to conventional grating coupled Bloch surface waves. However, azimuthally generated Bloch surface waves have enhanced angular sensitivity compared to conventional polar angle coupled modes, which makes them appropriate for detecting tiny variations in surface refractive index due to the addition of nano-particles such as protein molecules.


Introduction
Rigorous coupled wave analysis (RCWA) [1][2][3][4][5][6][7] , as suggested by its name, is a more general formulation of Maxwell's equation for structures with material variation in x-y plane as well. The assumption here is that the medium may be inhomogeneous in x-y plane, but it must be uniform in the z-direction for each layer. We start with Maxwell's equations describing the fields inside a single linear, homogeneous, and isotropic layer of the device which are given below The termH is the normalized magnetic field which is equal to − jη 0 H, where j = √ −1 and η 0 is the impedance of free space, k 0 is the free space wave number and is equal to 2π/λ 0 , where λ 0 is the free space wavelength, and µ r and ε r are relative permeability and permittivity of the material respectively. Eqn. (1) and (2) can be expanded into a set of six coupled partial differential equations as follows For RCWA, ε r and µ r are represented in terms of Fourier transforms along the x and y direction, the z-parameter remains analytical and unchanged.
Λx + 2πny The coefficients a m,n and b m,n are given as The terms Λ x and Λ y are the periods in x and y directions respectively.
The Fourier expansion of the fields are where Substituting the field expressions in eqn.
− jk x,m S y,m,n (z) + jk y,m S z,m,n (z) = Expanding each of these equations for every possible combination of m and n and putting them in matrix form, they can be compactly represented as where Eliminating the longitudinal components s z and u z by back-substitution and rearranging them, eqn. (27)-(32) can be further reduced down to These equations can be compactly written in matrix form as follows d dz where Taking the derivative of eqn. (38) w.r.t. z and then substituting eqn. (37) in the result, we get where Ω 2 = PQ. Eqn. (41) is the second order wave equation in matrix form. It has a general solution of the form The terms s + (0), and s − (0) are the initial values for this differential equation. The ± superscripts indicate whether they pertain to forward propagating waves (+) or backward propagating waves (-). The terms e −Ωz and e Ωz have a matrix as their exponents. These matrix exponentials can be computed using the eigen-vectors and eigen-values of the matrix Ω. Letting W and λ 2 as the eigen-vector and eigen-value matrix of Ω 2 , we can compute the matrix exponentials as where e ±λ z = diag(e The matrix W = 1 0 0 1 = I and e ±λ z = e ± jk z z 0 0 e ± jk z z . The magnetic field has a similar solution given below To compute V, eqn. (47) is differentiated w.r.t. z .
Eqn. (49) represents electric and magnetic field in a layer of linear, homogeneous, and isotropic material.

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Figure 2. Field representations in the i th layer of a 1D multilayer.

Scattering matrix for a layer
A 1D multilayer device has a stack of multiple layers of different material. Thus, for a certain i th layer in such a device, the solutions can be represented as Each layer has two interfaces with corresponding boundary conditions. Here, we considering each layer separately; thus, medium 1 (left region) and medium 2 (right region) do not strictly need to be layers of the 1D device in consideration. In fact, for the sake of numerical efficiency, we let both medium 1 and 2 be to free space of zero thicknesses. The boundary condition at the first (left) interface is The boundary condition at the second (right) interface is After some manipulations of eqn. (52) and (54), we can reduce them to a system of the following form where The element of matrix S (i) are calculated as The matrices A i , B i , and X i are computed as The matrix S (i) is the scattering matrix of the i th layer. It relates the input field to the output field. The elements S give reflection and transmission coefficients respectively. Because each layer is surrounded by free space in our formulation, the scattering matrices are symmetric. Thus only two of the matrix components have to be calculated for each layer.

Redheffer Star Product
In order to model a device with multiple layers, we need to combine multiple scattering matrices into a single scattering matrix. However, the scattering matrices cannot be combined directly by applying matrix multiplication. Also, the combined scattering matrix is not symmetric as the scattering matrix for a single layer so it becomes necessary to compute and store all four components of the combined scattering matrix. Two scattering matrices can be combined using the Redheffer star product 8,9 . The Redheffer star product of two scattering matrices S (A) = S The scattering matrix of a 1D multilayer with N layers can be computed by taking Redheffer star product of the scattering matrices of each layer.
The multilayer device is surrounded by the reflection region and transmission region at its two ends. It is connected to these external materials by "connection" scattering matrices that have zero-thicknesses. The global scattering matrix finally combines all the scattering matrices into a single matrix as The matrices S (Re f ) and S (Trn) are the reflection and transmission region scattering matrices. with with